Francesca Biagini

Bruno Bouchard

Luciano Campi

Caroline Hillairet

Ying Jiao

Huyên Pham

Peter I Tankov

Jean-François Chassagneux

Laure Elie

Noufel Frikha

Zorana Grbac

Idris Kharroubi

Gilles Pagès

Mathieu Rosenbaum

Simone Scotti

Beatrice Acciaio

Monique Jeanblanc

Kostas Kardaras

Qinghua Li

Bernt Øksendal

Shiqi Song

Agnès I Sulem

Chapter 3. Complete list of publications Part II. Information and arbitrage: a presentation of the main results Chapter 4. Information and no-arbitrage for asset price models with a change point Chapter 5. Arbitrages arising with honest times Chapter 6. Arbitrage of the first kind and filtration enlargements in semimartingale models Chapter 7. The strong predictable representation property in initially enlarged filtrations Chapter 8. Market viability and martingale measures under partial information Part III. Appendix Chapter 9. Detailed scientific CV Bibliography i This thesis contains an overview of my scientific activity in the years 2012-2018. During these years, I benefited enormously from intense collaborations, stimulating discussions and inspiring conferences and seminars, together with several colleagues and friends. This thesis would not have seen the light without them and, therefore, I take this opportunity to sincerely thank them all.

Let me start by expressing my gratitude to Martino Grasselli and Wolfgang J. Runggaldier, the two professors who first introduced me to the field of mathematical finance. In particular, I am thankful to Wolfgang for having been the most inspiring supervisor during my PhD studies. I am grateful to Monique Jeanblanc for having given me the opportunity to start working in Paris in 2012, as a post-doc at the University of Évry. Furthermore, I

Part I

General overview of the research activity CHAPTER 1

Introduction and outline of the thesis

In the present thesis, I aim at providing an outline of my research activities in the years 2012-2018, the period corresponding to the years after my PhD studies (PhD thesis defended on March 26th, 2012, at the University of Padova, Italy). As will be explained in more detail in Chapter 2, my research in the years 2012-2018 has been focused on three main areas, namely arbitrage theory, enlargement of filtrations and term structure modelling. These research activities have led to the publication of twelve refereed papers, listed in Chapter 3 of the present thesis.

The thesis is structured as follows. In Chapter 2, I give a general presentation of my research activities in the years 2012-2018. For each of three main directions of research mentioned above, I will present an outline of the main results obtained, also discussing the links between the different research lines and briefly illustrating ongoing and future research projects. Part I of the thesis concludes with Chapter 3, which contains a complete list of publications, separating the publications related to my PhD thesis from the publications originated from the research carried out after the PhD.

Part II consists of a detailed presentation of some of my more relevant contributions to the fields of arbitrage theory and enlargement of filtrations. More specifically, I choose to give an abridged presentation of the following five research papers, corresponding to the five chapters of Part II:

(1) C. Fontana, Z. Grbac I have chosen this selection of five papers as in my opinion it reflects my fascination and interest for the interplay between the notions of information and arbitrage in financial markets. Indeed, all these five papers are motivated by questions pertaining the role of different information structures on the no-arbitrages properties of financial market models. From a mathematical standpoint, the theory of enlargement of filtrations provides the probabilistic tools which enable us to give precise answers to such questions. For simplicity of presentation, only the more relevant results of the papers are discussed in this thesis, with the intent of giving to the reader a flavour of my research contributions, referring to the original papers for all the proofs of the results stated in this thesis. Finally, Part III of the thesis contains my up to date scientific Curriculum Vitae and a list of references.

CHAPTER 2

Description of the research activity in the years [2012][2013][2014][2015][2016][2017][2018] During my PhD studies, concluded in the year 2012, my research activity has been mostly devoted to credit risk modelling, mean-variance portfolio selection and arbitrage theory. In particular, the results obtained in the course of my doctoral studies have led to the articles [FR10, Fon12, FS12, FR13, FM14].

After my PhD, my research activity has been essentially concentrated along three main directions:

(1) arbitrage theory and weak no-arbitrage conditions;

(2) enlargement of filtrations and the modelling of information;

(3) interest rate and credit risk term structure modelling.

Besides these three main areas, I recently developed an interest for the applications of stochastic processes to financial economics, which is the focus of the monograph Financial Markets Theory (see [START_REF] Barucci | Financial Markets Theory: Equilibrium, Efficiency and Information[END_REF]) recently written together with Emilio Barucci. In the remaining part of this chapter, I will present an overview of my contributions related to the above three topics, also discussing recent and ongoing projects. In Part II of the thesis, I will give a more detailed presentation of some of the main results obtained at the intersection between arbitrage theory and enlargement of filtrations.

Arbitrage theory and weak no-arbitrage conditions

Arbitrage theory represents one of the cornerstones of modern stochastic finance and aims at deriving suitable probabilistic characterizations of viable financial markets. My interest in arbitrage theory, and in particular on no-arbitrage conditions that are weaker than the classical requirement of No Free Lunch with Vanishing Risk (NFLVR, see [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF][START_REF] Delbaen | The fundamental theorem of asset pricing for unbounded stochastic processes[END_REF]), started during my PhD with the writing of the survey paper [START_REF] Fontana | Diffusion-based models for financial markets without martingale measures[END_REF]. My contributions to arbitrage theory are contained in the papers [START_REF] Fontana | No-arbitrage conditions and absolutely continuous changes of measure[END_REF][START_REF] Fontana | A note on arbitrage, approximate arbitrage and the fundamental theorem of asset pricing[END_REF][START_REF] Fontana | Market viability and martingale measures under partial information[END_REF][START_REF] Fontana | Weak and strong no-arbitrage conditions for continuous financial markets[END_REF][START_REF] Chau | Optimal investment with intermediate consumption under no unbounded profit with bounded risk[END_REF][START_REF] Fontana | Sure profits via flash strategies and the impossibility of predictable jumps[END_REF]. Furthermore, as will be explained in the next section of this chapter, most of my contributions related to the theory of enlargement of filtrations (especially [START_REF] Fontana | On arbitrages arising with honest times[END_REF][START_REF] Acciaio | Arbitrage of the first kind and filtration enlargements in semimartingale financial models[END_REF]) also involve applications to questions arising from arbitrage theory. Before presenting an outline of my contributions, let me first point out that the No Arbitrage of the First Kind condition (NA 1 , see [START_REF] Kardaras | Finitely additive probabilities and the fundamental theorem of asset pricing[END_REF][START_REF] Kardaras | Market viability via absence of arbitrage of the first kind[END_REF]) has recently played a prominent role, among several no-arbitrage conditions proposed in the literature, as it represents the minimal condition ensuring the well-posedness of portfolio optimisation problems, as shown in [START_REF] Karatzas | The numeraire portfolio in semimartingale financial models[END_REF][START_REF] Choulli | How non-arbitrage, viability and numéraire portfolio are related[END_REF] (to this effect, see also Chapter 8 of this thesis).

The paper [START_REF] Fontana | No-arbitrage conditions and absolutely continuous changes of measure[END_REF] studies the behavior of several no-arbitrage conditions with respect to absolutely continuous but not equivalent changes of measure. From the financial standpoint, absolutely continuous changes of measure are motivated by the presence of agents with heterogeneous beliefs. In particular, it may happen that agents disagree on the existence of arbitrage opportunities, meaning that some agents can believe in illusory arbitrage opportunities (as considered also in [START_REF] Jarrow | Positive alphas, abnormal performance, and illusory arbitrage[END_REF]). A first example where an absolutely continuous change of measure induces arbitrage opportunities was proposed in [START_REF] Delbaen | Arbitrage possibilities in Bessel processes and their relations to local martingales[END_REF], based on a Bessel process. In [START_REF] Fontana | No-arbitrage conditions and absolutely continuous changes of measure[END_REF], two sets of results are obtained: first, in the context of continuous semimartingale models, we obtain an explicit characterization of the stability of several no-arbitrage conditions weaker than NFLVR. Second, in the context of general semimartingale models, we provide a necessary and sufficient condition ensuring that the NA 1 condition is stable under an absolutely continuous change of measure. This condition is related to the possibility that the density process of the two measures reaches zero due to a jump. A similar condition turns out to be relevant for the analysis of NA 1 stability under filtration enlargements, as will be discussed in the following section (see also Chapter 6 for more details). This similarity is not a coincidence, given the deep connection between absolutely continuous changes of measure and enlargement of filtrations formulae, as discussed in [START_REF] Yoeurp | Théorème de Girsanov généralisé et grossissement d'une filtration[END_REF].

In the subsequent paper [START_REF] Fontana | Weak and strong no-arbitrage conditions for continuous financial markets[END_REF], I gave a systematic presentation of the relevant no-arbitrage conditions in the context of continuous semimartingale models, with a specific focus on conditions which are weaker than NFLVR. The interesting aspect of such conditions is that they can be explicitly described in terms of the semimartingale characteristics of the discounted price process, unlike the stronger NFLVR condition. Moreover, one can distinguish between very weak and general no-arbitrage conditions (such as No Increasing Profit and No Strong Arbitrage), which only amount to exclude pathological forms of arbitrage, from more economically meaningful notions of no-arbitrage (such as NA 1 ). This distinction is illustrated in [START_REF] Fontana | Weak and strong no-arbitrage conditions for continuous financial markets[END_REF] by means of several examples and counterexamples. The short note [START_REF] Fontana | A note on arbitrage, approximate arbitrage and the fundamental theorem of asset pricing[END_REF] contains a critical study of the paper [START_REF] Wong | Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing[END_REF], whose goal was to propose a simple alternative proof of the fundamental theorem of asset pricing in the case of Itô-process models. The proof proposed in [START_REF] Wong | Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing[END_REF] is based on simple replication arguments. Unfortunately, as demonstrated in [START_REF] Fontana | A note on arbitrage, approximate arbitrage and the fundamental theorem of asset pricing[END_REF], the proof of [START_REF] Wong | Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing[END_REF] contains a critical gap, so that it can only work in the special case of complete markets, as already considered in [START_REF] Levental | A necessary and sufficient condition for absence of arbitrage with tame portfolios[END_REF].

The paper [START_REF] Fontana | Market viability and martingale measures under partial information[END_REF], which is presented in detail in Chapter 8, aims at establishing an equivalence between market viability (in the sense that a portfolio optimisation problem admits a solution) and the existence of a martingale measure whose density is given by the marginal utility of the optimal terminal wealth. In discrete-time models on a finite probability space, this equivalence is a classical result of financial economics (see [START_REF] Barucci | Financial Markets Theory: Equilibrium, Efficiency and Information[END_REF] Chapter IV]). In [START_REF] Fontana | Market viability and martingale measures under partial information[END_REF], we aim at establishing an analogous equivalence in the case of a jump-diffusion process under partial information. The generality of the setting requires the introduction of a suitable localizing sequence of stopping times, so that we can prove the desired equivalence locally. At a global level, a connection is established between market viability and the validity of the NA 1 condition, in line with the findings of [START_REF] Choulli | How non-arbitrage, viability and numéraire portfolio are related[END_REF]. Referring to Chapter 8 for a more detailed presentation, our results provide an explicit link between the densities of the martingale measures obtained locally and a local martingale deflator. Our proofs are based on constructive arguments and rely on stochastic maximum principles under partial information (see [START_REF] Baghery | A maximum principle for stochastic control with partial information[END_REF]). This line of research is continued in the subsequent paper [START_REF] Chau | Optimal investment with intermediate consumption under no unbounded profit with bounded risk[END_REF], albeit from a more abstract perspective. In [START_REF] Chau | Optimal investment with intermediate consumption under no unbounded profit with bounded risk[END_REF], we address the problem of optimal consumption (with respect to general random utilities and a stochastic clock) in a general semimartingale setup. Under the NA 1 hypothesis, we aim at proving a general duality theory for optimal consumption problems, under minimal assumptions. By generalizing the results of [START_REF] Mostovyi | Necessary and sufficient conditions in the problem of optimal investment with intermediate consumption[END_REF], we prove that the NA 1 condition is sufficient for establishing a complete duality theory and ensuring the existence of an optimal solution, as long as the dual value function is finite. The key step of the proof is represented by a dual characterization of admissible consumption plans, extending in a suitable way the optional decomposition theorem of [START_REF] Stricker | Some remarks on the optional decomposition theorem[END_REF]. As a corollary, in the classical case of maximization of expected utility from terminal wealth, our results extend the duality theory of [START_REF] Kramkov | The asymptotic elasticity of utility functions and optimal investment in incomplete markets[END_REF][START_REF] Kramkov | Necessary and sufficient conditions in the problem of optimal investment in incomplete markets[END_REF] to the case where NA 1 (but not necessarily NFLVR) holds.

In the recent work [START_REF] Fontana | Sure profits via flash strategies and the impossibility of predictable jumps[END_REF], we address a fundamental issue in stochastic asset pricing. From the financial economics literature (see, e.g., [START_REF] Barucci | Financial Markets Theory: Equilibrium, Efficiency and Information[END_REF] Chapter VII] for an overview), it is well-known that market efficiency implies that all relevant information is embedded in market prices, with the consequence that if asset prices jump at some scheduled date, then this can only be due to the release of unanticipated information. In [START_REF] Fontana | Sure profits via flash strategies and the impossibility of predictable jumps[END_REF], we provide a general formulation of this result in the modern language of mathematical finance. In particular, we do not impose any semimartingale restriction on the discounted asset price process and we only allow for realistic trading strategies, consisting of bounded buy-and-hold positions and high-frequency limits thereof. We aim at providing a noarbitrage characterization of predictable jumps, defined as jumps of predictable magnitude occurring at predictable times. Our main result shows that predictable jumps can occur if and only if the price process allows for sure profits generated by high-frequency limits of buy-and-hold strategies. Moreover, the result is robust with respect to the introduction of transaction costs. We furthermore show that right-continuity is a necessary requirement for any price process that does not admit sure profits, thus providing a sound financial justification for the ubiquitous assumption of right-continuity in mathematical finance. The results of [START_REF] Fontana | Sure profits via flash strategies and the impossibility of predictable jumps[END_REF] are closely related and motivated by several types of high-frequency strategies adopted in modern financial markets, such as "latency arbitrage strategies" and "directional event-based strategies" (see [Ald13]). As expected, in the semimartingale case the presence of predictable jumps leads to sure profits that can be realized via instantaneous strategies, and this is shown to be incompatible with the NA 1 condition.

Among the ongoing projects on arbitrage theory, I am currently studying the possibility of establishing a version of the fundamental theorem of asset pricing in the case of statistical arbitrage, along the lines of [START_REF] Bondarenko | Statistical arbitrage and securities prices[END_REF]. Unlike the classical notion of arbitrage (called "pure arbitrage opportunity" in [START_REF] Bondarenko | Statistical arbitrage and securities prices[END_REF]), a statistical arbitrage opportunity yields positive profits only with high probability, accepting a low probability of possible losses. In [START_REF] Bondarenko | Statistical arbitrage and securities prices[END_REF], the absence of statistical arbitrage opportunities is characterized in terms of the existence of a pricing functional with a certain structure. However, the setting considered in [START_REF] Bondarenko | Statistical arbitrage and securities prices[END_REF] is somehow restrictive from the standpoint of modern mathematical finance. Moreover, in the definition of statistical arbitrage there is an implicit notion of information, and this provides a clear suggestion that enlargement of filtration techniques can be useful to address the issue of statistical arbitrage. This project is part of an ongoing research collaboration with Christa Cuchiero (University of Vienna).

Enlargement of filtrations and its applications

The theory of enlargement of filtrations concerns the study of the behavior of stochastic processes with respect to en expansion of the underlying filtration. Historically, the theory started in the '70s with the study of several fundamental questions in the general theory of stochastic processes, in particular on the stability of the semimartingale property under an expansion of the reference filtration. Denoting by F = (F t ) t≥0 a given filtration, one can distinguish two canonical types of filtration enlargements. The first type is the progressive enlargement, where a filtration G = (G t ) t≥0 is constructed in such a way that F t ⊆ G t , for all t ≥ 0, and a given random time τ becomes a G-stopping time. The second type is known as the initial enlargement, where a filtration G = (G t ) t≥0 is constructed in such a way that F t ⊆ G t , for all t ≥ 0, and a given random variable L becomes G 0measurable. This corresponds respectively to adding information progressively over time and initially at the starting date t = 0. We refer the reader to the monograph [START_REF] Aksamit | Enlargement of Filtration with Finance in View[END_REF] for a recent presentation of the theory, as well as to the classic monograph [START_REF] Jeulin | Semi-martingales et Grossissement d'une Filtration[END_REF] and to [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF] Chapter VI] for a presentation of the main results. More recently, the theory of enlargement of filtrations gained new interest in view of financial applications, notably in the areas of credit risk and insider trading modelling.

My contributions to the theory and the applications of enlargement of filtrations are contained in the papers [START_REF] Fontana | Information, no-arbitrage and completeness for asset price models with a change point[END_REF][START_REF] Fontana | On arbitrages arising with honest times[END_REF][START_REF] Acciaio | Arbitrage of the first kind and filtration enlargements in semimartingale financial models[END_REF][START_REF] Fontana | The strong predictable representation property in initially enlarged filtrations under the density hypothesis[END_REF]. Referring to Chapter 4 for a detailed presentation of the main results of the paper [START_REF] Fontana | Information, no-arbitrage and completeness for asset price models with a change point[END_REF], let us mention that in this work we study an asset price model characterized by the fact that at some random time τ the drift, the volatility and the driving Brownian motions of the asset price process suddenly change (we call τ a random change point). We work under minimal assumptions on the random time τ and we perform a detailed study of the model under different filtrations (both progressively as well as initially enlarged) which arise naturally in this context, depending on the level of information associated with the random change point τ . In particular, we characterize the validity of the NFLVR and NA 1 no-arbitrage conditions in the different filtrations and we prove some new martingale representation results.

The paper [START_REF] Fontana | On arbitrages arising with honest times[END_REF] focuses on the case of a filtration F which is progressively enlarged with respect to a honest time τ , i.e., a random time τ such that for all t > 0 there exists an F t -measurable random variable ζ t such that τ = ζ t on {τ < t} (see, e.g., [Jeu80, Chapter V]). It is well-known that such a progressive enlargement is not compatible with the NFLVR requirement, as demonstrated in the papers [START_REF] Imkeller | Random times at which insiders can have free lunches[END_REF][START_REF] Zwierz | On existence of local martingale measures for insiders who can stop at honest times[END_REF]. The main issue that our work tries to address is the relation between the properties of the honest time τ and the validity of different notions of no-arbitrage on different time intervals in the enlarged filtration G. The most important results of [START_REF] Fontana | On arbitrages arising with honest times[END_REF] are reported in Chapter 5. Our work relies heavily on the results of [START_REF] Nikeghbali | Doob's maximal identity, multiplicative decompositions and enlargements of filtrations[END_REF], which give an explicit representation of the Azéma supermartingale associated to τ in terms of a positive F-local martingale N = (N t ) t≥0 . One of the main messages of [START_REF] Fontana | On arbitrages arising with honest times[END_REF] is that most of the (no-)arbitrage properties of the model (in the progressively enlarged filtration G) are encoded in the F-local martingale N . As an application, our results enable us to distinguish precisely at which point in time the activity of arbitrageurs (insider traders) can lead to a market failure.

The paper [START_REF] Acciaio | Arbitrage of the first kind and filtration enlargements in semimartingale financial models[END_REF] originated as a generalization of some of the results obtained in [START_REF] Fontana | On arbitrages arising with honest times[END_REF]. In particular, the setting of [START_REF] Fontana | On arbitrages arising with honest times[END_REF] was restricted to the case of continuous semimartingales in the presence of a honest times τ that avoids all F-stopping times, in the sense that P(τ = σ) = 0, for every F-stopping time σ. As explained in detail in Chapter 6, in [START_REF] Acciaio | Arbitrage of the first kind and filtration enlargements in semimartingale financial models[END_REF] we consider a general (possibly discontinuous) semimartingale model in the presence of a general random time τ . One should be aware that, at this level of generality, F-semimartingales can cease to be G-semimartingales. As demonstrated in [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF][START_REF] Kardaras | On the semimartingale property of discounted asset-price processes[END_REF], non-semimartingale processes (in a frictionless financial market) automatically lead to the existence of arbitrage opportunities. For this reason, we restrict our attention to the NA 1 condition and to the stochastic interval [[0, τ ]] since every F-semimartingale stopped at τ remains a G-semimartingale (see [START_REF] Jeulin | Semi-martingales et Grossissement d'une Filtration[END_REF]Chapter IV]). By relying on the multiplicative optional decomposition of the Azéma supermartingale of τ recently established in [START_REF] Kardaras | On the stochastic behaviour of optional processes up to random times[END_REF], we obtain new results on the behavior of F-local martingales stopped at τ in the progressively enlarged filtration G. In turn, these results enable us to explicitly construct local martingale deflators in the progressively enlarged filtration G, starting from local martingale deflators in the original filtration F. In the analysis, a crucial role is played by the possibility that the discounted asset price process jumps simultaneously to a jump to zero of the Azéma supermartingale, a condition whose importance has been first pointed out in [START_REF] Aksamit | Non-arbitrage up to random horizon for semimartingale models[END_REF]. One of the most important findings of [START_REF] Acciaio | Arbitrage of the first kind and filtration enlargements in semimartingale financial models[END_REF] is that the proposed approach is also applicable to the case of initially enlarged filtrations, as long as the classical Jacod's density hypothesis (see [START_REF] Jacod | Grossissement initial, hypothèse (H'), et théorème de Girsanov[END_REF]) holds true. In this case, as explained in Chapter 6, the family of F-conditional densities of the random variable L with respect to which the filtration F is enlarged plays an analogous role to the Azéma supermartingale in the case of a progressive enlargement.

The paper [START_REF] Fontana | The strong predictable representation property in initially enlarged filtrations under the density hypothesis[END_REF] is mostly theoretical and provides a novel martingale representation result in the case of initially enlarged filtrations, assuming the validity of Jacod's density hypothesis. In particular, the density hypothesis is assumed to hold in the form of an absolute continuity relation, and not necessarily in the form of an equivalence relation, as in most previous papers on the subject (see, e.g., [START_REF] Amendinger | Martingale representation theorems for initially enlarged filtrations[END_REF][START_REF] Amendinger | A monetary value for initial information in portfolio optimization[END_REF][START_REF] Callegaro | Carthaginian enlargement of filtrations[END_REF]). This generalisation is especially important in view of the modelling of insider trading phenomena and naturally arises when the additional information generates arbitrage opportunities. A detailed presentation of the main results of [START_REF] Fontana | The strong predictable representation property in initially enlarged filtrations under the density hypothesis[END_REF] is given in Chapter 7.

At the moment, I am working (in collaboration with Huy N. Chau, at the Hungarian Academy of Sciences, and Andrea Cosso, at the University of Bologna) on the problem of quantifying in monetary terms the value of an inside information which can be exploited to generate arbitrage profits. The problem is formulated in the setting of an initially enlarged filtration G, similarly as in [START_REF] Fontana | The strong predictable representation property in initially enlarged filtrations under the density hypothesis[END_REF] (see Chapter 7), where the random variable L generating the filtration expansion plays the role of the inside information. By relying on the results of [START_REF] Acciaio | Arbitrage of the first kind and filtration enlargements in semimartingale financial models[END_REF] (see Chapter 6), we work under a standing assumption ensuring that the filtration expansion preserves the NA 1 no-arbitrage condition, thus guaranteeing the well-posedness of portfolio optimisation problems in the presence of inside information. In order to define the value of information, we adopt an indifference pricing approach, in the spirit of [START_REF] Amendinger | A monetary value for initial information in portfolio optimization[END_REF]. However, we focus on the case where the inside information generates arbitrage opportunities, so that the NFLVR condition is violated. Such a situation is motivated by the possibility of insider trading phenomena in the presence of private information and was excluded in the analysis of [START_REF] Amendinger | A monetary value for initial information in portfolio optimization[END_REF]. This necessitates the development of an alternative approach to compute the indifference price of information and we rely on the duality approach of [START_REF] Chau | Optimal investment with intermediate consumption under no unbounded profit with bounded risk[END_REF] to optimal consumption problems in the presence of arbitrage opportunities (but under the NA 1 condition). For typical utility functions (power, logarithmic and exponential), the value of information can be computed explicitly and several comparative statics results can be obtained, thereby generalizing the analysis of [START_REF] Liu | Information, expected utility and portfolio choice[END_REF].

Interest rate and credit risk term structure modelling

In the last few years, I have also worked on multi-curve interest rate models. The appearance of multiple curves of interest rates (e.g., OIS, Libor/Euribor rates for different tenors) represents one of the most striking consequences of the last financial crises on fixed income markets and is due to the increased liquidity and credit risk in the interbank market. This phenomenon has led to violations of the classical no-arbitrage relations between risk-free rates and Libor rates associated to different tenors. We refer the reader to [START_REF] Grbac | Interest Rate Modeling: Post-Crisis Challenges and Approaches[END_REF] for a detailed overview of the features of multi-curve interest rate markets. Moreover, in the financial industry, a new paradigm in interest rate modelling has emerged, with the development of multi-curve models, where each interest rate curve is modelled in a separate way. From the standpoint of mathematical finance, a recent challenge is represented by the construction of general multi-curve models that are consistent with market data, are rooted on sound no-arbitrage principles and, finally, possess a satisfactory analytical tractability. This has been the main objective of a research collaboration with Christa Cuchiero (University of Vienna) and Alessandro Gnoatto (BayernLB), which has led to the articles [START_REF] Cuchiero | A general hjm framework for multiple yield curve modeling[END_REF][START_REF] Cuchiero | Affine multiple yield curve models[END_REF].

The main goal of [START_REF] Cuchiero | A general hjm framework for multiple yield curve modeling[END_REF] 1 was the development of a general framework, inspired by the Heath-Jarrow-Morton (HJM) philosophy, for modelling multiple yield curves, with a focus on a precise characterization of absence of arbitrage. In [START_REF] Cuchiero | A general hjm framework for multiple yield curve modeling[END_REF], we have considered a general semimartingale setting for the modelling of the term structures of OIS rates (considered as the best proxy for a risk-free rate, consistently with market practice) and of multiplicative spreads between FRA (Forward Rate Agreements) rates and simply compounded OIS rates. This modelling choice is motivated by the fact that FRAs represent the essential building blocks of fixed income derivatives written on Libor/Euribor rates. Moreover, such multiplicative spreads admit a clear economic interpretation and can be easily inferred from market data. In this context, we have derived drift and consistency conditions ensuring absence of arbitrage, in the sense of No Asymptotic Free Lunch with Vanishing Risk (NAFLVR, see [START_REF] Cuchiero | A new perspective on the fundamental theorem of asset pricing for large financial markets[END_REF]). We rely on an abstract representation of the HJM modelling framework, inspired by the recent paper [START_REF] Kallsen | On a Heath-Jarrow-Morton approach for stock options[END_REF]. We also address the issue of the existence of an arbitrage-free multiple curve model. To this effect, we prove existence and uniqueness of the SPDEs associated to the forward curves when translated to the Musiela parametrization and show how to guarantee the drift and consistency conditions by constructing an appropriate pure jump process whose compensator solves a generalized moment problem. The generality of the setting considered in [START_REF] Cuchiero | A general hjm framework for multiple yield curve modeling[END_REF] is highlighted by the fact that most of the multiple curve models proposed so far in the literature can be recovered as suitable specifications of our general framework.

While the focus of the paper [START_REF] Cuchiero | A general hjm framework for multiple yield curve modeling[END_REF] is on the theoretical aspects, the subsequent paper [START_REF] Cuchiero | Affine multiple yield curve models[END_REF] is more applied. In particular, we aim at providing a general and tractable framework under which all multiple yield curve modelling approaches based on affine processes can be consolidated. We adopt an abstract formulation, with respect to a generic numéraire -martingale measure couple, as this enables us to simultaneously consider short rate models, market models as well as rational and benchmark models (the latter under the physical probability measure). Similarly as in [START_REF] Cuchiero | A general hjm framework for multiple yield curve modeling[END_REF], we choose multiplicative spreads between FRA rates and simply compounded OIS rates as the main modelling quantities and we assume that they are given as functions of an underlying affine process, taking values in a general convex state space. Besides allowing for ordered spreads and an exact fit to the initially observed term structures, this general framework leads to tractable valuation formulas for caplets and swaptions and embeds all existing multi-curve affine models. The proposed approach also gives rise to new developments, such as a short rate model driven by a Wishart process, for which we prove a closed-form pricing formula for caplets. Finally, we have tested the empirical performance of two specifications of our framework with respect to market data, demonstrating a satisfactory accuracy in fitting the implied volatilities extracted from caps and floors market quotes.

Concerning credit risk modelling, my interest in this research area started at the beginning of my PhD studies, with the incomplete information models developed in [START_REF] Fontana | Credit risk and incomplete information: filtering and EM parameter estimation[END_REF][START_REF] Fontana | Credit risk and incomplete information: a filtering framework for pricing and risk-management[END_REF] and the hybrid equity-credit model of [START_REF] Fontana | A unified approach to pricing and risk management of equity and credit risk[END_REF]. More recently, in the context of an ongoing collaboration with Thorsten Schmidt (University of Freiburg), we studied a general HJM framework for defaultable term structures, where the distinguishing feature is the presence of dynamic information on the future likelihood of default. More specifically, referring the reader to [START_REF] Fontana | General dynamic term structures under default risk[END_REF] for a detailed presentation of the framework, we model the term structure of defaultable 0-coupon bonds, with possibly non-zero recovery at the time of default. We work under minimal assumptions on the random time τ representing the default time, in particular we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. This requires the introduction of an additional term in the HJM setting, which is driven by a random measure encoding information about those times where default can happen with positive probability. In the model, this information evolves in a dynamic way and absence of arbitrage implies a certain consistency relation between the dynamics of the forward rate describing the term structure and the information process. We derive necessary and sufficient conditions for a reference probability measure to be a local martingale measure for the large financial market of defaultable bonds. This is also related to the impossibility of predictable jumps in the dynamics of defaultable bond prices, coherently with the analysis of [START_REF] Fontana | Sure profits via flash strategies and the impossibility of predictable jumps[END_REF].

At the moment, I am working on a generalisation of the approach of [START_REF] Fontana | General dynamic term structures under default risk[END_REF] in the context of multi-curve interest rate models, thereby generalizing the existing approaches. The motivation comes from the existence of discontinuities in maturity in the term structure of interest rates, generated for instance by political events and monetary policy decisions. This is a project developed in collaboration with Zorana Grbac (Paris Diderot University) and funded by a research grant of the Europlace Institute of Finance. Another ongoing research project consists in the construction of a tractable affine multicurve model based on α-stable Lévy processes, in the spirit of the α-CIR model recently proposed in [START_REF] Jiao | Alpha-CIR model with branching processes in sovereign interest rate modeling[END_REF], also performing a calibration analysis to market data. I am currently working on this project together with my PhD student Guillaume Szulda (Paris Diderot University).

CHAPTER 3

Complete list of publications

Book

(1) E. Barucci

Part II

Information and arbitrage: a presentation of the main results

Introduction

The behavior of asset prices is often subject to certain random events that result in abrupt changes in their dynamics. The random time when such an event occurs is called a change point. For example, a sudden adjustment in interest rates, a default of a major financial institution, or the release of some political news could all have an impact on asset prices. Although these events are not caused by the price evolution of the individual asset, their occurrence may change the asset price dynamics. In this case the change point is said to be exogenous to the model. On the other side, events linked to the assets themselves can also cause a change in their dynamics (e.g., an asset price crossing a certain threshold). Such a change point is said to be endogenous to the model.

In this chapter we study a general class of models with a change point, which are able to capture both of the above-mentioned situations. The dynamics of the asset price is modeled as a stochastic exponential of a continuous semimartingale, whose drift, volatility and driving Brownian motions change at a random time. We impose minimal assumptions on the non-negative random variable representing the change point: if such assumptions are violated, then the semimartingale property of the driving Brownian motions is lost when changing filtrations and pathological forms of arbitrage are possible. The drift and the volatility are stochastic and depend on time and on the current state of the process and the Brownian motions might be correlated. In particular, the random time is allowed to depend on the Brownian motions.

We aim at understanding the structure and the behavior of this class of models in several filtrations which naturally arise in this setting. Such filtrations represent different levels of information and are obtained as progressive enlargements of a reference filtration with respect to the random time. Using enlargement of filtration techniques, we study the no-arbitrage properties of the model in each filtration. In particular, we characterize the absence of arbitrages of the first kind, which is equivalent to a square-integrability property of the market price of risk process. In turn, the latter condition allows us to obtain martingale representation theorems. Combining these results we give for all filtrations a complete characterization of equivalent local martingale measures and of market completeness. Even though we are not specifically concerned with the detection of the change point, we find that in the case when the two volatility regimes are distinct, the random time is actually a stopping time with respect to the price process filtration. In the present chapter, we limit our attention to the case of progressively enlarged filtrations, referring the reader to the original paper [START_REF] Fontana | Information, no-arbitrage and completeness for asset price models with a change point[END_REF] for the analysis of the initially enlarged filtrations related to the random change point.

Our study is inspired by the work of [START_REF] Cawston | F -divergence minimal equivalent martingale measures and optimal portfolios for exponential Lévy models with a change-point[END_REF], where an exponential asset price model driven by two independent Lévy processes and with an independent change point is developed and analyzed in the context of utility maximization. In comparison with that paper, we refrain from imposing any independence assumption. We refer to [START_REF] Cawston | F -divergence minimal equivalent martingale measures and optimal portfolios for exponential Lévy models with a change-point[END_REF] for a comprehensive literature review on change point models (which have been extensively studied by A. N. Shiryaev and co-authors, see [START_REF] Shiryaev | On stochastic models and optimal methods in the problems of the quickest detection[END_REF]), as well as their applications.

The model presented in this chapter can also be seen as a regime switching model. By regime switching models we mean models in which the drift and the volatility of the price process are functions of a process taking finitely many values, which are interpreted as states of the economy. This underlying state process is usually assumed to be a Markov chain in order to ensure analytical tractability. A special case of our model, in which the two Brownian motions are assumed to be the same and the random time is the first jump time to an absorbing state of a Markov process with two states {0, 1}, is interpreted as a regime switching model according to the above definition. Regime switching models have been widely employed in statistics and financial modeling, see for instance the volume [START_REF] Elliott | Hidden Markov Models in Finance[END_REF], the survey paper [START_REF] Guo | A regime switching model: statistical estimation, empirical evidence, and change point detection[END_REF] and, for econometric applications, [START_REF] Hamilton | Time Series Analysis[END_REF] and the references therein.

This chapter is organized as follows. In Section 2 we introduce the notation and the setting of the model. Section 3 discusses the well-posedness of the SDE defining the model and studies the properties of the model in two progressively enlarged filtrations. In Section 4 we analyze the model in its own filtration. Finally, Section 5 concludes by pointing out possible generalizations and applications.

Setting and preliminaries

Let (Ω, A, P) be a given probability space, with P denoting the physical probability measure, and let T ∈ (0, +∞) be a fixed time horizon. We assume that all random variables and stochastic processes introduced in the following are measurable with respect to the σ-fields A and A ⊗ B([0, T ]), respectively. Let F = (F t ) 0≤t≤T be a filtration on (Ω, A, P), assumed to satisfy the usual conditions of right-continuity and P-completeness. For a given stochastic process Y = (Y t ) 0≤t≤T on (Ω, A, P) we denote by F Y = (F Y t ) 0≤t≤T the right-continuous P-augmented natural filtration of Y . The random change point is represented by a random time τ , i.e., an A-measurable random variable τ : Ω → R + which is not necessarily an F-stopping time. Furthermore, we let W 1 = (W 1 t ) 0≤t≤T and W 2 = (W 2 t ) 0≤t≤T be two independent Brownian motions on (Ω, F, P) and, for i = 1, 2, we denote by F W i the natural P-augmented filtration of W i .

We consider a financial market with one risky asset and one riskless asset. As usual in the literature, we take the riskless asset as the numéraire and directly pass to discounted quantities. We denote by S = (S t ) 0≤t≤T the discounted price process of the risky asset and suppose that S can be represented as follows, for some initial value S 0 ∈ (0, +∞):

S = S 0 E(X), (2.1)
where E(•) denotes the stochastic exponential and X is described by the SDE

dX t = 1 {t≤τ } µ 1 (t, X t ) + 1 {t>τ } µ 2 (t, X t ) dt + 1 {t≤τ } σ 1 (t, X t )dW 1 t + 1 {t>τ } σ 2 (t, X t ) ρdW 1 t + 1 -ρ 2 dW 2 t , (2.2) X 0 = 0, with ρ ∈ [-1, 1].
The well-posedness of the above SDE, as well as the existence and uniqueness of a solution, will be shown in Section 3.1 on a suitable filtered probability space (see Proposition 3.2). Observe that F S coincides with the filtration F X generated by the process X. The functions µ i : [0, T ] × R → R and σ i : [0, T ] × R → (0, +∞), for i = 1, 2, are Borel-measurable and are assumed to satisfy the following condition.

Condition 2.1. The functions µ i : [0, T ] × R → R and σ i : [0, T ] × R → (0, +∞), for i = 1, 2, satisfy the following conditions:

(1) there exists a constant K > 0 such that:

|µ i (t, x) -µ i (t, y)| ≤ K|x -y|, ∀t ∈ [0, T ], ∀x, y ∈ R, for i = 1, 2; |σ i (t, x) -σ i (t, y)| ≤ K|x -y|, ∀t ∈ [0, T ], ∀x, y ∈ R, for i = 1, 2;
(2) the function (t, x) → σ i (t, x) is jointly continuous in (t, x) ∈ [0, T ] × R, for i = 1, 2.

Part (1) of Condition 2.1 consists of the usual global Lipschitz conditions on the functions µ i and σ i appearing in the SDE (2.2), while part (2) is needed for technical reasons.

We aim at studying the properties of the model (2.1)-(2.2) with respect to different levels of information, mathematically represented by different filtrations on (Ω, A, P). In particular, we restrict our attention to the study of the behavior of the process X in the following filtrations:

(i) the filtration F X ; (ii) the filtration G X , obtained as the progressive enlargement of F X with respect to τ and defined as G X t := s>t (F X s ∨ σ(τ ∧ s)) for all t ∈ [0, T ]; (iii) the filtration G, obtained as the progressive enlargement of F with respect to τ and defined as

G t := s>t (F s ∨ σ(τ ∧ s)) for all t ∈ [0, T ].
The filtrations G X and G are the smallest right-continuous filtrations which contain F X and F, respectively, and make τ a G X -stopping time and a G-stopping time, respectively. It holds that:

F X ⊆ G X ⊆ G.
Intuitively, in the special case where F = F W 1 ∨ F W 2 , the different filtrations introduced above correspond to market participants having access to different information sets:

(i) F X t : the knowledge of only the price process of the risky asset up to time t; (ii) G X t : the knowledge of the price process of the risky asset up to time t plus the knowledge of the random time τ if the latter has occurred before time t; (iii) G t : the knowledge of the two driving Brownian motions W 1 and W 2 up to time t plus the knowledge of the random time τ if the latter has occurred before time t.

We shall denote by p Y the predictable projection of a process Y = (Y t ) 0≤t≤T onto one of the filtrations introduced above. The filtration onto which we will take projections changes throughout the chapter, but this will be made clear in the text. Let A = (A t ) 0≤t≤T be a generic filtration on (Ω, A, P) with respect to which the process S is a semimartingale and L(S, A) be the set of all S-integrable A-predictable processes. We denote by hdS the stochastic integral process ( t 0 h u dS u ) 0≤t≤T , for h = (h t ) 0≤t≤T ∈ L(S, A), and by M loc (A) the family of all A-local martingales. Note that, if S = M + B denotes the canonical decomposition of S into M ∈ M loc (A) and a continuous predictable process of finite variation B, it holds that hdS = hdM + hdB.

In order to study the no-arbitrage properties of the model (2.1)-(2.2), we recall the characterizations of two important notions of arbitrage which have been considered in the literature. Definition 2.2.

(1) We say that No Arbitrage of the First Kind (NA 1 ) holds in the filtration A if there exists an A-local martingale deflator, i.e., a process L = (L t ) 0≤t≤T ∈ M loc (A) with L 0 = 1 and L T > 0 P-a.s. and such that SL ∈ M loc (A); (2) We say that No Free Lunch with Vanishing Risk (NFLVR) holds in the filtration A if there exists an A-martingale deflator, i.e., a process L = (L t ) 0≤t≤T ∈ M loc (A) with L 0 = 1, L T > 0 P-a.s. and E[L T ] = 1 and such that SL ∈ M loc (A).

Part (1) of Definition 2.2 is due to [START_REF] Kardaras | Finitely additive probabilities and the fundamental theorem of asset pricing[END_REF], while part (2) goes back to [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF]. In particular, the NA 1 condition is weaker than NFLVR and, moreover, can be shown to be the minimal condition for market viability. Note also that martingale deflators correspond to density processes of Equivalent Local Martingale Measures (ELMMs). We refer to [START_REF] Fontana | Diffusion-based models for financial markets without martingale measures[END_REF] for a study of the two no-arbitrage conditions introduced above in the context of general diffusion-based models.

The progressively enlarged filtrations G and G X

In this section we study the progressively enlarged filtrations G and G X . We shall make no assumption on the random time τ apart from a very weak semimartingale-preservation hypothesis (see Condition 3.1). We start our analysis with the progressively enlarged filtration G, which is easier to describe than the filtration G X .

3.1. The progressively enlarged filtration G. The filtration G is the smallest filtration satisfying the usual conditions which contains F and makes τ a G-stopping time. However, the F-Brownian motions W 1 and W 2 may fail to be G-semimartingales. Therefore, we impose the following condition.

Condition 3.1. There exist two G-predictable processes θ 1 = (θ 1 t ) 0≤t≤T and θ 2 = (θ 2 t ) 0≤t≤T and two G-Brownian motions W 1 = ( W 1 t ) 0≤t≤T and W 2 = ( W 2 t ) 0≤t≤T such that:

W i t = W i t + t 0 θ i u du
, for all t ∈ [0, T ] and for i = 1, 2.

Condition 3.1 can be regarded as a weak form of the (H )-hypothesis from the theory of enlargement of filtrations, which assumes that all F-semimartingales are G-semimartingales ([Jeu80], Chapter II). Condition 3.1 can be shown to hold for almost all random time models considered in financial and insurance mathematics (in particular, it is always trivially satisfied in the common case when τ is a doubly stochastic random time, as well as when the density hypothesis holds, see [START_REF] Karoui | What happens after a default: The conditional density approach[END_REF]).

3.1.1. Existence and uniqueness of the solution to the SDE (2.2). As long as Condition 3.1 holds, equation (2.2) makes sense as a semimartingale SDE on (Ω, A, G, P). This provides a good setting for establishing the existence and uniqueness of a solution. We say that a G-semimartingale X = (X t ) 0≤t≤T is a solution to the SDE (2.2) on (Ω, A, G, P) if X 0 = 0 and X satisfies equation (2.2) with respect to the G-semimartingales W 1 and W 2 . This corresponds to the notion of strong solution of a semimartingale-driven SDE, as considered in Chapter V of [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF] (see also [START_REF] Jacod | Calcul Stochastique et Problèmes de Martingales[END_REF], Chapter XIV).

Proposition 3.2. Suppose that Conditions 2.1 and 3.1 hold. Then there exists a unique continuous G-semimartingale X = (X t ) 0≤t≤T which is a solution to the SDE (2.2) on (Ω, A, G, P).

3.1.2. Canonical decomposition and no-arbitrage properties in G. Let us now investigate the noarbitrage properties of the financial market where the asset S is traded with respect to the information contained in the progressively enlarged filtration G. As a preliminary, we write the canonical decomposition of the process X = (X t ) 0≤t≤T in the filtration G:

(3.1) X t = t 0 μu du + t 0 V u d W u , for all t ∈ [0, T ],
where the processes μ = (μ t ) 0≤t≤T , V = (V t ) 0≤t≤T and W = ( W t ) 0≤t≤T are defined as 1 :

(3.2) μt := 1 {t≤τ } µ 1 (t, X t ) + σ 1 (t, X t )θ 1 t + 1 {t>τ } µ 2 (t, X t ) + σ 2 (t, X t ) ρθ 1 t + 1 -ρ 2 θ 2 t , (3.3) V t := 1 {t≤τ } σ 1 (t, X t ) + 1 {t>τ } σ 2 (t, X t ), W t := W 1 t∧τ + ρ ( W 1 t∨τ -W 1 τ ) + 1 -ρ 2 ( W 2 t∨τ -W 2 τ ). (3.4)
1 Note that Condition 3.1 implicitly requires that T 0 |θ i u |du < +∞ P-a.s., for i = 1, 2. In turn, due to Condition 2.1-(2) together with the continuity of X, this implies that t 0 θ i u σ i (u, Xu)du is well-defined, for all t ∈ [0, T ] and i = 1, 2.

Since [ W ] t = t, for all t ∈ [0, T ], the continuous G-local martingale W is a G-Brownian motion. Equation (3.1) gives the canonical decomposition of X in G and leads to the next proposition, which characterizes the no-arbitrage properties of the model (2.1)-(2.2) in the progressively enlarged filtration G. We define the G-predictable process θ = ( θt ) 0≤t≤T as:

(3.5) θ := 1 [ [0,τ ] ] θ 1 + 1 ] ]τ,T ] ] ρ θ 1 + 1 -ρ 2 θ 2 .
Proposition 3.3. Suppose that Conditions 2.1 and 3.1 hold. Then the following assertions hold for the model (2.1)-(2.2) considered with respect to the filtration G:

(1) NA 1 holds if and only if

= (N t ) 0≤t≤T ∈ M loc (G) with N 0 = 0, ∆N > -1 P-a.s., [N, W ] = 0 such that E[E(-(μ/V )d W + N ) T ] = 1.
Note that, if the Brownian motions W 1 and W 2 are G-semimartingales but their finite variation parts are not absolutely continuous with respect to dt (i.e., Condition 3.1 is violated), one can then obtain the most egregious form of arbitrage (i.e., an increasing profit) in the filtration G, see [FR13, Section 4.3]. In this sense, Condition 3.1 represents a minimal assumption for the study of the noarbitrage properties of the model (2.1)-(2.2).

Proposition 3.3 shows that the process θ defined in (3.5) plays a crucial role in determining the noarbitrage properties of the model (2.1)-(2.2) in G. In turn, this implies that the existence of arbitrages in G crucially depends on the properties of τ . For instance, the condition T 0 θ2

t dt < +∞ P-a.s. may fail in the cases considered in [START_REF] Imkeller | Random times at which insiders can have free lunches[END_REF]. This is also related to arbitrages possibly arising with honest times, as considered in Chapter 5.

3.2.

The progressively enlarged price process filtration G X . This section studies the properties of the model (2.1)-(2.2) when considered in the filtration G X .

3.2.1. Canonical decomposition and no-arbitrage properties in G X . The next lemma gives the canonical decomposition of X with respect to the filtration G X . Lemma 3.4. Suppose that Conditions 2.1 and 3.1 hold. Then the process X admits the following canonical decomposition with respect to the filtration G X :

(3.6) X t = t 0 µ u du + t 0 V u dB u , for all t ∈ [0, T ],
where the G X -predictable process µ = (µ t ) 0≤t≤T is defined as:

(3.7) µ t := 1 {t≤τ } µ 1 (t, X t ) + σ 1 (t, X t ) p θ 1 t + 1 {t>τ } µ 2 (t, X t ) + σ 2 (t, X t ) ρ p θ 1 t + 1 -ρ 2 p θ 2 t ,
with p θ i denoting the G X -predictable projection of θ i , for i = 1, 2, and where the process

B = (B t ) 0≤t≤T is a G X -Brownian motion and the G X -predictable process V = (V t ) 0≤t≤T is defined as in (3.3).
We can now answer the question of whether the model (2.1)-(2.2), viewed in the filtration G X , allows for arbitrage profits. We denote by p θ the G X -predictable projection of the process θ =

1 [ [0,τ ] ] θ 1 + 1 ] ]τ,T ] ] (ρ θ 1 + 1 -ρ 2 θ 2 ) introduced in (3.5).
Proposition 3.5. Suppose that Conditions 2.1 and 3.1 hold. Then the following assertions hold for the model (2.1)-(2.2) considered with respect to the filtration G X :

(1) NA 1 holds if and only if

= (N t ) 0≤t≤T ∈ M loc (G X ) with N 0 = 0, ∆N > -1 P-a.s., [N, B] = 0 such that E E -(µ/V )dB + N T = 1.
3.2.2. Martingale representation property in G X . We now study in more detail the structure of the filtration G X . In particular, we aim at proving a martingale representation result (see Proposition 3.7). In turn, this will lead to an explicit characterization of all G X -martingale deflators.

As a preliminary, observe that the process 1/V is well-defined, G X -predictable and locally bounded, being left-continuous by part (2) of Condition 2.1. Hence, we can define the G X -adapted continuous process Y = ( Y t ) 0≤t≤T as follows, for all t ∈ [0, T ]:

(3.8) Y t := t 0 V -1 u dX u = t 0 µ u V u du + B t ,
where the processes µ and B are as in Lemma 3.4. Let us denote by

F Y = (F Y t ) 0≤t≤T the right- continuous P-augmented natural filtration of Y and by G Y = (G Y t ) 0≤t≤T the progressive enlargement of F Y with respect to τ , meaning that G Y t = s>t (F Y s ∨ σ(τ ∧ s)) for all t ∈ [0, T ].
We can now state a useful lemma which describes the structure of the filtration G X . Lemma 3.6. Suppose that Conditions 2.1 and 3.1 hold. Then G X = G Y .

We are now in position to state a martingale representation result for the filtration G X . It is important to note that we do not make any assumption on τ nor on the underlying filtration F (in particular, we do not assume that

F = F W 1 ∨ F W 2 ). We denote by A G X the G X -predictable compensator of τ and by M G X := 1 {τ ≤•} -A G X the associated G X -martingale.
Proposition 3.7. Suppose that Conditions 2.1 and 3.1 hold and assume in addition that NA 1 holds in the filtration G X . Then every G X -local martingale L = (L t ) 0≤t≤T admits a representation of the form:

(3.9) L t = L 0 + t 0 ϕ u dB u + t 0 ψ u dM G X u , for all t ∈ [0, T ],
for some G X -predictable processes ϕ = (ϕ t ) 0≤t≤T and ψ = (ψ t ) 0≤t≤T with T 0 ϕ 2 t dt < +∞ P-a.s. and

T 0 |ψ t ||dA G X t | < +∞ P-a.s.
3.2.3. Stability of no-arbitrage properties with respect to filtration shrinkage. At this point, one may wonder whether the absence of arbitrage in G already implies the absence of arbitrage in the smaller filtration G X . The answer to such a question is expected to be affirmative, because any outcome of a G X -trading strategy should also be realized as the outcome of a G-trading strategy, since G X ⊂ G. However, one has to prove that stochastic integrals defined in G X can also be viewed as stochastic integrals in the larger filtration G (see [Jeu80, Theorem 3.23] for a counterexample).

Proposition 3.8. Suppose that Conditions 2.1 and 3.1 hold. Then the following assertions hold for the model (2.1)-(2.2):

(1) NA 1 in the filtration G implies NA 1 in the filtration G X ;

(2) NFLVR in the filtration G implies NFLVR in the filtration G X .

Conversely, starting from a smaller filtration that satisfies NA 1 /NFLVR and passing to a larger filtration, it may well happen that arbitrage possibilities are introduced. As a simple example, consider the case where X = W 1 , so that NFLVR (and, hence, NA 1 as well) trivially holds in F X = F W 1 . If τ is an honest time (see [START_REF] Jeulin | Semi-martingales et Grossissement d'une Filtration[END_REF], Chapter V) that avoids all F X -stopping times, then both NFLVR and NA 1 will fail to hold in the enlarged filtrations G = G X , as shown in Chapter 5.

Special cases.

In this section we analyze two special cases of the general setting described so far, which are important in view of financial applications.

Immersion property between F and G.

Let us suppose that the filtrations F and G satisfy the immersion property (or (H)-hypothesis, see [START_REF] Brémaud | Changes of filtrations and of probability measures[END_REF]) with respect to the random time τ , meaning that all F-martingales are also G-martingales. This situation is rather interesting in view of the fact that many random time models considered in financial and insurance mathematics satisfy this property. For instance, many popular credit risk models assume that τ is a doubly stochastic random time (see e.g. [START_REF] Bielecki | Credit Risk: Modeling, Valuation and Hedging[END_REF], Section 8.2). In this case, the immersion property holds between F and G and τ is a random change point that occurs in an unpredictable way.

The immersion property considerably simplifies the analysis of the model (2.1)-(2.2). Indeed, Condition 3.1 trivially holds with θ i ≡ 0 for i = 1, 2, and Proposition 3.3 immediately implies that the model (2.1)-(2.2) satisfies NA 1 in G. However, we cannot a priori exclude the existence of free lunches with vanishing risk. Since τ is a G X -stopping time, it is easy to deduce from (3.1)-(3.4) and Lemma 3.4 that the canonical decomposition of X in the filtration G X coincides with its canonical decomposition in G. Furthermore, given that θ ≡ 0 (and hence p θ ≡ 0), Proposition 3.5 implies that NA 1 holds for the model (2.1)-(2.2) considered in the filtration G X .

3.3.2. Stopping times with respect to the filtration F. Let us now consider the case where τ is a stopping time with respect to F. For instance, τ could be defined as the first passage time of one of the two Brownian motions W 1 and W 2 at some given level. This is the case where τ is a random change point which is endogenous to the model, in the sense that its occurrence is determined by the same stochastic processes which drive the dynamics of S. If τ is an F-stopping time, it is evident that G = F. Hence, Condition 3.1 is trivially satisfied with θ i ≡ 0 for i = 1, 2. In this case, as discussed in Section 3.3.1, Proposition 3.3 implies that NA 1 holds in F = G. However, we cannot exclude a priori the existence of free lunches with vanishing risk. Also, when considering the model (2.1)-(2.2) in the filtration G X , we are in a situation analogous to that discussed at the end of Section 3.3.1.

The price process filtration F X

In this section, we study the model (2.1)-(2.2) with respect to its own filtration F X , which is the smallest among the filtrations introduced at the beginning of this chapter.

4.1. Canonical decomposition and no-arbitrage properties in F X . The next lemma gives the canonical decomposition of X in its own filtration F X . Lemma 4.1. Suppose that Conditions 2.1 and 3.1 hold. Then the process X admits the following canonical decomposition with respect to the filtration F X :

(4.1) X t = t 0 μu du + t 0 V u d Bu , for all t ∈ [0, T ],
where the F X -predictable process μ = (μ t ) 0≤t≤T is defined as follows, for all t ∈ [0, T ]:

(4.2) μt := p (1 [ [0,τ ] ] ) t µ 1 (t, X t ) + p 1 [ [0,τ ] ] θ 1 t σ 1 (t, X t ) + p (1 ] ]τ,T ] ] ) t µ 2 (t, X t ) + p 1 ] ]τ,T ] ] ρ θ 1 + 1 -ρ 2 θ 2 t σ 2 (t, X t ),
with p denoting the F X -predictable projection and where the process B = ( Bt ) 0≤t≤T is an F X -Brownian motion and V = (V t ) 0≤t≤T is as in (3.3).

The next proposition answers the question of whether the model (2.1)-(2.2), considered now with respect to its own filtration F X , allows for arbitrage profits.

Proposition 4.2. Suppose that Conditions 2.1 and 3.1 hold. Then the following assertions hold for the model (2.1)-(2.2) considered with respect to the filtration F X :

(1) NA 1 holds if and only if

T 0 (μ t /V t ) 2 dt < +∞ P-a.s.
(2) NFLVR holds if and only if NA 1 holds and there exists N

= (N t ) 0≤t≤T ∈ M loc (F X ) with N 0 = 0, ∆N > -1 P-a.s., [N, B] = 0, such that E[E(-(μ/V )d B + N ) T ] = 1.
4.2. The F X -martingale representation property. This section will provide the martingale representation property with respect to the price filtration F X . We shall see that the representation formula and thus the market completeness or incompleteness are determined by the relationship of the two volatility functions σ 1 and σ 2 . Condition 4.3.

(1) Distinct volatility functions

σ 1 (t, x) = σ 2 (t, x), for any (t, x) ∈ [0, T ] × R.
(2) Identical volatility functions

σ 1 (t, x) = σ 2 (t, x) =: σ(t, x), for all (t, x) ∈ [0, T ] × R.

Distinct volatility functions.

Let us first analyze the case where σ 1 and σ 2 differ everywhere.

Proposition 4.4. Suppose that Conditions 2.1, 3.1 and 4.3-(1) hold. Then τ is a stopping time for the filtration F X and, consequently, the filtrations F X and G X coincide.

As long as Condition 4.3-(1) holds, all the results of Section 3.2 for the progressively enlarged filtration G X also hold for the filtration F X . In particular, Lemma 3.6 gives an explicit description of the filtration F X = G X as the progressive enlargement of the filtration F Y generated by the drifted F X -Brownian motion Y . Furthermore, any F X -local martingale admits the representation obtained in Proposition 3.7.

Identical volatility functions. Let us now analyze the case of Condition 4.3-(2)

, where the two volatility functions σ 1 and σ 2 coincide everywhere. Lemma 4.1 gives the following canonical decomposition of X in its own filtration F X : (4.3)

X t = t 0 μu du + t 0 σ(u, X u )d Bu , for all t ∈ [0, T ],
where the process μ = (μ t ) 0≤t≤T is as in (4.2). Since the continuous process σ(•, X • ) is F X -adapted and never attains zero, it is F X -predictable and bounded away from zero. Hence, we can define a drifted F X -Brownian motion Ȳ = ( Ȳt ) 0≤t≤T as follows, for all t ∈ [0, T ]:

(4.4) Ȳt := t 0 1 σ(u, X u ) dX u = t 0 μu σ(u, X u ) du + Bt .
Denote by F Ȳ = (F Ȳ t ) 0≤t≤T the right-continuous P-augmented natural filtration Ȳ . We can easily obtain the next lemma, which shows that F X coincides with the filtration F Ȳ generated by the drifted Brownian motion Ȳ (see also Section 3 of [START_REF] Pham | Optimal portfolio in partially observed stochastic volatility models[END_REF] for related results).

Lemma 4.5. Suppose that Conditions 2.1, 3.1 and 4.3-(2) hold. Then F X = F Ȳ .

As in Section 3.2, we can now state a martingale representation result for the filtration F X in the special case where the two volatility functions σ 1 and σ 2 coincide. Proposition 4.6. Suppose that Conditions 2.1, 3.1 and 4.3-(2) hold and assume in addition that NA 1 holds in the filtration F X . Then every F X -local martingale L = (L t ) 0≤t≤T admits a representation of the form:

L t = L 0 + t 0 ϕ u d Bu , for all t ∈ [0, T ],
for some F X -predictable process ϕ = (ϕ t ) 0≤t≤T with T 0 ϕ 2 t dt < +∞ P-a.s.

Remark 4.7. By relying on Proposition 4.6, we can show that the financial market where S is traded with respect to the information contained in its own filtration F X is complete, in the sense that for any bounded F X T -measurable non-negative random variable H there exists a couple (v

H , h H ) ∈ [0, +∞) × L(S, F X ) such that H = v H + T 0 h H t dS t P-a.s.
Note that this result only requires the NA 1 condition for the model (2.1)-(2.2) in the filtration F X and does not depend on the validity of NFLVR.

Conclusion and further developments

In this chapter we have studied a class of asset price models with a change point, imposing only minimal assumptions on the random time and on the driving Brownian motions. We have characterized the model by its properties of martingale representation, completeness or incompleteness and two notions of arbitrage.

The model can be generalized to incorporate multiple change points by enlargement of filtrations with a sequence of random times and to incorporate discontinuous semimartingales as the driving processes by techniques for jump processes. If a martingale representation property holds in a filtration generated by the two underlying semimartingales, then martingale representation theorems can also be obtained for various filtrations related to the model. Similarly, our results can also be generalized to the case where the coefficients µ and σ in (2.2) are only F-progressively measurable processes. The extension of other results from this chapter, in particular the characterization of the no-arbitrage properties, depends on the specific class of semimartingales which are used as driving processes.

Besides the obvious application to the modeling of financial asset prices, the results of the present chapter can also be of interest in view of credit risk, interest rate modeling and energy markets, where sudden changes in the dynamics of the underlying processes are naturally observed. A further line of research concerns stochastic control problems under incomplete information, either in the manner of utility maximization like in, for example, [START_REF] Björk | Optimal investment under partial information[END_REF], [START_REF] Lakner | Utility maximization with partial information[END_REF], and [START_REF] Pham | Optimal portfolio in partially observed stochastic volatility models[END_REF], or extending a method lately introduced by [START_REF] Abbas-Turki | Impulse control of a diffusion with a change point[END_REF].

Introduction and motivation

The study of insider trading behavior represents a classical issue in mathematical finance and financial economics. Loosely speaking, insider trading phenomena occur when agents having access to different information sets operate in the same financial market. In particular, the better informed agents may try to realise profits by relying on their deeper private knowledge and trading with the less informed agents. Typically, the presence of two distinct layers of information is mathematically represented by two filtrations F = (F t ) t≥0 and G = (G t ) t≥0 with F t ⊆ G t for all t ≥ 0. Intuitively, the filtration G represents the information in possession of the insider trader. Assuming that the less informed agents cannot realise arbitrage profits by trading in the market, the fundamental question can be formulated in the following terms: is it possible for the insider trader to realise arbitrage profits by making use of the information contained in the larger filtration G? And, if yes, what is the appropriate notion of "arbitrage profit" and what is the arbitrage strategy?

The main goal of the present chapter is to give complete and precise answers to the above questions in the context of a general continuous financial market model where the information of the insider is associated to an honest time τ . Referring to Section 2 for a precise definition of the notion of honest time (which seems to have been first introduced by [MSW72]), we would like to quote the following passage from [START_REF] Dellacherie | Probabilités et Potentiel. Chapitres de XVII à XXIV. Processus de Markov (fin), Compléments de Calcul Stochastique[END_REF] (page 137):

Par exemple S t peut reprsenter le cours d'une certaine action l'instant t, et τ est le moment idal pour vendre son paquet d'actions. Tous les spculateurs cherchent connaître τ sans jamais y parvenir, d'o son nom de variable aléatoire honnête.

We consider a filtered probability space (Ω, F, F, P) and let S = (S t ) t≥0 represent the discounted price process of some risky assets. The filtration G = (G t ) t≥0 is constructed as the progressive enlargement of F with respect to an honest time τ , which is assumed to avoid all F-stopping times. For a detailed account of the theory of progressive enlargement of filtrations, we refer the reader to Chapitres IV-V of [START_REF] Jeulin | Semi-martingales et Grossissement d'une Filtration[END_REF] (see also [JYC09, Section 5.9.4], [Nik06, Section 9.2] and [Pro05, Section VI.3] and the book [START_REF] Yor | Random Times and Enlargements of Filtrations in a Brownian Setting[END_REF] for a presentation of the theory in the case where F is a Brownian filtration). In this context, the investors who have access only to the information contained in the filtration F represent the "spculateurs" referred to in the passage quoted above. In particular, they are not allowed to construct portfolio strategies based on τ , simply because τ is not an F-stopping time. In contrast, an insider trader having access to the full information of the filtration G can rely on his private knowledge about τ when trading in the market and, hence, may have the possibility of realising arbitrage profits.

In the present chapter, we provide a complete analysis of the different kinds of arbitrage that can be realised by an insider trader having access to the additional information generated by an honest time. We do not confine ourselves to the classical no-arbitrage theory based on the notions of Arbitrage Opportunity and Free Lunch with Vanishing Risk, as developed by [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF], but we also consider several stronger notions of arbitrage, namely Unbounded Increasing Profits, Arbitrages of the First Kind and Unbounded Profits with Bounded Risk, which are of current interest in mathematical finance, as documented by the recent papers [START_REF] Hulley | M 6 -on minimal market models and minimal martingale measures[END_REF], [START_REF] Karatzas | The numeraire portfolio in semimartingale financial models[END_REF], [START_REF] Kardaras | Finitely additive probabilities and the fundamental theorem of asset pricing[END_REF][START_REF] Kardaras | Market viability via absence of arbitrage of the first kind[END_REF], [START_REF] Takaoka | A note on the condition of no unbounded profit with bounded risk[END_REF] and [START_REF] Kabanov | No arbitrage of the first kind and local martingale numéraires[END_REF]. We make precise the severity of the arbitrages induced by an honest time τ . Furthermore, we carefully distinguish what kinds of arbitrage can be realised before, at and after time τ and show that arbitrage profits incompatible with market viability can only be realised by trading as soon as τ occurs.

It is already known that an honest time τ induces arbitrage opportunities in the progressively enlarged filtration G immediately after time τ , see [START_REF] Imkeller | Random times at which insiders can have free lunches[END_REF] and [START_REF] Zwierz | On existence of local martingale measures for insiders who can stop at honest times[END_REF]. In comparison with these papers, our results provide two main contributions. On the one hand, we show that an insider trader can always realise an arbitrage opportunity not only after τ but also exactly at time τ . On the other hand, we can explicitly exhibit in a simple way the trading strategies which yield the arbitrage profits. This contrasts with the approach adopted in [START_REF] Imkeller | Random times at which insiders can have free lunches[END_REF] and [START_REF] Zwierz | On existence of local martingale measures for insiders who can stop at honest times[END_REF], where the existence of arbitrage opportunities is proved by relying on the abstract results of [START_REF] Delbaen | The existence of absolutely continuous local martingale measures[END_REF]. Moreover, our approach permits to recover the results obtained in [START_REF] Imkeller | Random times at which insiders can have free lunches[END_REF] and [START_REF] Zwierz | On existence of local martingale measures for insiders who can stop at honest times[END_REF] in a very simple way. A key tool in our analysis is represented by the multiplicative decomposition of the Azéma supermartingale Z = (Z t ) t≥0 associated to an honest time τ , established in [START_REF] Nikeghbali | Doob's maximal identity, multiplicative decompositions and enlargements of filtrations[END_REF].

Example 1.1. We illustrate the framework of the present chapter in the simplest possible setting (detailed proofs and sharper results in a general setting will be given in Sections 4-6). Let W = (W t ) t≥0 be a one-dimensional Brownian motion on the filtered probability space (Ω, F, F W , P), where F W = (F W t ) t≥0 denotes the natural filtration of W (augmented by the P-nullsets of F W ∞ ) and F := F W ∞ . Let the process S = (S t ) t≥0 represent the discounted price of a risky asset and be given as the solution to the following SDE, for some σ = 0 and s ∈ (0, +∞):

(1.1) dS t = S t σ dW t , S 0 = s.

We define the finite random time τ := sup t ≥ 0 : S t = sup u≥0 S u and the filtration G = (G t ) t≥0 as the progressive enlargement of F with respect to τ (see Section 2 for precise definitions).

We call informed agent an agent who can invest in the risky asset S and has access to the information contained in the enlarged filtration G. The main results of the present chapter can then be essentially summarised as follows: Furthermore, we can explicitly construct the trading strategies which realise the arbitrage profits for the informed agent in (a) and (c): it will be enough to hold appropriate long and short positions, respectively, in the portfolio which replicates the non-negative F-local martingale N = (N t ) t≥0 appearing in the multiplicative decomposition of the Azéma supermartingale Z = (Z t ) t≥0 of the random time τ (see Lemma 2.9). In the specific case of Example 1.1, these positions will simply reduce to long and short buy-and-hold positions, respectively, in the risky asset S itself.

The study of the impact of the additional information associated to a random time on the noarbitrage-type properties of a financial market and on the behavior of market participants has already attracted attention in the mathematical finance literature. In particular, [START_REF] Imkeller | Random times at which insiders can have free lunches[END_REF] and [START_REF] Zwierz | On existence of local martingale measures for insiders who can stop at honest times[END_REF] are the closest precursors to our work (related results also appear in [START_REF] Ankirchner | Finite utility on financial markets with asymmetric information and structure properties of the price dynamics[END_REF]). In the context of credit risk modelling, a study of the no-arbitrage-type properties of a market model with a filtration progressively enlarged with respect to a random time has also been undertaken in [START_REF] Coculescu | Default times, no-arbitrage conditions and changes of probability measures[END_REF]. We also want to mention that, in the case of initially enlarged filtrations, the possibility of realising arbitrage profits has been studied in [START_REF] Grorud | Asymmetrical information and incomplete markets[END_REF] and [START_REF] Imkeller | Free lunch and arbitrage possibilities in a financial market model with an insider[END_REF]. We refer the interested reader to [START_REF] Nikeghbali | On honest times in financial modeling[END_REF][START_REF] Nikeghbali | A reading guide for last passage times with financial applications in view[END_REF] for a detailed analysis of the role of honest times in financial modelling. In Chapter 6, we will generalise the analysis to the case of a general random time τ in a general semimartingale model, restricting our attention to the no-arbitrage properties up to time τ . This chapter is structured as follows. Section 2 describes the general setting and recalls several no-arbitrage-type conditions as well as some key technical results from the theory of progressive enlargement of filtrations. Sections 3 studies the existence and the properties of local martingale deflators in the progressively enlarged filtration G up to different random time horizons, i.e., at, before and after an honest time τ . Sections 4, 5 and 6 contain the main results on the existence of arbitrage profits with respect to the filtration G at τ , before τ and after τ , respectively. Section 7 concludes the chapter by discussing the role played by Assumptions 2.6-2.8 introduced in Section 2.

General setting and preliminary results

Let (Ω, F, P) be a given probability space endowed with a filtration F = (F t ) t≥0 satisfying the usual conditions, where P denotes the physical probability measure and F := F ∞ . We consider a financial market comprising d+1 assets, with prices described by the R d+1 -valued process S = ( St ) t≥0 . To allow for greater generality, we consider a financial market model on an infinite time horizon. Of course, financial markets on a finite time horizon [0, T ] can be embedded by simply considering the stopped process ( S) T . We assume that S0 represents a numéraire or reference asset and is P-a.s. strictly positive. Without loss of generality, we express the prices of all d + 1 assets in terms of S0 -discounted quantities, thus obtaining the R d -valued process S = (S t ) t≥0 , with S i := Si / S0 for each i = 1, . . . , d. We assume that the process S is a continuous semimartingale on (Ω, F, F, P).

Let the random time τ : Ω → [0, +∞] be a P-a.s. finite honest time on (Ω, F, F, P). This means that τ is an F-measurable random variable such that, for all t > 0, there exists an F t -measurable random variable ζ t with τ = ζ t on {τ < t} (see [Jeu80, Chapitre V]). We define the filtration G = (G t ) t≥0 as the progressive enlargement of F with respect to τ , i.e., G t := s>t (F s ∨ σ (τ ∧ s)) for all t ≥ 0, augmented by the P-nullsets of F = F ∞ = G ∞ . It is well-known that G is the smallest filtration satisfying the usual conditions which contains F and makes τ a G-stopping time. Furthermore, the (H )-hypothesis holds between F and G, meaning that any F-semimartingale is also a G-semimartingale (see [Jeu80, Theorem 5.10]). In particular, this implies that the discounted price process S is also a G-semimartingale.

In order to model the activity of trading, we need to define the notion of admissible trading strategy, following [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF]. Let H denote a generic filtration, i.e., in our setting H ∈ {F, G}. We denote by L H (S) the set of all R d -valued H-predictable processes θ = (θ t ) t≥0 which are S-integrable in H and we write θ • S for the corresponding stochastic integral process.

Definition 2.1. Let H ∈ {F, G}. For a ∈ R + , an element θ ∈ L H (S) is said to be an a-admissible H-strategy if (θ • S) ∞ := lim t→+∞ (θ • S) t exists and (θ • S) t ≥ -a P-a.s. for all t ≥ 0. We denote by A H a the set of all a-admissible H-strategies. We say that an element θ ∈ L H (S) is an admissible

H-strategy if θ ∈ A H := a∈R + A H a .
We assume that there are no frictions or trading constraints and that trading is done in a selffinancing way. This implies that the wealth process generated by trading according to an admissible H-strategy θ starting from an initial endowment of x ∈ R is given by V (x, θ) := x + θ • S, for H ∈ {F, G}. We call restricted financial market the tuple M F := Ω, F, F, P ; S, A F , as opposed to the enlarged financial market M G := Ω, F, G, P ; S, A G . Intuitively, agents operating in the enlarged financial market are better informed than agents operating in the restricted financial market, due to the additional information generated by the random time τ .

Remark 2.2. Note that, since F ⊆ G and all F-semimartingales are also G-semimartingales, it holds that L F (S) ⊆ L G (S), as can be deduced from [Jeu80, Proposition 2.1]. In turn, this implies that A F ⊆ A G , thus reflecting the fact that agents in the enlarged financial market are allowed to use a richer information set to construct their portfolios.

As mentioned in the introduction, we aim at answering the following question: how does the additional information associated to the honest time τ give rise to arbitrage profits? To this end, let us first recall three important notions of arbitrage which have appeared in the literature.

Definition 2.3. Let H ∈ {F, G}.
(1) A non-negative F-measurable random variable ξ with P (ξ > 0) > 0 yields an Arbitrage of the First Kind if for all x > 0 there exists an element θ x ∈ A H x such that V (x, θ x ) ∞ ≥ ξ P-a.s. If there exists no such random variable we say that the financial market M H satisfies the No Arbitrage of the First Kind (NA 1 ) condition.

(2) An element θ ∈ A H yields an Arbitrage Opportunity if V (0, θ) ∞ ≥ 0 P-a.s. and P(V (0, θ) ∞ > 0) > 0. If there exists no such θ ∈ A H we say that the financial market M H satisfies the No Arbitrage (NA) condition. (3) A sequence {θ n } n∈N ⊂ A H yields a Free Lunch with Vanishing Risk if there exist an ε > 0 and an increasing sequence {δ n } n∈N with 0 ≤ δ n 1 such that P(V (0,

θ n ) ∞ > -1 + δ n ) = 1 and P(V (0, θ n ) ∞ > ε) ≥ ε.
If there exists no such sequence we say that the financial market M H satisfies the No Free Lunch with Vanishing Risk (NFLVR) condition.

For a (possibly infinite-valued) H-stopping time , we say that NA 1 /NA/NFLVR holds in the financial market M H on the time horizon [0, ] if the financial market Ω, F, H, P ; S , A H satisfies NA 1 /NA/NFLVR, where S denotes the stopped process (S t∧ ) t≥0 .

The notion of No Arbitrage of the First Kind has been introduced by [START_REF] Kardaras | Finitely additive probabilities and the fundamental theorem of asset pricing[END_REF] and can be shown to be equivalent to the boundedness in probability of the set

V (1, θ) ∞ : θ ∈ A H 1 , see [Kar10, Proposi- tion 1].
The latter condition has appeared under the name of No Unbounded Profit with Bounded Risk (NUPBR) in [START_REF] Karatzas | The numeraire portfolio in semimartingale financial models[END_REF] and its importance was first recognised by [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF][START_REF] Kabanov | On the FTAP of Kreps-Delbaen-Schachermayer[END_REF] (see also [START_REF] Fontana | Weak and strong no-arbitrage conditions for continuous financial markets[END_REF] for a discussion of the relations between the different conditions introduced in Definition 2.3). The NA 1 and NFLVR conditions can both be characterised in purely probabilistic terms. As a preliminary, let us recall the following definition.

Definition 2.4. Let H ∈ {F, G} and a (possibly infinite-valued) H-stopping time.

(1) A strictly positive H-local martingale L = (L t ) t≥0 with L 0 = 1 and L ∞ > 0 P-a. and NA hold in the financial market M H .

In particular, the above theorem implies that NFLVR holds in the financial market M H on the time horizon [0, ] if and only if there exists a local martingale deflator L in H such that L is a uniformly integrable H-martingale. In the present chapter, we shall work under the following standing assumption, which ensures that the financial market M F does not allow for any kind of arbitrage.

Assumption 2.6. The restricted financial market M F satisfies NFLVR.

We aim at studying the no-arbitrage-type properties (or the lack thereof) of the enlarged financial market M G . In the remaining part of the chapter, we shall give a clear answer to this issue under the two following standing assumptions, where we denote by M = (M t ) t≥0 the F-local martingale part in the canonical decomposition of the semimartingale S in the filtration F.

Assumption 2.7. The random time τ avoids all F-stopping times: for every F-stopping time T we have P (τ = T ) = 0.

Assumption 2.8. The continuous F-local martingale M = (M t ) t≥0 has the F-predictable representation property in the filtration F.

Assumption 2.7 is classical when dealing with progressive enlargements of filtrations (see [JYC09, Section 5.9.4]). Assumption 2.8 means that any F-local martingale U = (U t ) t≥0 with U 0 = 0 can be represented as U = ϕ • M , where ϕ = (ϕ t ) t≥0 is an R d -valued F-predictable process such that t 0 ϕ s d M, M s ϕ s < +∞ P-a.s. for all t ≥ 0, see [JS03, Chapter III]. In particular, Assumption 2.8 implies that all F-martingales are continuous. We shall discuss the importance of Assumptions 2.6-2.8, and of possible extensions thereof, in Section 5 of the present chapter.

We close this section by recalling two technical results obtained by [START_REF] Nikeghbali | Doob's maximal identity, multiplicative decompositions and enlargements of filtrations[END_REF] under the hypothesis that all F-local martingales are continuous and Assumption 2.7 holds. Recall that a P-a.s. finite random time τ is an honest time if and only if it is the end of an F-optional set (see [Jeu80, Proposition 5.1]) and note that, due to Assumption 2.8 together with the continuity of S, the F-optional sigma field coincides with the F-predictable sigma field. In the following, we denote by Z = (Z t ) t≥0 the Azéma supermartingale of the random time τ , i.e., Z t = P (τ > t|F t ), for all t ≥ 0.

Lemma 2.9 ([NY06], Theorem 4.1). There exists a continuous non-negative F-local martingale N = (N t ) t≥0 with N 0 = 1 and lim t→+∞ N t = 0 P-a.s. such that Z admits the following multiplicative decomposition, for all t ≥ 0:

Z t = P (τ > t|F t ) = N t N * t
where N * t := sup s≤t N s . Furthermore, we have that:

τ = sup {t ≥ 0 : N t = N * t } = sup {t ≥ 0 : N t = N * ∞ } .
Lemma 2.10 ([NY06], Proposition 2.5). Every F-local martingale X = (X t ) t≥0 has the following canonical decomposition as a semimartingale in G:

X t = X t + t∧τ 0 d X, N s N s - t∨τ τ d X, N s N * ∞ -N s 29
where X = ( X t ) t≥0 is a G-local martingale and N = (N t ) t≥0 is as in Lemma 2.9.

In the specific case of Example 1.1, the local martingale N appearing in Lemma 2.9 is equal to S/S 0 itself (see Example 4.2). The importance of Lemma 2.9 consists in the possibility of reducing the study of the existence of arbitrage profits in a general enlarged financial market M G to the simple situation considered in Example 1.1.

Existence and properties of local martingale deflators in G

In view of Theorem 2.5, local martingale deflators play a fundamental role in characterising the validity of the NA 1 and NFLVR conditions. The goal of the present section consists in studying the existence and the properties of local martingale deflators in the progressively enlarged filtration G.

Without any loss of generality, we may and do assume that P is already an ELMM F for the restricted financial market M F . Indeed, Assumption 2.6 together with part (2) of Theorem 2.5 ensures the existence of an ELMM F Q. Since Q ∼ P, it is easy to verify that all the properties of the general setting described in Section 2 still hold under Q. More precisely, the random time τ is still an honest time which avoids F-stopping times under any ELMM F Q and the (Q, F)-local martingale S = (S t ) t≥0 has the F-predictable representation property under the measure Q (see [JS03, Theorem III.5.24]). Hence, Assumptions 2.6, 2.7 and 2.8 hold under every ELMM F Q. Finally, observe that the notion of admissible strategy given in Definition 2.1 is stable under an equivalent change of measure. As a consequence, all the no-arbitrage-type conditions introduced in Definition 2.3 hold for the enlarged financial market M G under the measure P if and only if they hold under the measure Q.

3.1.

Local martingale deflators in G on the time horizon [0, τ ]. Note first that, due to Assumption 2.8, the F-local martingale N appearing in Lemma 2.9 admits the stochastic integral representation N = 1 + ϕ • S, where ϕ = (ϕ t ) t≥0 is an R d -valued F-predictable process in L F (S). By Lemma 2.10, the stopped process S τ admits the following canonical decomposition in the filtration G:

(3.1) S τ t = S τ t + t∧τ 0 d S, N s N s = S τ t + t∧τ 0 d S, S s ϕ s N s = S τ t + t 0 d S τ , S τ s ϕ s N s
where S = ( S t ) t≥0 is a continuous G-local martingale.

Proposition 3.1. The process 1/N τ = (1/N t∧τ ) t≥0 is a local martingale deflator in G on the time horizon [0, τ ]. Furthermore, the process 1/N τ fails to be a uniformly integrable G-martingale.

L σ∧τ = 1 N σ∧τ exp - k 1 [ [0,σ] ] N * • N * 1 + 1 {τ ≤σ} k τ 1 [ [τ,+∞[ [ + η1 [ [τ,+∞[ [
where k = (k t ) t≥0 is an F-predictable process such that

(3.3) E L σ∧τ = E 1 -exp - τ 0 1 + k s N * s dN * s 1 {ν≤σ}
where k = (k t ) t≥0 is the F-predictable process appearing in the representation (3.2) and τ 0 1+ks N * s dN * s > 0 P-a.s. The stopped process L σ∧τ is a uniformly integrable G-martingale if and only if P(ν ≤ σ) = 0.

Remark 3.4 (On martingales and strict local martingales). Let L = (L t ) t≥0 be a local martingale deflator in G on the time horizon [0, τ ]. By Fatou's lemma, the strictly positive G-local martingale L τ is also a G-supermartingale. As a consequence, L τ is a (true) G-martingale if and only if it has constant expectation, i.e., E [L t∧τ ] = 1 for all t ≥ 0. Due to Lemma 3.3 (with σ = t), the latter condition holds if and only if P (ν = +∞) = 1. This means that, as soon as P (ν < +∞) > 0, any local martingale deflator L in G on [0, τ ] is a strict G-local martingale in the sense of [START_REF] Elworthy | The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes[END_REF], being a G-local martingale which fails to be a (true) G-martingale. It is interesting to note that Lemma 3.2 gives then a recipe for constructing a whole class of possibly discontinuous strict G-local martingales. As a preliminary, recall that Lemma 2.10 gives the following canonical decomposition of S = (S t ) t≥0 in the enlarged filtration G:

(3.4) S t = S t + t∧τ 0 d S, N s N s - t∨τ τ d S, N s N * ∞ -N s = S t + t 0 d S, S s αs =: S t + A t where S = ( S t ) t≥0 is a G-local martingale, αt := 1 {t≤τ } ϕ t /N t -1 {t>τ } ϕ t /(N * ∞ -N t )
and where the process ϕ = (ϕ t ) t≥0 ∈ L F (S) ⊂ L G (S) is the integrand in the stochastic integral representation N = 1 + ϕ • S, with N as in Lemma 2.9.

Proposition 3.5. The enlarged financial market M G does not admit any local martingale deflator in G on the global time horizon [0, +∞]. Proposition 3.5 represents a negative result, since it shows that there exists no local martingale deflator in G on [0, +∞]. We close this section by showing that, if we restrict our attention to the time horizon [ρ, +∞], for any G-stopping time ρ with ρ > τ P-a.s., then there still exists a local martingale deflator in G. For every such G-stopping time ρ, let us introduce the process ρ S := S -S ρ = S ρ∨• -S ρ .

Proposition 3.6. For every G-stopping time ρ such that ρ > τ P-a.s., the process ρ L = ( ρ L t ) t≥0 defined by, for all t ≥ 0,

ρ L t := N * ∞ -N ρ N * ∞ -N ρ∨ t
, is a local martingale deflator for the process ρ S with respect to the filtration G. The process ρ L is a uniformly integrable G-martingale if and only if P(ρ < ν) = 0, with ν = inf{t ≥ 0 : N t = 0}.

Arbitrages on the time horizon [0, τ ]

The goal of this section is to determine whether the information associated to an honest time τ does give rise to arbitrage profits in the enlarged financial market M G on the time horizon [0, τ ]. Unless mentioned otherwise, we shall always suppose that Assumptions 2.6, 2.7 and 2.8 are satisfied.

Recall first that Lemma 2.9 gives the existence of a non-negative F-local martingale N with N 0 = 1 and lim t→+∞ N t = 0 P-a.s. such that τ = sup {t ≥ 0 :

N t = N * ∞ }.
It is clear that N τ ≥ 1 P-a.s. as well as P (N τ > 1) > 0. Furthermore, due to Assumption 2.8, there exists an R d -valued F-predictable process ϕ = (ϕ t ) t≥0 ∈ L F (S) such that N = 1+ϕ•S. By relying on these arguments, we can construct an admissible G-strategy which yields an arbitrage opportunity in the enlarged financial market M G .

Theorem 4.1.

(1) NA 1 (or, equivalently, NUPBR) holds in M G on the time horizon [0, τ ];

( Despite of its simplicity, the result of part (2) of Theorem 4.1 is quite interesting. Indeed, it shows that, as long as Assumptions 2.6, 2.7 and 2.8 hold, one can explicitly construct an admissible Gstrategy which realises an arbitrage opportunity at the honest time τ . To the best of our knowledge, this result is new: as mentioned in the introduction, all previous works in the literature have only shown the existence of arbitrage opportunities immediately after τ (see [START_REF] Imkeller | Random times at which insiders can have free lunches[END_REF] and [START_REF] Zwierz | On existence of local martingale measures for insiders who can stop at honest times[END_REF]).

Example 4.2 (Discussion of Example 1.1). Let d = 1 and suppose that the real-valued process S = (S t ) t≥0 is given as the solution of the SDE (1.1) on the filtered probability space Ω, F, F W , P , where F W is the (P-augmented) natural filtration of W . Since S is a (P, F)-martingale, Assumption 2.6 is trivially satisfied and, clearly, Assumption 2.8 holds as well. Furthermore, since lim t→+∞ S t = 0 P-a.s. (due to the law of large numbers for Brownian motion), [NY06, Corollary 2.4] implies that the random time τ = sup t ≥ 0 : S t = sup u≥0 S u is an honest time which avoids all F-stopping times, meaning that Assumption 2.7 is also satisfied. Note that, in the context of this example, we have S/S 0 = N , as can be deduced from [NY06, Proposition 2.2], and ν = +∞ P-a.s. Then, Theorem 4.1 directly imply claim (a) after Example 1.1. Observe, furthermore, that, in the context of this simple example, the arbitrage opportunity reduces to a simple buy-and-hold position on the risky asset S until time τ .

We close this section with an example, which in particular shows how the arbitrage strategy φ appearing in part (2) Theorem 4.1 can be explicitly calculated.

Example 4.3 (An arbitrage opportunity arising from a last passage time). As in Example 1.1, let the discounted price process S = (S t ) t≥0 of a risky asset be modeled as in (1.1) on the filtered probability space (Ω, F, F W , P) and define the random time τ := sup{t ≥ 0 : S t = a}, for some a ∈ (0, S 0 ). Since lim t→+∞ S t = 0 P-a.s., the random time τ is easily seen to be a P-a.s. finite honest time. Moreover, following [JYC09, Section 5.6], the associated Azéma supermartingale Z = (Z t ) t≥0 is given by Z t = (S t /a) ∧ 1, for all t ≥ 0. By Tanaka's formula (see e.g. [JYC09, Section 4.1.8]), the Doob-Meyer decomposition of Z can be computed as:

Z t = 1 + 1 a t 0 1 {Su<a} dS u - 1 2a L a t , for all t ≥ 0,
where L a = (L a t ) t≥0 denotes the local time of S at the level a. Note that, in view of Remark 1.2 in [START_REF] Yor | Random Times and Enlargements of Filtrations in a Brownian Setting[END_REF] together with the continuity of L a , the honest time τ satisfies Assumption 2.7. Since Z t > 0 P-a.s. for all t ≥ 0, [JS03, Theorme II.8.21] implies that the Azéma supermartingale Z admits a multiplicative decomposition of the form Z = N/D, where:

D = exp L a 2a and N = E 1 aZ 1 {S<a} dS = 1 + D a 1 {S<a} dS
using the fact that, for almost all ω ∈ Ω, the measure dL a • (ω) is supported by {t ≥ 0 : S t (ω) = a} = {t ≥ 0 : Z t (ω) = 1}, due to [Pro05, Theorem IV.69]. Since lim t→+∞ Z t = 0 P-a.s., the continuous F-local martingale N satisfies lim t→+∞ N t = 0 P-a.s. Furthermore, Skorohod's reflection lemma (see [JYC09, Lemma 4.1.7.1]) and the arguments used in the proof of [NY06, Theorem 4.1] allow to check that D = N * , with N * denoting the running supremum of N . This gives a complete and explicit description of the multiplicative decomposition Z = N/N * appearing in Lemma 2.9. Since τ = sup{t ≥ 0 :

Z t = 1} = sup{t ≥ 0 : N t = N * ∞ }, the strategy φ := 1 [ [0,τ ] ] 1 a exp( L a 2a )1 {S<a} ∈ A G 1
yields an arbitrage opportunity in the enlarged financial market M G .

Arbitrages on the time horizon

[0, σ ∧ τ ]
In the present section, we study whether it is possible to exploit the information of the progressively enlarged filtration G in order to obtain arbitrage opportunities before the honest time τ . The answer to such a question is given by the following theorem, which relies on Lemma 3.3.

Theorem 5.1. Let σ be an F-stopping time. Then NFLVR holds in the enlarged financial market M G on the time horizon [0, σ ∧ τ ] if and only if P (σ ≥ ν) = 0.

In particular, due to Theorem 5.1, the NFLVR condition holds in the enlarged financial market M G on the time horizon [0, τ ∧ T ] for any T ∈ (0, +∞) satisfying P (ν ≤ T ) = 0. We also have the following corollary, which shows that one can never obtain arbitrage opportunities in the enlarged financial market M G strictly before the honest time τ .

Arbitrages on the time horizon [0, +∞]

In this section, we study the existence of arbitrage profits in the enlarged financial market M G on the global time horizon [0, +∞], taking into account especially what can happen after the honest time τ . We shall always suppose that Assumptions 2.6, 2.7 and 2.8 are satisfied and that, without loss of generality, P is already an ELMM F for S (see the beginning of Section 3).

The question of whether the NA 1 condition holds in the enlarged financial market M G on the time horizon [0, +∞] is negatively answered by the following proposition, which explicitly exhibits an arbitrage of the first kind, in the sense of part (1) of Definition 2.3. Proposition 6.1. The random variable ξ := N τ -1 yields an arbitrage of the first kind. As a consequence, NA 1 (or, equivalently, NUPBR) fails to hold in the enlarged financial market M G on the global time horizon [0, +∞].

It has already been shown in [START_REF] Imkeller | Random times at which insiders can have free lunches[END_REF] and [START_REF] Zwierz | On existence of local martingale measures for insiders who can stop at honest times[END_REF] that NFLVR fails to hold after τ in the enlarged financial market M G . However, the proofs given in those papers are abstract and technical. In contrast, the proof of Proposition 6.1 is extremely simple and explicitly shows the trading strategy which realises the arbitrage profit. Furthermore, we have shown that not only NA and NFLVR but also the weaker NA 1 no-arbitrage-type condition fails to hold in the enlarged financial market M G . Remark 6.2. 1) As shown in the proof of Proposition 6.1, the strategy φ ∈ A G 0 satisfies φ = φ1 ] ]τ,+∞[ [ and ( φ • S) t > 0 for all t > τ . According to [DS95b, Definition 3.2], the strategy φ generates an immediate arbitrage opportunity at the G-stopping time τ . This intuitively means that one can realise an arbitrage profit immediately after the G-stopping time τ has occurred, i.e., on the time interval [τ, τ + ε], for every ε > 0. This possibility has been also pointed out in [START_REF] Zwierz | On existence of local martingale measures for insiders who can stop at honest times[END_REF].

2) Proposition 6.1 can be seen as a counterpart to part (2) of Theorem 4.1. Indeed, part (2) of Theorem 4.1 shows that one can realise an arbitrage opportunity at time τ by taking a position (up to τ ) in the strategy ϕ which replicates N , while Proposition 6.1 shows that one can realise an arbitrage of the first kind (as well as an immediate arbitrage opportunity) after time τ by taking a position in the strategy -ϕ. Note also that the admissibility constraint prevents the arbitrage of the first kind ξ = N τ -1 to be realised on the time horizon [0, τ ].

3) As considered in Example 1.1, let d = 1 and suppose that the real-valued process S = (S t ) t≥0 is given as the solution of the SDE (1.1) on (Ω, F, F W , P), where F W is the (P-augmented) natural filtration of W . In view of Example 4.2, the random time τ is an honest time which avoids every F-stopping time. Hence, Proposition 6.1 directly implies claim (c) after Example 1.1.

As shown in Proposition 6.1, an agent can realise an arbitrage of the first kind in the enlarged financial market M G by adopting a suitable trading strategy as soon as the honest time τ occurs. Motivated by this result, let us now consider what happens in the enlarged financial market M G if all market participants are only allowed to trade on the time horizon [ρ, +∞], where ρ is a G-stopping time with ρ > τ P-a.s. To this effect, let us recall the process ρ S = S -S ρ introduced at the end of Section 3.3 and define the ρ-shifted enlarged financial market ρ M G := Ω, F, G, P; ρ S, ρ A G , where ρ A G denotes the set of all elements in L G ( ρ S) which are admissible, in the sense of Definition 2.1. Proposition 6.3.

(1) For every G-stopping time ρ such that ρ > τ P-a.s., NA 1 holds in the ρ-shifted enlarged financial market ρ M G ; (2) there exists a constant δ > 0 such that the strategy φ := -1 ] ]τ,+∞[ [ ϕ/N * τ yields an arbitrage opportunity in the (τ + ε)-shifted enlarged financial market τ +ε M G , for every ε ∈ (0, δ), and, hence, NA and NFLVR fail to hold in the (τ + ε)-shifted enlarged financial market τ +ε M G , for every ε ∈ (0, δ).

In particular, Proposition 6.1 together with part (1) of the above proposition shows that the possibility of realising arbitrages of the first kind in the enlarged financial market M G crucially depends on the possibility of trading as soon as τ has occurred. Indeed, from time τ + ε onwards, the additional knowledge of the information of the filtration G can only allow for arbitrage opportunities, since NA 1 holds in the (τ + ε)-shifted enlarged financial market τ +ε M G . In other words, the potential loss of the viability (in the sense of [START_REF] Kardaras | Finitely additive probabilities and the fundamental theorem of asset pricing[END_REF]) of the enlarged financial market M G on the time horizon [0, +∞] is only due to the arbitrage profits that can be realised by trading as soon as the honest time τ occurs. Hence, preventing agents from trading on the time horizon (τ, τ + ε), for every ε > 0, will preserve the viability of the enlarged financial market M G , even though arbitrage opportunities may still exist. Note also that, in the special context considered in Example 1.1 (see also Example 4.2), Proposition 6.1 and Proposition 6.3 together imply claim (c) after Example 1.1.

We close this section by showing that the completeness of the restricted financial market M F (see Assumption 2.8) or, more specifically, the existence of a stochastic integral representation of the form N = 1 + ϕ • S, for some ϕ ∈ L F (S), is crucial for our results to hold (in particular, see the proofs of part (2) of Theorem 4.1, Proposition 6.1 and part (2) of Proposition 6.3), as we are now going to illustrate by means of an explicit counterexample.

Example 6.4. Let W 1 = (W 1 t ) t≥0 and W 2 = (W 2 t ) t≥0 be two independent Brownian motions and denote by F i = (F i t ) t≥0 the P-augmented natural filtration of W i , for i = 1, 2. Define F := F 1 ∨F 2 and let the discounted price process S = (S t ) t≥0 of a risky asset be given as the solution to the following SDE on the filtered probability space (Ω, F, F, P):

(6.1) dS t = S t f (W 1 t ) dW 2 t , S 0 = s ∈ (0, +∞) ,
where f : R → (0, +∞) is such that the above SDE admits a unique strong solution. Clearly, since S is an F-local martingale, NFLVR holds in the financial market M F (Assumption 2.6). Let τ be any Pa.s. finite honest time with respect to the filtration F 1 and denote by G 1 = (G 1 t ) t≥0 and G = (G t ) t≥0 the progressive enlargements of F 1 and F, respectively, with respect to τ . Since W 1 and W 2 are independent and τ is

F 1 ∞ -measurable, the Brownian motion W 2 is independent of G 1 ∞ = F 1 ∞ . Noting that G = G 1 ∨ F 2 ,
this implies that W 2 remains a Brownian motion in the filtration G and, hence, the process S given by (6.1) is also a G-local martingale. Due to Theorem 2.5, this implies that NA 1 , NA and NFLVR all hold in the enlarged financial market M G on the global time horizon [0, +∞].

In the context of the present example, it is easy to show that the replication arguments used in the proofs of part (2) of Theorem 4.1, Proposition 6.1 and part (2) of Proposition 6.3 break down. Indeed, if τ avoids all F 1 -stopping times, Lemma 2.9 applied to the filtration F 1 gives the existence of an F 1local martingale N = (N t ) t≥0 with N 0 = 1 and lim t→+∞ N t = 0 P-a.s. such that P(τ > t|F 1 t ) = N t /N * t for all t ≥ 0. By the predictable representation property in the Brownian filtration F 1 , there exists an F 1 -predictable process ϕ = (ϕ t ) t≥0 ∈ L 2 loc (W 1 ) such that N = 1 + ϕ • W 1 . Moreover, due to the independence of W 1 and W 2 , it is also easy to check that P(τ = σ) = 0 for every F-stopping time σ (Assumption 2.7) and Z t = N t /N * t , for all t ≥ 0, where N is also an F-local martingale. However, we cannot replicate the F-local martingale N by trading in the risky asset S, since the two processes N and S are driven by the independent Brownian motions W 1 and W 2 , respectively.

Conclusions

We want to conclude by commenting on the role of Assumptions 2.6-2.8 and discussing some possible extensions. In the present chapter, we aimed at understanding the impact of an honest time on the validity of suitable no-arbitrage-type conditions in the enlarged financial market M G and, hence, we assumed from the beginning that the restricted financial market M F is free from any kind of arbitrage, in the classical sense of NFLVR (Assumption 2.6). However, we want to point out that analogous results can be obtained if the restricted financial market M F satisfies NA 1 (or, equivalently, NUPBR) but the stronger NFLVR condition does not necessarily hold. In that case, Theorem 4.1 and Proposition 6.1 continue to hold, provided that Assumptions 2.7-2.8 are still satisfied. Indeed, due to Remark 2.2, if NFLVR fails to hold in the restricted financial market M F , then it fails in the enlarged financial market M G as well (and can be also shown to fail on the time horizon [0, τ ]). Moreover, by relying on part (1) of Theorem 2.5 together with Lemma 2.10 and Assumptions 2.7-2.8, one can show that NA 1 holds in the enlarged financial market M G on the time horizon [0, τ ] but fails on the global time horizon [0, +∞].

The assumption that the honest time τ avoids all F-stopping times (Assumption 2.7) seems to be crucial. Indeed, if NFLVR holds in the restricted financial market M F (Assumption 2.6) but Assumption 2.7 does not hold, then an honest time τ does not necessarily give rise to arbitrage opportunities in the enlarged financial market M G on the time horizon [0, τ ]. As an example, let τ be an honest time which avoids all F-stopping times and let σ be any F-stopping time such that Zσ > 0 P-a.s., where Z = ( Zt ) t≥0 is the Azéma supermartingale of τ . Then τ := τ ∧ σ is easily seen to be an honest time (which does not avoid F-stopping times) and, as can be deduced from Theorem 5.1, NFLVR still holds in the enlarged financial market M G on [0, τ ].

Observe that our results have been obtained under Assumption 2.8, which implies that, under any ELMM F Q, the (Q, F)-local martingale N appearing in the multiplicative decomposition of the Azéma Q-supermartingale of τ (see Lemma 2.9) can be written as N = 1 + ϕ • S. As can be easily checked (see in particular the proofs of part (2) of Theorem 4.1 and Proposition 6.1 and Example 6.4), only the latter condition is necessary and, hence, the assumption that all F-local martingales can be represented as stochastic integrals of S can be significantly relaxed, provided that all F-local martingales are continuous.

In the following chapter, we will study the validity of NA 1 under a progressive filtration enlargement with respect to a general random time τ for a general semimartingale model. However, the fact that in general F-semimartingales may fail to be G-semimartingales after time τ implies that a general no-arbitrage analysis can only be performed on the time interval [0, τ ].

Introduction

In financial mathematics, market models with different sets of information have been widely studied, especially in relation to insider trading and credit risk modeling, as discussed in the introduction to Chapter 5. Typically, one starts by postulating a model with respect to a given information set and then enlarges that set with some additional information not originally present in the market. From a mathematical point of view, this corresponds to considering an enlargement of the original filtration on a given filtered probability space. Since the model aims at representing a financial market, a fundamental question is whether the additional information allows for arbitrage profits.

In this chapter, we aim at answering the above question in the context of models driven by general semimartingales, both in the case where the additional information is added in a progressive way through time, and in the case where the additional information is fully added at the initial time. Referring to the terminology of the theory of enlargement of filtrations (see [START_REF] Jeulin | Semi-martingales et Grossissement d'une Filtration[END_REF] for a complete account of the theory and [JYC09, Section 5.9] and [Pro05, Ch. VI] for a presentation of the main results), this corresponds to considering a filtration obtained as a progressive or as an initial enlargement, respectively, of the original filtration. In particular, this extends the analysis performed in Chapter 5 in the case of a continuous semimartingale model for an honest time avoiding all stopping times of the original filtration.

Our analysis focuses on the No Arbitrage of the First Kind (NA 1 ) condition (see [START_REF] Kardaras | Finitely additive probabilities and the fundamental theorem of asset pricing[END_REF]), already introduced in Chapters 4 and 5. Mathematically, condition NA 1 is equivalent to existence of strictly positive local martingale deflators, and can be shown to be the minimal condition ensuring the wellposedness of expected utility maximisation problems (see [KK07, Proposition 4.19]). In the case of a progressive enlargement with respect to a random time τ , we study the stability of NA 1 on the random time horizon [0, τ ], showing that the existence of arbitrages of the first kind in the enlarged filtration is crucially linked to the possibility of the asset-price process exhibiting a jump at the same time when a particular nonnegative local martingale in the original filtration jumps to zero. In turn, we show that the possibility of the latter event is intimately related to how local martingales from the original filtration behave in the enlarged filtration, up to a suitable normalisation. In the case of an initial enlargement of the original filtration, and under the classical density hypothesis of [START_REF] Jacod | Grossissement initial, hypothèse (H'), et théorème de Girsanov[END_REF], we establish an analogous set of results.

In both cases of progressive and of initial enlargement, these results allow us to provide an easy sufficient condition ensuring the NA 1 stability for a fixed semimartingale model, as well as to explicitly characterise the stability of NA 1 for all semimartingale models. Our proofs (for which we refer the reader to the original paper [START_REF] Acciaio | Arbitrage of the first kind and filtration enlargements in semimartingale financial models[END_REF]) are based on a hands-on approach: using local martingale deflators in the original filtration, we explicitly construct local martingale deflators in the enlarged filtration in order to show validity of condition NA 1 . In the process, we obtain some interesting new results on progressive as well as initial filtration enlargement, showing how the super/local martingale property of a process can be transferred from the original filtration to the enlarged one.

For progressive filtration enlargement with respect to an honest time τ (see [Pro05, Ch. VI]), examples of arbitrage profits are provided in [START_REF] Imkeller | Random times at which insiders can have free lunches[END_REF], [START_REF] Zwierz | On existence of local martingale measures for insiders who can stop at honest times[END_REF] and [START_REF] Fontana | On arbitrages arising with honest times[END_REF]. In the context of continuous semimartingale models, as we have shown in Chapter 5 (see also [Kre17, Lemma 6.7]), condition NA 1 is always valid in the enlarged filtration on the random time horizon [0, τ ]. In the case of general semimartingale models, this is no longer true (see the example in Section 2.5.1). In that context, the recent paper [START_REF] Aksamit | Non-arbitrage up to random horizon for semimartingale models[END_REF] addresses the issue of NA 1 stability in progressively enlarged filtrations and represents one of the sources of inspiration for the present work. In particular, the key role of conditions equivalent to those given in Theorem 2.3 and Remark 2.4 has been first pointed out and proved in [START_REF] Aksamit | Non-arbitrage up to random horizon for semimartingale models[END_REF] (see Remark 2.5) and the characterisation we obtain in Theorem 2.6 turns out to be equivalent to the one already established in [START_REF] Aksamit | Non-arbitrage up to random horizon for semimartingale models[END_REF]. However, in comparison with the latter paper, we follow here a totally different approach and provide original and rather simple proofs to those results, avoiding the use of the compensated stochastic integral (see [HWY92, Definition 9.7]) and, somewhat surprisingly, not relying on the classical Jeulin-Yor decomposition formula (see [Jeu80, Proposition 4.16]). In contrast, we exploit the properties of an optional decomposition of the Azéma supermartingale associated to τ recently established in [START_REF] Kardaras | On the stochastic behaviour of optional processes up to random times[END_REF]. We also want to mention that, in the case of the classical No Free Lunch with Vanishing Risk (NFLVR) condition (see [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF][START_REF] Delbaen | The fundamental theorem of asset pricing for unbounded stochastic processes[END_REF]), a study of its stability and of the relation with the preservation of the martingale property in progressively enlarged filtrations has been carried out in [START_REF] Coculescu | Default times, no-arbitrage conditions and changes of probability measures[END_REF].

In the initial filtration enlargement case, the possibility of realising arbitrage profits in the enlarged filtration has been studied in [START_REF] Grorud | Insider trading in a continuous time market model[END_REF], [START_REF] Grorud | Asymmetrical information and incomplete markets[END_REF] and [START_REF] Imkeller | Free lunch and arbitrage possibilities in a financial market model with an insider[END_REF], among others. Concerning the classical NFLVR condition, it is well-known that it is stable under an initial enlargement with respect to a random variable L if the conditional law of L for all times is equivalent to the unconditional one (see e.g. [START_REF] Grorud | Insider trading in a continuous time market model[END_REF]). However, to the best of our knowledge, the issue of NA 1 stability with respect to an initial enlargement has never been studied so far. Interestingly, we show that both the progressive and the initial case can be treated by relying on the same methodological approach.

This chapter is organised as follows. Section 2 contains the framework and statements of our main results. In Section 3 we consider progressive enlargement of filtrations. We study the crucial stopping times that will be then used to pinpoint local martingales and to prove stability of the NA 1 condition in the enlarged filtrations. We refer the reader to [START_REF] Acciaio | Arbitrage of the first kind and filtration enlargements in semimartingale financial models[END_REF] for a detailed presentation of the initial enlargement case, which is similar in spirit to the progressive enlargement case.

Main results

2.1. Probabilistic set-up. In all that follows, we work on a filtered probability space (Ω, A, F, P), where F = (F t ) t∈R + is a filtration satisfying the usual hypotheses of right-continuity and saturation by P-null sets. In general, F ∞ ⊆ A holds, with the last set-inclusion being potentially strict.

We shall be using standard notation from the general theory of stochastic processes. For any unexplained notation and results, the reader can consult [START_REF] He | Semimartingale Theory and Stochastic Calculus[END_REF] or [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF].

2.2. The market model. Fix d ∈ N = {1, 2, . . .}, and let S ≡ (S i ) i∈{1,...,d} be a collection of nonnegative semimartingales on (Ω, F, P) 1 . Each S i , i ∈ {1, . . . , d}, models the price process of an asset, discounted by a baseline security in the market. Starting with initial capital x ∈ [0, ∞) and following a d-dimensional, F-predictable and S-integrable strategy H, an investor's discounted wealth process is given by X x,H := x + • 0 H t dS t . It should be noted that we are using vector 1 We want to mention that the nonnegativity assumption is not crucial for the following results to hold, provided that the notion of local martingale is suitably replaced by the notion of sigma-martingale (see [START_REF] Delbaen | The fundamental theorem of asset pricing for unbounded stochastic processes[END_REF] and [START_REF] Takaoka | A note on the condition of no unbounded profit with bounded risk[END_REF]). stochastic integration throughout. Define X (F, S) to be the class of all nonnegative processes X x,H in the previous notation. (In the definition of the class X (F, S), the initial capital x ∈ [0, ∞) and d-dimensional, F-predictable and S-integrable strategies H are arbitrary, as long as X x,H ≥ 0.) Definition 2.1. For T ∈ (0, ∞), an arbitrage of the first kind with information F and assets S on [0, T ] is a non-negative F T -measurable random variable χ T with P(χ T > 0) > 0 and with the property that for all x ∈ (0, ∞) there exists X ∈ X (F, S) with X 0 = x (where the wealth process X may depend on x) such that P(X T ≥ χ T ) = 1. If no arbitrage of the first kind with information F and assets S exists on any interval [0, T ] for T ∈ (0, ∞), we say that condition NA 1 (F, S) holds.

(F, S, Q) = Y (Y Q 0 /Y Q ) | Y ∈ Y(F, S, P) holds. As a consequence of [TS14, Theorem 2.6], condition NA 1 (F, S) is equivalent to Y(F, S, Q) = ∅
(where, of course, Q ∼ P is arbitrary). For our purposes, we need a more precise statement.

Theorem 2.2. Condition NA 1 (F, S) holds if and only if there exist Q ∼ P and strictly positive X ∈ X (F, S) such that (1/ X) ∈ Y(F, S, Q).

In the present chapter, we aim at studying the stability of the NA 1 condition when enlarging the filtration F in a progressive or initial way. The first issue to be settled is the preservation of the semimartingale property of processes, which is typically referred to in the literature as the Hhypothesis. In the case of progressive filtration enlargement by a random time τ , it comes as a consequence of the Jeulin-Yor theorem that this always holds up to time τ (and that for honest times it holds on all [0, ∞), as discussed in Chapter 5); see [START_REF] Jeulin | Grossissement d'une filtration et semi-martingales: formules explicites[END_REF]. For the case of initial filtration expansion, one well-known situation where the preservation of the semimartingale property holds is when Jacod's density hypothesis is satisfied; see [START_REF] Jacod | Grossissement initial, hypothèse (H'), et théorème de Girsanov[END_REF]. We want to remark that these facts will also come as consequences of our analysis.

Main results under progressive filtration enlargement.

We first study the stability of the NA 1 condition under a progressive enlargement of the filtration F with respect to an A-measurable random time τ : Ω → [0, ∞] such that P(τ = +∞) = 0. The progressively enlarged filtration G = (G t ) t∈R + is defined via

(2.1) G t = {B ∈ A | B ∩ {τ > t} = B t ∩ {τ > t} for some B t ∈ F t } , ∀t ∈ R + .
In particular, G is a right-continuous filtration that contains F and makes τ a stopping time, but note that G is not the smallest right-continuous filtration that contains F and makes τ a stopping time (compare with the discussion in [START_REF] Guo | Intensity process and compensator: a new filtration expansion approach and the Jeulin-Yor theorem[END_REF]). It comes as a consequence of the Jeulin-Yor theorem that S τ := (S τ ∧t ) t∈R + is a semimartingale on (Ω, G, P). Then, the class X (G, S τ ) can be defined exactly in the same way as the corresponding class X (F, S) of Section 2.2. The notation NA 1 (G, S τ ) used in the sequel of this chapter refers to absence of arbitrage of the first kind with information G and assets S τ .

A key role in the study of progressive enlargement of filtrations is played by the Azéma supermartingale associated with τ (given by the optional projection of I [ [0,τ [ [ on (Ω, F, P), see [START_REF] Jeulin | Semi-martingales et Grossissement d'une Filtration[END_REF] and references therein), that we denote by Z. This means that P(τ > σ | F σ ) = Z σ for all finite stopping times σ on (Ω, F), and note that Z ∞ := lim t→+∞ Z t = 0 holds in view of P(τ = +∞) = 0 (note that the limit Z ∞ always exists due to the supermartingale convergence theorem). Furthermore, if A denotes the dual optional projection of I [ [τ,+∞[ [ , it follows that µ := A + Z is a nonnegative uniformly integrable martingale on (Ω, F, P) with

µ t = E [A ∞ |F t ], for all t ≥ 0 (see [Nik06, Section 8.2]).
Moreover, by the general properties of the dual optional projection (see [START_REF] He | Semimartingale Theory and Stochastic Calculus[END_REF]Theorem 5.27]), for any stopping time σ on (Ω, F), it holds that ∆A σ = P(τ = σ | F σ ) on {σ < +∞}.

For all n ∈ N, let

ζ n := inf {t ∈ R + | Z t < 1/n}. Furthermore, set (2.2) ζ := lim n→∞ ζ n = inf {t ∈ R + | Z t-= 0 or Z t = 0} = inf {t ∈ R + | Z t = 0} ,
where the last equality holds from the fact that Z is a nonnegative supermartingale on (Ω, F, P). We now introduce a stopping time that will be of major importance in the this chapter. Consider the F ζ -measurable event Λ := {ζ < +∞, Z ζ-> 0, ∆A ζ = 0}, and define

(2.3) η := ζ Λ = ζI Λ + ∞I Ω\Λ .
Clearly, η is a stopping time on (Ω, F), and it satisfies P(η > τ ) = 1. Indeed, P(τ > η|F η ) = Z η = 0 and

P(τ = η < +∞|F η ) = ∆A η I {η<+∞} = ∆A ζ I Λ = 0 (remember that P(τ = +∞) = 0 by assumption).
In Section 2.5, it is shown that η may be totally inaccessible or accessible. However, Lemma 3.4 shows that P(η = σ < +∞ | F σ-) < 1 holds for all predictable times σ on (Ω, F).

The results below establish stability of condition NA 1 in the current setting of progressive filtration enlargement. Together with their counterparts for initially enlarged filtrations (Theorems 2.9 and 2.10), they are the main results reported in this chapter. The first result is concerned with stability of the NA 1 condition for a fixed semimartingale model.

Theorem 2.3. If NA 1 (F, S) holds and P(η < +∞, ∆S η = 0) = 0, then NA 1 (G, S τ ) holds.

Remark 2.4. The message of the above theorem is that, to ensure the preservation of NA 1 under progressive filtration enlargement, one only needs to check whether the price process jumps at time η. It is then clear that, if NA 1 (F, S) holds for S := S η-= S η -∆S η I [ [η,+∞[ [ , then NA 1 (G, S τ ) holds as well, since P(η > τ ) = 1. In order to have NA 1 (G, S τ ), it is sufficient that NA 1 (F, S ζn ) holds for all n ∈ N. Indeed, note that NA 1 (F, S ζn ) implies NA 1 (G, S τ ∧ζn ), and that the intervals [[0, τ ∧ ζ n ]] exhaust [[0, τ ]], since P(ζ ≥ τ ) = 1. The claim follows since the NA 1 condition can be given locally.

Remark 2.5. Define Z to be the optional projection of I [ [0,τ ] ] on (Ω, F, P) (see also [Jeu80, Section IV.1]); in other words, for any stopping time σ on (Ω, F), Z σ = P(τ ≥ σ | F σ ) holds on {σ < +∞}, so that Z = Z + ∆A. It is then straightforward to see that condition P(η < +∞, ∆S η = 0) = 0 is equivalent to evanescence of the set {Z -> 0, Z = 0, ∆S = 0}. Hence, Theorem 2.3 corresponds exactly to the result proved in [ACDJ14, Corollary 2.20, part (b)], by means of different techniques. Moreover, when S is a quasi-left-continuous semimartingale (see [JS03, Definition I.2.25]), [ACDJ14, Theorem 2.8] shows that the validity of NA 1 (F, S ζn ), for all n ∈ N, is actually necessary and sufficient for the preservation of the NA 1 property in G (see also [ACDJ14, Remark 2.9]).

Theorem 2.3 recovers the already-known fact that condition NA 1 is stable under progressive enlargement for all continuous semimartingales; see Chapter 5 as well as [START_REF] Kreher | Change of measure up to a random time: details[END_REF]. Moreover, it implies that the condition P(η < +∞) = 0 is sufficient to guarantee NA 1 stability for any collection of assetprice processes. In the next result we show that this condition is also necessary in order to have this general stability. In fact, for P(η < +∞) > 0, we provide an explicit example of arbitrage of the first kind, which further shows how condition P(η < +∞, ∆S η = 0) = 0 in Theorem 2.3 cannot be dropped; see also Section 2.5.1.

Theorem 2.6. The following statements hold true:

(1) If P(η < +∞) = 0, then for any S such that NA 1 (F, S) holds, NA 1 (G, S τ ) also holds.

(2) Suppose that P(η < +∞) > 0. Then, with D being the predictable compensator of I [ [η,+∞[ [ on (Ω, F, P), the nonnegative process S := E(-D) -1 I [ [0,η[ [ is a local martingale on (Ω, F, P), and S τ is nondecreasing with P(S τ > 1) > 0. In particular, condition NA 1 (F, S) holds but condition NA 1 (G, S τ ) fails.

Note that separability is a mild technical assumption that only allows us to use the results of [SY78, Proposition 4]. In Section 2.5.3 we will provide an example showing how condition P(η x < +∞, ∆S η x = 0) = 0, for γ-a.e. x ∈ E, cannot be dropped.

Theorem 2.9 has the following consequence: if P(η x < +∞) = 0 for γ-a.e. x ∈ E, condition NA 1 (F, S) implies condition NA 1 (G, S) for any asset-price process S. In order to formulate the counterpart to statement (2) of Theorem 2.6 (regarding stability of the NA 1 condition for all semimartingale models) in the case of initially enlarged filtrations, we have to slightly depart from our original setting. More precisely, the explicit example of an arbitrage of the first kind in the enlarged filtration when P(η x < +∞) > 0 will involve a potentially infinite collection of semimartingales. To wit, with D x denoting the predictable compensator of I [ [η x ,+∞[ [ on (Ω, F, P) for all x ∈ E, define the collection (S x ) x∈E via (2.6)

S x := E(-D x ) -1 I [ [0,η x [ [ , ∀x ∈ E.
Under separability assumption on the space L 1 (Ω, A, P), we can obtain a version of the function

E × Ω × R + (x, ω, t) → S x t (ω) which is B E ⊗ O(F)-measurable. The process S L defined via S L (ω, t) := S L(ω) t
(ω) for all (ω, t) ∈ Ω × R + is a semimartingale on (Ω, G, P), and has the following financial interpretation: an insider with knowledge of L and unit initial capital takes at time zero a position on a single unit of the stock with index L, and keeps it indefinitely. Although this strategy may involve an infinite number of assets, it is of the simplest possible buy-and-hold nature.

Theorem 2.10. Under Assumption 2.7, the following statements hold true:

(1) If P(η x < +∞) = 0 holds for γ-a.e x ∈ E, then for any S such that NA 1 (F, S) holds, NA 1 (G, S) also holds.

(2) Suppose that the space L 1 (Ω, A, P) is separable and that E P(η x < +∞)γ [dx] > 0. Then, the family (S x ) x∈E in (2.6) consists of local martingales on (Ω, F, P), and S L is nondecreasing with P(S L t = S L 0 , ∀t ∈ R + ) < 1. In particular, NA 1 (F, S x ) holds, for every x ∈ E, but NA 1 (G, S L ) fails.

Loosely speaking, in part (2) of Theorem 2.10, the insider identifies from the beginning a single asset in the family (S x ) x∈E which will not default and can therefore arbitrage.

Remark 2.11. It is interesting to observe that the necessary and sufficient conditions given in Theorem 2.6 and in Theorem 2.10 for the preservation of the NA 1 property under filtration enlargements bear resemblance to the necessary and sufficient condition obtained in [START_REF] Fontana | No-arbitrage conditions and absolutely continuous changes of measure[END_REF] for the preservation of the NA 1 property under absolutely continuous (but not necessarily equivalent) changes of measure. This similarity is not a coincidence, given the deep link existing between filtration enlargements and non-equivalent changes of measure, as shown in [START_REF] Yoeurp | Théorème de Girsanov généralisé et grossissement d'une filtration[END_REF].

Examples. The first two examples are in the progressive filtration enlargement framework.

In the first one, the stopping time η is totally inaccessible and assertion (2) of Theorem 2.6 is illustrated by explicit computations; the second example contains a set-up where η is accessible. The last example shows how condition P(η x < +∞, ∆S η x = 0) = 0, for γ-a.e. x ∈ E, cannot be dropped in Theorem 2.9.

2.5.1. An example under progressive filtration enlargement where η is totally inaccessible. Let (Ω, F, P) be a complete probability space supporting an A-measurable random variable ζ : Ω → R + such that P(ζ > t) = exp(-t) holds for all t ∈ R + . Set F = (F t ) t∈R + to be the smallest filtration that satisfies the usual hypotheses and makes ζ a stopping time. Define τ := ζ/2, and consider the filtration G obtained as the progressive enlargement of F with respect to τ .

Note that Z t = 0 holds on {ζ ≤ t}, while Z t = exp(-t) holds on {t < ζ}, the last fact following from τ = ζ/2 and the memoryless property of the exponential law. Therefore,

Z t = exp(-t)I {t<ζ} is true for all t ∈ R + . Similarly, ∆A σ = P(τ = σ | F σ ) = P(ζ = 2σ | F σ ) = 0 is
true for all bounded stopping times σ on (Ω, F), which implies that ∆A = 0. Note that ζ = inf {t ∈ R + | Z t-= 0 or Z t = 0} and Z ζ-= exp (-ζ) > 0. Since ∆A = 0, for η defined as in (2.3), we obtain that η = ζ. The predictable compensator of I [ [η,+∞[ [ on (Ω, F, P) is equal to D := (η ∧ t) t∈R + ; in particular, ζ = η is totally inaccessible on (Ω, F, P).

Here we have P(η < +∞) = 1, hence we can proceed to construct a local martingale S as in Theorem 2.6-(2). To wit, S := E(-D)

-1 I [ [0,η[ [ = exp(D)I [ [0,η[ [ , that is, S t = exp(t)I {t<ζ} for t ∈ R + .
Note that S is a quasi-left-continuous nonnegative martingale on (Ω, F, P), so that NA 1 (F, S) trivially holds. However, since S is strictly increasing up to τ , NA 1 (G, S τ ) fails.

2.5.2. An example under progressive filtration enlargement where η is accessible. Let (Ω, F, P) be a complete probability space that supports an A-measurable random variable ζ : Ω → N such that p k := P(ζ = k) ∈ (0, 1) holds for all k ∈ N, where ∞ k=1 p k = 1. Set F = (F t ) t∈R + to be the smallest filtration that satisfies the usual hypotheses and makes ζ a stopping time. Since ζ is N-valued, it is an accessible time on (Ω, F, P). Define τ := ζ -1, and consider the progressively enlarged filtration G. Let Z and A be defined as in Section 2.3.

Again, one may compute Z explicitly. In fact, Z t = 0 holds on {ζ ≤ t}; furthermore, upon defining q k = ∞ n=k+1 p n for all k ∈ {0, 1, . . .}, and denoting by • the integer part, we have

Z t = P(τ > t | F t ) = P(ζ > t + 1 | F t ) = P(ζ > t + 1 | F t ) = q t+1 q t , on {t < ζ} . Note that ζ = inf {t ∈ R + | Z t-= 0 or Z t = 0} and Z ζ-= q ζ /q ζ-1 > 0. Furthermore, ∆A ζ = P(τ = ζ | F ζ ) = 0 holds true.
It follows that η = ζ and η is accessible on (Ω, F, P). 2.5.3. An example under initial filtration enlargement. Let us consider a probability space (Ω, F, P) supporting a Poisson process N with intensity λ > 0 stopped at time T ∈ (0, +∞). Let F be the rightcontinuous filtration generated by N and consider the random variable L := N T . As in [GVV06, Section 4.2] (compare also with [GP01, Section 4.3]), it can be checked that

q x t = e -λt λ(T -t)
x-Nt (λT ) x

x! (x -N t )! I {Nt≤x} , for all t ∈ [0, T ), and q x T = e -λT x!/(λT ) x I {N T =x} , so that Jacod's criterion (Assumption 2.7) is satisfied. Consider then the process S defined by S t := exp N t -λt(e-1) , for all t ∈ [0, T ]. The process S is a strictly positive F-martingale (see e.g. [JYC09, Proposition 8.2.2.1]), so that NA 1 (F, S) holds. However, NA 1 (G, S) does not hold. To see this, define the G-stopping time σ := inf {t ∈ [0, T ] | N t = N T } and consider the strategy -I ] ]σ,T ] ] . Then, for all t ∈ [0, T ], we get

(-I ] ]σ,T ] ] • S) t = I {t>σ} exp N σ -λσ(e -1) 1 -exp -λ(t -σ)(e -1) .
In particular, the process -I ] ]σ,T ] ] •S is nondecreasing and P(σ < T ) = 1, thus implying that NA 1 (G, S) fails to hold. Indeed, in the context of the present example, the processes q x have a positive probability to jump to zero and this event occurs exactly in correspondence of the jump times of the Poisson process N , thus showing that the condition P(η x < +∞, ∆S η x = 0) = 0 for γ-a.e. x ∈ E fail to hold.

Arbitrage of the first kind in progressively enlarged filtrations

In this section, the proof of Theorem 2.3 and Theorem 2.6 will be given. In the process, we will also obtain certain interesting results concerning the behaviour (up to the random time τ ) of nonnegative super/local martingales on (Ω, F, P) in the enlarged filtration G (see Section 3.3). In particular, these results do not follow from classical results of enlargement of filtrations theory. Theorem 3.1. For any random time τ on (Ω, F) satisfying P(τ = +∞) = 0 there exists a pair of processes (K, L) with the following properties:

(1) K is F-adapted, right-continuous, nondecreasing, with 0 ≤ K ≤ 1.

(2) L is a nonnegative local martingale on (Ω, F, P), with L 0 = 1.

(3) For any nonnegative optional processes V on (Ω, F), we have

(3.1) E[V τ ] = E R + V t L t dK t .
(4) R + I {K t-=1} dL t = 0 and R + I {Lt=0} dK t = 0 hold P-a.s. It also comes as part of the results in [Kar15, Section 1.1] that Z = L(1 -K), which gives a particular multiplicative optional decomposition of Z. In general, there are many possible optional multiplicative decompositions; the properties described in Theorem 3.1 specify the pair (K, L) in a unique way. Note also that, in the special case where P(τ = σ) = 0 for every stopping time σ on (Ω, F, P), the decomposition Z = L(1 -K) coincides with the multiplicative Doob-Meyer decomposition of the supermartingale Z (see [Kar15, Remark 1.6]).

Remark 3.2. Let σ be a stopping time on (Ω, F). For any B ∈ F σ , (3.1) applied to the process

V = I B I ] ]σ,+∞[ [ , combined with Z = L(1 -K) and the definition of Z, implies that E [L σ (1 -K σ )I B ] = E [Z σ I B ] = E [V τ ] = E I B (σ,∞) L t dK t .
Since the above equality holds for all B ∈ F σ , it follows that

(3.2) L σ (1 -K σ ) = E (σ,∞) L t dK t F σ .
Remark 3.3. Another use of (3.1) gives

P(L τ = 0) = E R + I {Lt=0} L t dK t = 0.
Since L is a nonnegative local martingale on (Ω, F, P), it follows that [[0, τ ]] ⊆ {L > 0}.

3.2. Results regarding the stopping time η. Recall that η = ζI Λ + ∞I Ω\Λ , where Λ :

= {ζ < +∞, Z ζ-> 0, ∆A ζ = 0}. It holds that Λ = {ζ < +∞, K ζ-< 1, L ζ-> 0, ∆K ζ = 0}.
In the proof of the next result, it is established inter alia that η is not predictable, when finite. Lemma 3.4. Let D be the predictable compensator of I [ [η,+∞[ [ on (Ω, F, P). Then:

(1) ∆D < 1, P-a.s.; in particular, E(-D) is nonincreasing and strictly positive;

(2) the nonnegative process E(-D)

-1 I [ [0,η[ [ is a local martingale on (Ω, F, P).
We write Q ∼ P whenever Q is a probability that is equivalent to P on F. Note that all the quantities that we have defined and depend on τ (in particular, η) depend on the underlying probability measure. For establishing Theorem 2.3, it is important that η remains invariant under equivalent changes of probability. The next result ensures that this is indeed the case. Lemma 3.5. Let Q ∼ P, and let η Q be the stopping time on (Ω, F) defined under Q in analogy to η ≡ η P defined in (2.3) under P. Then η Q = η holds almost surely (under both P and Q).

3.3. Super/local martingales in the progressively enlarged filtration. The next result, which will be key in the development, is also of independent interest. Proposition 3.6. The following statements hold true:

(1) Let X be a nonnegative supermartingale on (Ω, F, P). Then, the process X τ /L τ is a supermartingale on (Ω, G, P).

(2) Let X be a nonnegative local martingale on (Ω, F, P) such that [[η, +∞[[ ⊆ {X = 0} holds (modulo evanescence). Then, the process X τ /L τ is a local martingale on (Ω, G, P).

Proposition 3.6 shows that, up to a normalisation with respect to 1/L τ , the supermartingale property can always be transferred from the original filtration F to the enlarged filtration G and provides a sufficient criterion for transforming F-local martingales into G-local martingales. This result will play a key role in proving Theorem 2.3.

In the rest of this section we provide an interesting side result which is intimately connected to Proposition 3.6. The first one provides a characterisation of the local martingale property of X τ /L τ on (Ω, G, P) for every nonnegative local martingale X on (Ω, F, P).

Proposition 3.7. The following statements are equivalent:

(1) For every nonnegative local martingale X on (Ω, F, P), the process X τ /L τ is a local martingale on (Ω, G, P).

(2) The process 1/L τ is a local martingale on (Ω, G, P).

(3) P(η < +∞) = 0.

3.4. Condition NA 1 in the progressively enlarged filtration. As a consequence of Proposition 3.6, a sufficient condition for NA 1 (G, S τ ) to hold is immediate. The proof of the following result is straightforward, hence omitted. The notation Y(G, S τ , P) is self-explanatory.

Proposition 3.8. Suppose that there exists a local martingale deflator M for S on (Ω,

F, P) such that {M > 0} = [[0, η[[. Then, M τ /L τ ∈ Y(G, S τ , P).
In particular, observe that Proposition 3.8 provides an explicit procedure for transforming a local martingale deflator for S on (Ω, F, P) into a local martingale deflator for S τ on (Ω, G, P). We are now ready to present the proofs of our results on NA 1 stability under progressive filtration enlargement.

Proof of Theorem 2.3. In view of Lemma 3.5 and Theorem 2.2, we may assume without loss of generality (replacing P with Q if necessary) the existence of a strictly positive X ∈ X (F, S) such that Y := (1/ X) ∈ Y(F, S, P). Since P(η < +∞, ∆S η = 0) = 0 holds, we obtain P(η < +∞, ∆Y η = 0) = 0; in particular, P(η < +∞, ∆(Y S) η = 0) = 0 holds. In the notation of Lemma 3.4, define

M := Y E(-D) -1 I [ [0,η[ [ . Note that M 0 = 1 and {M > 0} = [[0, η[[. By Lemma 3.4, it follows that M S i -E(-D) -1 I [ [0,η[ [ , Y S i
is a local martingale on (Ω, F, P) for all i ∈ {1, . . . , d}. Furthermore,

E(-D) -1 I [ [0,η[ [ , Y S i = E(-D) -1 , Y S i -E(-D) -1 I [ [η,+∞[ [ , Y S i = E(-D) -1 , Y S i , where E(-D) -1 I [ [η,+∞[ [ , Y S i = 0 follows from the fact that E(-D) -1 I [ [η,+∞[ [ = E(-D) -1 η I [ [η,+∞[ [ is a single-jump process, jumping at η. Since E(-D) -1 is predictable, it follows that E(-D) -1 I [ [0,η[ [ , Y S i = E(-D) -1 , Y S i = • 0 ∆E(-D) -1 t d Y S i t
is a local martingale on (Ω, F, P) for all i ∈ {1, . . . , d}. Therefore, M S i is a local martingale on (Ω, F, P) for all i ∈ {1, . . . , d}, and Theorem 2.3 follows from Proposition 3.8.

Proof of Theorem 2.6. Statement (1) follows directly from Theorem 2.3. For statement (2), let D be as in Lemma 3.4, and define S = E(-D) -1 I [ [0,η[ [ . Then S 0 = 1 and S is a nonincreasing process up to τ , thus S τ ≥ 1. Moreover, by Lemma 3.4, S is a local martingale on (Ω, F, P), hence NA 1 (F, S) holds. From (3.1) and Z = L(1 -K), and using integration by parts and the definition of D, we have

E[D τ ] = E ∞ 0 D t L t dK t = -E ∞ 0 D t dZ t = E ∞ 0 Z t-dD t = E ∞ 0 Z t-dI {η≤t} = E[Z η-I {η<+∞} ].

Introduction

In this chapter we address the following fundamental question: suppose that every F-local martingale can be written as a stochastic integral of a given F-local martingale S = (S t ) t≥0 , does there exist a G-local martingale S G = (S G t ) t≥0 (and, if yes, how is it related to S) such that every G-local martingale can be written as a stochastic integral of S G ? In other words, is it possible to transfer the strong predictable representation property from the original filtration F to the initially enlarged filtration G? Assuming the validity of Jacod's density hypothesis, already introduced in Chapter 6, we shall give an answer to this question in full generality.

Given the fundamental importance of martingale representation results, the above question has already been studied in several papers. In particular, martingale representation theorems in initially enlarged filtrations have been established first by [START_REF] Grorud | Insider trading in a continuous time market model[END_REF] in a Brownian setting and then by [Ame00, ABS03, CJZ13] in more general settings. However, to the best of our knowledge, the existing martingale representation results always assume a stronger version of Jacod's density hypothesis, namely the equivalence between the F t -conditional law and the unconditional law of L, for all t ∈ R + . In contrast, we shall only assume an absolute continuity relation, as in the original paper [START_REF] Jacod | Grossissement initial, hypothèse (H'), et théorème de Girsanov[END_REF]. This is a seemingly slight generalization of the existing literature, but on the contrary it requires a different approach to the martingale representation property. Moreover, it allows to study several interesting examples which are not covered by the existing results and are of special importance for the modeling of insider information (see, e.g., the examples given in Chapter 6). If Jacod's density hypothesis holds as an equivalence between the conditional and the unconditional laws of L, as previously assumed in the literature, then the key tool is represented by an equivalent probability measure which makes the random variable L independent of the original filtration F and under which every F-martingale is also a G-martingale. The idea of such a measure, called martingale preserving probability measure in [START_REF] Amendinger | A monetary value for initial information in portfolio optimization[END_REF], goes back to early works on enlargement of filtrations and also appears in [START_REF] Föllmer | Anticipation cancelled by a Girsanov transformation: a paradox on Wiener space[END_REF]. Together with Girsanov's theorem, this measure permits to easily move between F and G, thus allowing to transfer the predictable representation property from F onto G. In contrast, if Jacod's density hypothesis is only assumed to hold in the absolutely continuous form of [START_REF] Jacod | Grossissement initial, hypothèse (H'), et théorème de Girsanov[END_REF], then a martingale preserving probability measure may not exist (at least on the original probability space) and one has to rely on a different methodology.

Let us briefly describe our approach to a general martingale representation in initially enlarged filtrations, assuming the validity of Jacod's density hypothesis as stated in [START_REF] Jacod | Grossissement initial, hypothèse (H'), et théorème de Girsanov[END_REF] and the existence of an F-local martingale having the martingale representation property in F. First, as a preliminary step, we shall study the general structure of the initially enlarged filtration G, establishing its rightcontinuity and a useful characterization of G-martingales in terms of parameterized families of Fmartingales. As a second step, we obtain a representation result which holds simultaneously for all the elements of a parameterized family of F-martingales. Finally, by relying on the results of [START_REF] Stricker | Calcul stochastique dépendant d'un paramètre[END_REF] on stochastic integration depending on a parameter, we go back to the enlarged filtration G in order to obtain the desired martingale representations. In this last step, a crucial ingredient is represented by the G-optional decomposition of F-local martingales recently established in [START_REF] Aksamit | On an optional semimartingale decomposition and the existence of a deflator in an enlarged filtration[END_REF] together with the results of [START_REF] Acciaio | Arbitrage of the first kind and filtration enlargements in semimartingale financial models[END_REF] on the behavior of F-local martingales in initially enlarged filtrations.

Our approach to establish the martingale representation property in initially enlarged filtrations crucially exploits Jacod's density hypothesis. Even though a martingale preserving probability measure does not necessarily exist on the original probability space, we show that there exists a process playing a similar role and providing a precise link between F-martingales and G-martingales. This latter fact is also related to the local solution method developed in [START_REF] Song | Grossissement d'une filtration et problèmes connexes[END_REF], based on the insight of constructing locally a martingale preserving probability measure on an auxiliary probability space. Since Jacod's density hypothesis can be embedded in the local solution method (see [Son15, Section 6.1]), that method could also provide a strategy, alternative to the one described in the present chapter, to prove the martingale representation property in initially enlarged filtrations. We also mention that, when F is progressively enlarged with a non-negative random variable satisfying Jacod's density hypothesis and under the additional assumption that all F-martingale are continuous, a martingale representation result has been obtained in [START_REF] Jeanblanc | Immersion property and credit risk modelling[END_REF].

The results presented in the present chapter are naturally related to the ones reported in Chapter 6 in the initial enlargement case. In particular, from the mathematical point of view, a key role is played by the possibility that the densities of the F t -conditional law of L jump to zero.

This chapter is structured as follows. Section 2 presents the probabilistic framework, the statements of the main results and two examples. In particular, we give several alternative martingale representation results depending on whether and how the conditional densities of the random variable L are allowed to reach zero (see Subsection 2.4). Section 3 presents a simple application to the hedging of contingent claims under insider information. For simplicity, in this chapter we will only present the main martingale representation results established in [START_REF] Fontana | The strong predictable representation property in initially enlarged filtrations under the density hypothesis[END_REF], referring the reader to the original paper for a detailed study of the structure of martingales in initially enlarged filtrations.

Setting and main results

2.1. Notation and preliminaries. In this chapter we shall be working on several stochastic bases. Hence, we introduce the following notation for a generic probability space (Ω , A , P ) endowed with a filtration F = (F t ) t≥0 satisfying the usual conditions of right-continuity and P -completeness.

• M(P , F ) (M loc (P , F ), resp.) denotes the set of all martingales (local martingales, resp.) on (Ω , F , P ); • if X = (X t ) t≥0 is an R d -valued process in M loc (P , F ), we denote by L m (X; P , F ) the set of all R d -valued F -predictable processes which are integrable with respect to X under the measure P in the sense of local martingales; • if X = (X t ) t≥0 is an R d -valued F -semimartingale, we denote by L(X; P , F ) the set of all R d -valued F -predictable processes which are integrable with respect to X under the measure P in the sense of semimartingales.

Adopting the notation of [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF], we denote by (H • X) t := (0,t] H u dX u the stochastic integral of H with respect to X, for all t ∈ R + , with (H • X) 0 = 0. We denote by O(F ) and by P(F ) the F -optional and the F -predictable σ-fields, respectively, on Ω × R + .

Let us recall the notion of strong predictable representation property (see [START_REF] Jacod | Calcul Stochastique et Problèmes de Martingales[END_REF] Chapter IV] as well as [HWY92, Chapter 13] for the one-dimensional case), here formulated with respect to an R d -valued local martingale X = (X t ) t≥0 on a generic filtered probability space (Ω , A , F , P ).

Definition 2.1. A local martingale X = (X t ) t≥0 is said to have the strong predictable representation property on (Ω , F , P ) if

M loc (P , F ) = ζ + ϕ • X : ζ ∈ L 0 (F 0 ) and ϕ ∈ L m (X; P , F ) ,
with L 0 (F 0 ) denoting the set of all F 0 -measurable random variables.

In other words, a local martingale X has the strong predictable representation property on (Ω , F , P ) if and only if every local martingale on that space null at zero can be written as a stochastic integral with respect to X. In this case, the local martingale X is also said to have the (strong) martingale representation property on (Ω , F , P ).

Setting.

As the first main ingredient of our framework, we consider a probability space (Ω, A, P) endowed with a filtration F = (F t ) t≥0 satisfying the usual conditions of right-continuity and P-completeness. We do not necessarily assume that the initial σ-field F 0 is trivial. We let S = (S t ) t≥0 be a given R d -valued local martingale on (Ω, F, P).

As the second main ingredient of our framework, we consider an A-measurable random variable L taking values in a Lusin space (E, B E ), where B E denotes the Borel σ-field of E. Let γ : B E → [0, 1] be the (unconditional) law of L, so that γ(B) = P(L ∈ B) holds for all B ∈ B E . We then enlarge the filtration F by adding the information of the random variable L to the initial σ-field F 0 , i.e., we consider the filtration G = (G t ) t≥0 given by the right-continuous augmentation of the filtration G 0 = (G 0 t ) t≥0 defined as G 0 t := F t ∨ σ(L), for all t ≥ 0.

2.3. The conditional densities of L. For all t ∈ R + , let γ t : Ω × B E → [0, 1] be a regular version of the F t -conditional law of L (which always exists since the space (E, B E ) is Lusin). The following assumption, which has already appear in Chapter 6, will play a central role in our analysis.

Assumption 2.2. For all t ∈ R + , γ t γ holds in the P-a.s. sense.

As shown in [Jac85, Proposition 1.5] (see also [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF]Theorem VI.11]), Assumption 2.2 holds if and only if, for all t ∈ R + , there exists a positive σ-finite measure γ t on (E, B E ) such that γ t γ t holds in the P-a.s. sense. Lemma 2.8 in Chapter 6 gives the existence of a good version of the conditional densities q x . Similarly as in Chapter 6, for every x ∈ E and n ∈ N, we define the following F-stopping times:

ζ x n := inf{t ∈ R + | q x t < 1/n} and ζ x := inf{t ∈ R + | q x t = 0}.
As in Chapter 6, for every x ∈ E, we consider the F ζ x -measurable event Λ x := {ζ x < ∞, q x ζ x -> 0} and define

η x := ζ x Λ x = ζ x 1 Λ x + ∞1 Ω\Λ x
, which is an F-stopping time and represents the time at which q x jumps to zero.

It is a fundamental result of [Jac85] that, under Assumption 2.2, the canonical decomposition in G of an arbitrary F-local martingale can be written in terms of the conditional densities (evaluated at x = L). However, a different type of decomposition of F-local martingales in G turns out to be better suited to our analysis. The following proposition has been recently established in [ACJ15, Theorem 5] and provides an optional decomposition (as opposed to the canonical decomposition).

Proposition 2.3. Suppose that Assumption 2.2 holds and that the space L 1 (Ω, A, P) is separable. Then the process S G = (S G t ) t≥0 defined as

(2.1) S G := S - 1 q L • [S, q L ] + ∆S η x 1 [ [η x ,∞[ [ p,F x=L is a local martingale on (Ω, G, P), with (∆S η x 1 [ [η x ,∞[ [ ) p,F denoting a B E -measurable version of the dual F-predictable projection of the process ∆S η x 1 [ [η x ,∞[ [ .
Remark 2.4. The last term appearing in decomposition (2.1) is due to the possibility that the densities q x jump to zero simultaneously to a jump of S. As we have seen in Chapter 6, this is precisely the condition which leads to a failure of the NA 1 condition in the enlarged filtration G. The separability assumption in Proposition 2.3 is only needed to ensure the existence of a version of the dual F-predicable projection of ∆S

η x 1 [ [η x ,∞[ [ which is measurable in x, see [SY78, Proposition 4].
2.4. The strong predictable representation property in G. This section contains the statements of the main results. We start with the following theorem, which represents the central result.

Theorem 2.5. Suppose that Assumption 2.2 holds and that the space L 1 (Ω, A, P) is separable. If S = (S t ) t≥0 has the strong predictable representation property on (Ω, F, P), then the process S G = (S G t ) t≥0 defined in (2.1) has the strong predictable representation property on (Ω, G, P).

In view of Definition 2.1, Theorem 2.5 shows that, if S has the strong predictable representation property on (Ω, F, P), then every local martingale M = (M t ) t≥0 on (Ω, G, P) admits the stochastic integral representation (2.2)

M t = M 0 + (ϕ • S G ) t , P-a.s. for all t ∈ R + ,
where (ϕ t ) t≥0 is a G-predictable process admitting a rather explicit characterization. As mentioned in the introduction, Theorem 2.5 generalizes the martingale representation results previously obtained in the literature on initially enlarged filtrations. In particular, no assumption is made on the family {q x : x ∈ E} of conditional densities of L, apart from its existence.

In the remaining part of this subsection, we present some alternative martingale representation results under additional assumptions on the conditional densities of L. We start with the following corollary which shows that, if the F-martingale q x can only reach zero continuously (and not due to a jump), for γ-a.e. x ∈ E, then the strong predictable representation property of S on (Ω, F, P) can be easily transferred to (Ω, G, P) up to a suitable "change of numéraire".

Corollary 2.6. Suppose that Assumption 2.2 holds and that P(η x < ∞) = 0 for γ-a.e. x ∈ E. If the process S = (S t ) t≥0 has the strong predictable representation property on (Ω, F, P), then the R d+1valued process (1/q L t , S t /q L t ) t≥0 is a G-local martingale and has the strong predictable representation property on (Ω, G, P).

Remark 2.7. Theorem 2.5 and Corollary 2.6 show that, as long as P(η x < ∞) = 0 for γ-a.e.

x ∈ E, the martingale representation property in G can be expressed in terms of both S G and (1/q L , S/q L ). The representation result of Theorem 2.5 is obviously more general and holds with respect to a d-dimensional process (i.e., of the same dimension of the original process S). The representation result of Corollary 2.6 is less general and requires a (d + 1)-dimensional process. However, the process (1/q L , S/q L ) admits an important interpretation, especially in the context of financial modeling. Indeed, under the assumptions of Corollary, the process q L represents the numéraire portfolio for S in the enlarged filtration G (see [START_REF] Karatzas | The numeraire portfolio in semimartingale financial models[END_REF]). In this sense, Corollary 2.6 shows that the martingale representation property can be transferred from F onto G by changing the numéraire, choosing the numéraire portfolio in G as the baseline asset. Note also that, in view of applications, the process (1/q L , S/q L ) can be immediately deduced from a model's fundamental ingredients and does not require any computation, unlike the process S G appearing in Theorem 2.5.

As mentioned in the introduction, the existing martingale representation results in initially enlarged filtrations have been obtained under the stronger assumption that γ t ∼ γ (see [START_REF] Grorud | Insider trading in a continuous time market model[END_REF]Theorem 4.3], [Ame00, Theorem 4.2], [ABS03, Theorem 3.2] and [CJZ13, Proposition 5.3]). This case corresponds to the following proposition, which can be easily deduced from our general approach.

Proposition 2.8. Suppose that γ t ∼ γ holds P-a.s. for all t ∈ [0, T ], for some fixed T < ∞. If S = (S t ) t∈[0,T ] has the strong predictable representation property on (Ω, F, P), then d P := (q L 0 /q L T )dP defines a probability measure P ∼ P such that S ∈ M loc ( P, G) and S has the strong predictable representation property on (Ω, G, P).

We close this subsection with a last martingale representation result, under the same assumptions of Corollary 2.6. As a preliminary, we recall that, in view of [Jac85, Theorem 2.5], the process S, q x F x=L is well-defined1 and the process S -1 q L -• S, q x F x=L is a local martingale on (Ω, G, P). The following result has been established in [CJZ13, Proposition 5.5] under the stronger assumption that γ t ∼ γ P-a.s. for all t ∈ R + .

Corollary 2.9. Suppose that Assumption 2.2 holds and that P(η x < ∞) = 0 for γ-a.e. x ∈ E. If S = (S t ) t≥0 has the strong predictable representation property on (Ω, F, P), then the process SG := S -1 q L -• S, q x F x=L has the strong predictable representation property on (Ω, G, P). We can remark that, while Corollary 2.6 relates the martingale representation property to a change of numéraire (see Remark 2.7), Corollary 2.9 relates the martingale representation property to a locally equivalent change of measure.

Two examples.

We now present two simple examples of processes having the strong predictable representation property in the initially enlarged filtration G. For simplicity, we consider a fixed time horizon T < ∞. In both examples, Assumption 2.2 is satisfied but the equivalence γ t ∼ γ does not hold. Hence, the following martingale representation results are not covered by the existing literature. We first present an example where the conditional densities can reach zero due to a jump and then an example where the conditional densities can only reach zero continuously (i.e., P(η x < ∞) = 0 for γ-a.e. x ∈ E). In a financial context, with S representing the prices of d risky assets (see Section 3), these two examples admit interesting interpretations in relation to the modeling of insider information and have been considered in [START_REF] Acciaio | Arbitrage of the first kind and filtration enlargements in semimartingale financial models[END_REF] and [START_REF] Chau | Market models with optimal arbitrage[END_REF], respectively.

2.5.1. An example based on the Poisson process. Let N = (N t ) t∈[0,T ] be a standard Poisson process and F = (F t ) t∈[0,T ] its P-augmented natural filtration, denoting by {τ n } n∈N the jump times of N . We let A = F T and consider the random variable L = N T together with the corresponding initially enlarged filtration G. Similarly as in [AFK16, Example 1.5.3] (see also [START_REF] Gasbarra | Enlargement of filtration and additional information in pricing models: Bayesian approach[END_REF]§4.3]), it can be easily checked that, for every n ∈ N,

q n t = P(L = n|F t ) P(L = n) = e t (T -t) n-Nt T n n! (n -N t )!
1 {Nt≤n} , for all t < T, and q n T = e T T -n n!1 {N T =n} , thus showing that Assumption 2.2 is satisfied. Observe also that the conditional density (q n t ) t∈[0,T ] jumps to zero at τ n+1 (if τ n+1 ≤ T ), meaning that η n = τ n+1 . The compensated Poisson process S := (N t -t) t∈[0,T ] has the strong predictable representation property on (Ω, F, P) (see e.g. [JYC09, Proposition 8.3.5.1]). Hence, the assumptions of Theorem 2.5 are satisfied and the process S G = (S G t ) t∈[0,T ] defined in (2.1) has the strong predictable representation property on (Ω, G, P). Moreover, S G can be explicitly represented as follows:

(2.3)

S G t = (N t -t) - N T n=1 1 {τn≤t} 1 - T -τ n N T -n + 1 + (t -τ N T ∧ t).
Indeed, noting that [S,

q L ] = 0<u≤• ∆N u ∆q L u = N T n=1 ∆q L τn 1 {τn≤•} and that ∆q L τn = e τn (T -τ n ) N T -n T N T N T ! (N T -n)! 1 - T -τ n N T -n + 1 , it holds that 1 q L • [S, q L ] = N T n=1 1 {τn≤•} 1 - T -τ n N T -n + 1 . Moreover, observe that ∆S η n 1 [ [η n ,T ] ] = 1 [ [τ n+1 ,T ] ]
, for all n ∈ N. Hence, in view of Proposition 2.3, in order to prove (2.3) it remains to compute the dual F-predictable projection of the increasing process (1 {τ n+1 ≤t} ) t∈[0,T ] . Observe that {τ n+1 ≤ t} = {N t ≥ n + 1}, so that 1 {τ n+1 ≤t} = f n (N t ), with f n (x) := 1 {x≥n+1} . By [JYC09, Proposition 8.2.3.1], the infinitesimal generator L of N is given by L(f )(•) = f (• + 1) -f (•), for any bounded measurable function f . Hence, the process

f n (N t ) - t 0 L(f n )(N u )du = 1 {τ n+1 ≤t} - t 0 1 {n≤Nu<n+1} du = 1 {τ n+1 ≤t} -(τ n+1 ∧ t -τ n ∧ t)
is a local martingale on (Ω, F, P), for each n ∈ N. Note that, according to the analysis presented in Chapter 6, if we interpret the process S as a price process, then there exist arbitrages of the first kind in the initially enlarged filtration G.

2.5.2. An example based on the Brownian motion. As in [CT15, Section 5.1], let define the process S = (S t ) t∈[0,T ] by S := 1 + W , where W = (W t ) t∈[0,T ] is a standard Brownian motion and F its P-augmented natural filtration. We let A = F T and consider the discrete random variable L = 1 {inf t∈[0,T ] St>0} . Since L is a discrete random variable, Assumption 2.2 holds (see e.g. [Pro05, Corollary 2 to Theorem VI.10]) and, following [START_REF] Chau | Market models with optimal arbitrage[END_REF], we can compute

q 1 t = P(inf u∈[0,T ] S u > 0|F t ) P(inf u∈[0,T ] S u > 0) = 1 + 1 P(inf u∈[0,T ] S u > 0) 2 π σ∧t 0 1 √ T -s e - S 2 s 2(T -s) dW s , q 0 t = P(inf u∈[0,T ] S t ≤ 0|F t ) P(inf u∈[0,T ] S u ≤ 0) = 1 - 1 P(inf u∈[0,T ] S u ≤ 0) 2 π σ∧t 0 1 √ T -s e - S 2 s 2(T -s) dW s ,
where σ := inf{t ∈ R + | S t = 0}. Observe that q 1 and q 0 can reach zero in a continuous way with a strictly positive probability. Since W has the strong predictable representation property on (Ω, F, P), the assumptions of Corollary 2.9 are satisfied. Noting that q L t = q 1 t 1 {σ>T } + q 0 t 1 {σ≤T } , the strong predictable representation property on (Ω, G, P) holds with respect to the process

SG t = S t - t 0 1 q L s d S, q x F s x=L = S t - 1 {inf u∈[0,T ] Su>0} P(inf u∈[0,T ] S u > 0) 2 π t 0 1 q 1 s √ T -s e - S 2 s 2(T -s) ds + 1 {inf u∈[0,T ] Su≤0} P(inf u∈[0,T ] S u ≤ 0) 2 π σ∧t 0 1 q 0 s √ T -s e - S 2 s 2(T -s) ds.
As a consequence of the results reported in Chapter 6, the present example does not allow for arbitrages of the first kind, even though it admits classical arbitrage opportunities.

Hedging under insider information

In this section, we study the implications of the martingale representation property in the context of an abstract financial market with insider information. We consider a fixed time horizon T < ∞ and an R d -valued local martingale S = (S t ) t∈[0,T ] having the strong predictable representation property on (Ω, F, P). In the present section, we also assume that the initial σ-field F 0 is trivial. As in Section 2.2, we consider a random variable L together with the associated initially enlarged filtration G and suppose that Assumption 2.2 is satisfied, so that the family {q x : x ∈ E} is well-defined and

A := G T = F T ∨ σ(L).
The interpretation of the above setting is as follows. The process S represents the prices (discounted with respect to some baseline security) of d risky assets traded in the market and the filtration F represents the information publicly available to every market participant. On the contrary, the random variable L represents an additional information (insider information) which is only available to some better informed agents, having access to the information flow G. The better informed agents are allowed to trade on the same set of securities as the uninformed agents but can rely on their private information when choosing their strategies. For H ∈ {F, G}, we denote by H(H) the set of admissible strategies based on the information flow H, i.e., H(H) := H ∈ L(S; P, H) : (H • S) t ≥ -a P-a.s. for all t ∈ [0, T ], for some a ∈ R + , which amounts to exclude trading strategies requiring an unlimited line of credit.

We interpret any A-measurable non-negative random variable ξ as a contingent claim. The hedging problem, with respect to a filtration H ∈ {F, G}, consists in finding a strategy H ∈ H(H) such that ξ = v H (ξ) + (H • S) T holds P-a.s., for some initial wealth v H (ξ). If this is possible, then the strategy H is said to replicate the contingent claim ξ and v H (ξ) represents the initial cost of replicating ξ having access to the information flow H. If every bounded contingent claim can be replicated, then the financial market is said to be complete.

Since S has the strong predictable representation property on (Ω, F, P), every contingent claim ξ ∈ L 1 + (F T ) can be replicated. Indeed, it suffices to consider the non-negative F-martingale M = (M t ) t∈[0,T ] defined by M t := E[ξ|F t ], for all t ∈ [0, T ]. Then, in view of Definition 2.1, there exists an

R d -valued process H ∈ L m (S; P, F) such that ξ = M T = M 0 + (H • S) T P-a.s. Moreover, it holds that (H • S) t = M t -M 0 ≥ -E[ξ], thus showing that H ∈ H(F). The initial cost of replicating ξ is given by v F (ξ) = M 0 = E[ξ]
. This solves the hedging problem of an F T -measurable contingent claim from the perspective of an uninformed agent having access to the information flow F. In general, there does not exist an F-predictable hedging strategy for a G T -measurable contingent claim.

The following proposition shows that Assumption 2.2 together with the completeness of the financial market based on (Ω, F, P) suffices to ensure that the financial market based on (Ω, G, P) is also complete (up to a σ(L)-measurable initial wealth).

Proposition 3.1. Suppose that Assumption 2.2 holds and that S = (S t ) t∈[0,T ] has the strong predictable representation property on (Ω, F, P). Let ξ be a bounded non-negative G T -measurable random variable. Then there exists a strategy H ∈ H(G) which replicates ξ with initial cost v

G (ξ) = E[ξ/q L T |σ(L)].
In particular, the above proposition can be applied to an F T -measurable contingent claim ξ. In this case, both for the uninformed agent and for the informed agent there exists a hedging strategy. However, since the two agents have access to different information flows, the hedging strategy is not necessarily the same and the initial cost of replicating ξ will depend on the available information. This is the content of the following corollary, which shows that the better informed agent can always take replicate any F T -measurable contingent claim at a lower cost.

Corollary 3.2. Suppose that Assumption 2.2 holds and that S = (S t ) t∈[0,T ] has the strong predictable representation property on (Ω, F, P). Then, for every bounded non-negative F T -measurable random variable ξ, it holds that v G (ξ) ≤ v F (ξ).

In the context of the above corollary, for a given contingent claim ξ, the difference between the two values v F (ξ) and v G (ξ) can be regarded as a monetary value of the additional information contained in the random variable L when replicating ξ. integrability properties of the processes involved. Since such integrability conditions are not satisfied in general, we need to resort to a localization procedure, as explained in Section 3.

In the present chapter, we report the main contributions of [START_REF] Fontana | Market viability and martingale measures under partial information[END_REF], which can be summarized as follows:

(1) we show that the financial market under partial information is locally viable, in the sense that a portfolio optimization problem admits a solution up to a stopping time, if and only if there exists a partial information equivalent martingale measure (PIEMM) up to a stopping time. Furthermore, the density of such PIEMM is given by the (normalised) marginal utility of the optimal terminal wealth, thus recovering the classical result of financial economics;

(2) we prove that, if the financial market under partial information is globally viable, in the sense that it is locally viable for a sequence {τ n } n∈N of increasing stopping times, then there exists a partial information local martingale deflator (PILMD). Furthermore, we show that such PILMD can be constructed by aggregating the densities of all local PIEMMs if and only if the locally optimal portfolios satisfy a consistency condition; (3) as a special case, if the price process has bounded coefficients, we prove that the financial market is viable on the global time horizon if and only if the (normalised) marginal utility of the optimal terminal wealth defines a PIEMM on the global time horizon; (4) by means of a simple and classical example (see Section 5) we show that, even for regular utility functions and continuous-path processes with good integrability properties but unbounded coefficients, a PIEMM may fail to exist globally but, nevertheless, a PILMD exists.

To the best of our knowledge, the issue of linking the viability of the financial market to the existence of weaker counterparts of equivalent martingale measures such as PIEMMs and PILMDs has never been dealt with in the partial information case. Furthermore, we go beyond a pure existence result, since our approach allows us to obtain a precise and explicit connection between the solution to an optimal portfolio problem and the density of an equivalent martingale measure / local martingale deflator. In that regard, our results contribute to the literature dealing with utility maximization problems under partial information, see [START_REF] Björk | Optimal investment under partial information[END_REF][START_REF] Lakner | Utility maximization with partial information[END_REF][START_REF] Lakner | Optimal trading strategy for an investor: the case of partial information[END_REF][START_REF] Pham | Optimal portfolio in partially observed stochastic volatility models[END_REF][START_REF] Stettner | Asymptotics of utility from terminal wealth for partially observed portfolios[END_REF].

Some of our results are reminiscent of the dual approach to portfolio optimization, developed in the general semimartingale setting in [START_REF] Kramkov | The asymptotic elasticity of utility functions and optimal investment in incomplete markets[END_REF][START_REF] Schachermayer | Optimal investment in incomplete markets when wealth may become negative[END_REF], where optimal portfolios (solutions to the primal problem) are linked to supermartingale deflators (solutions to the dual problem). In the general setting of [START_REF] Kramkov | The asymptotic elasticity of utility functions and optimal investment in incomplete markets[END_REF][START_REF] Schachermayer | Optimal investment in incomplete markets when wealth may become negative[END_REF], it is shown that the marginal utility of optimal terminal wealth defines a martingale measure with respect to a price system which uses as numéraire the optimal wealth process itself. In contrast, in the present chapter we focus on characterizing the validity of the marginal utility measure for the original price system, without the need of changing the numéraire.

This chapter is also closely related to the recent strand of literature that deals with financial models going beyond the traditional setting based on EMMs, relaxing the NFLVR requirement, along the lines of the previous chapters. One of the first studies in this direction is [START_REF] Loewenstein | Local martingales, arbitrage, and viability[END_REF], where the authors are concerned with the viability of a complete Itô-process model. In particular, they show that the financial market can be viable, in the sense that portfolio optimization problems can be meaningfully solved, even if the NFLVR condition does not necessarily hold (in a related context, see also [START_REF] Fontana | Diffusion-based models for financial markets without martingale measures[END_REF] for an analysis of pricing and hedging problems in the absence of EMMs). In a general semimartingale framework, Proposition 4.19 of [START_REF] Karatzas | The numeraire portfolio in semimartingale financial models[END_REF] shows that the minimal no-arbitrage requirement in order to solve expected utility maximization problems amounts to the no unbounded profit with bounded risk (NUPBR) condition, the latter being weaker than NFLVR. Moreover, it has been recently proven in [START_REF] Choulli | How non-arbitrage, viability and numéraire portfolio are related[END_REF] that the NUPBR condition is equivalent to the local solvability of portfolio optimization problems in a general semimartingale setting. by the utility function U , we define the family A U F of admissible strategies as follows:

(2.2) A U F := all P F -measurable processes φ = (φ t ) t∈[0,T ] s.t. X φ ∈ S 2 and E U X φ T 2 < +∞ , where φ t represents the number of units of the risky asset held in the portfolio at time t, for all t ∈ [0, T ], with associated wealth process X φ = (X φ t ) t∈[0,T ] , and where S 2 denotes the family of all G-semimartingales Y = (Y t ) t∈[0,T ] satisfying E[sup t∈[0,T ] |Y t | 2 ] < +∞. The requirement of Fpredictability amounts to ensuring that agents trade by relying only on the partial information at their disposal, while the square-integrability requirement is an indispensable integrability condition in order to solve utility maximization problems via BSDE techniques.

As usual, we assume that trading is done in a self-financing way, so that the wealth process associated to a given strategy φ ∈ A U F starting from an initial endowment x ∈ R is given by

(2.3)      dX φ t = φ t dS t = φ t b t dt + σ t dB t + R γ(t, ζ) N (dt, dζ) , t ∈ [0, T ], X φ 0 = x.
We want to emphasize that we do not impose any no-arbitrage a-priori restriction on the financial market model. In the rest of this chapter, the no-arbitrage properties of the model will be inferred as consequences of the (local) solvability of a portfolio optimization problem.

Local market viability under partial information

In the present section we show the equivalence between the concept of local market viability, introduced below in Definition 3.2, and the local existence of a partial information equivalent martingale measure (PIEMM; see Definition 3.6) such that its density is expressed in terms of the marginal utility of terminal wealth. To this effect, as a first step (Section 3.1), we shall characterize the solutions to portfolio optimization problems by applying suitable versions of the maximum principles developed in [START_REF] Baghery | A maximum principle for stochastic control with partial information[END_REF] for stochastic control problems under partial information. As a second step (Section 3.2), we provide a characterization of the density processes of PIEMMs. From a technical point of view, the need of a local analysis arises from the integrability properties required in the above two steps. Hence, in order to have good integrability properties, we shall rely on the local boundedness assumption (see part (1) of Assumption 2.1; under stronger assumptions, a direct global result will be proved in Section 4.1). Until the end of Section 3, we fix an element n ∈ N and let τ n be the corresponding F-stopping time from the sequence {τ n } n∈N introduced after Assumption 2.1. We denote by b n the stopped process b n := (b n t∧τn ) t∈[0,T ] , with an analogous notation for σ n and γ n .

Problem 3.1 (Partial information locally optimal portfolio problem). For a fixed n ∈ N, for a given utility function U and an initial endowment x ∈ R, find an element φ

* ,n ∈ A U F (n) such that sup φ∈A U F (n) E U X φ T ∧τn = E U X φ * ,n T ∧τn < +∞, where A U F (n) := all P F -measurable processes φ = (φ t ) t∈[0,T ] s.t. φ1 [ [0,τn] ] ∈ A U F .
Definition 3.2 (Local market viability). Let U be a utility function. The financial market is said to be locally viable up to τ n if Problem 3.1 admits an optimal solution φ * ,n ∈ A U F (n).

3.1.

A BSDE characterization of locally optimal portfolios. As a first step, we provide a characterization of the locally optimal portfolio which solves Problem 3.1 in terms of the solution to a backward stochastic differential equation (BSDE), by relying on necessary and sufficient maximum principles for stochastic control under partial information, see [START_REF] Baghery | A maximum principle for stochastic control with partial information[END_REF][START_REF] Øksendal | Forward-backward stochastic differential games and stochastic control under model uncertainty[END_REF].

We define the Hamiltonian H

n : Ω × [0, T ] × R 3 × R → R as follows: (3.1) H n ω, t, φ, p, q, r(•) := φ b n (ω, t) p + φ σ n (ω, t) q + φ R r(ζ) γ n (ω, t, ζ)ν(dζ),
where R is defined as the class of functions r : R \ {0} → R such that the integral in (3.1) converges.

To the Hamiltonian H n we associate a BSDE for the adjoint processes p n = (p n t ) t∈[0,T ] , q n = (q n t ) t∈[0,T ] and for the function

r n : Ω × [0, T ] × R → R as follows, for any φ ∈ A U F (n): (3.2)      dp n t = q n t dB t + R r n (t, ζ) N (dt, dζ), t ∈ [0, T ],
p n T = U X φ T ∧τn . In order to study the BSDE (3.2), we need to introduce the following classes of processes:

M 2 := all G-martingales M = (M t ) t∈[0,T ] s.t. E sup t∈[0,T ] |M t | 2 < +∞ , L 2 (B) := all P G -measurable processes q = (q t ) t∈[0,T ] s.t. E T 0 q 2 t dt < +∞ , G 2 ( N ) := all P G ⊗ B(R)-measurable functions r s.t. E T 0 R r 2 (t, ζ)ν(dζ)dt < +∞ . Lemma 3.3. For any fixed n ∈ N and φ ∈ A U F (n), the BSDE (3.2) admits a unique solution (p n , q n , r n ) ∈ M 2 × L 2 (B) × G 2 ( N ). Furthermore, the G-martingale p n satisfies E[ T 0 (p n t ) 2 dt] < +∞. Proposition 3.4. For any fixed n ∈ N, an element φ ∈ A U F (n) solves Problem 3.1 if and only if the solution (p n , q n , r n ) ∈ M 2 ×L 2 (B)×G 2 ( N ) to the corresponding BSDE (3.2) satisfies the following condition P-a.s. for a.a. t ∈ [0, T ∧ τ n ]: (3.3) E ∂H ∂φ t, φ t , p n t , q n t , r n t F t = E b n t p n t + σ n t q n t + R γ n (t, ζ) r n (t, ζ) ν(dζ) F t = 0 .
Condition (3.3) also admits an alternative formulation, in terms of the generalised Malliavin derivatives of the marginal utility U . To this effect, recall the generalised Clark-Ocone theorem (see [START_REF] Aase | White noise generalizations of the Clark-Haussmann-Ocone theorem, with application to mathematical finance[END_REF] for the Brownian motion case and [DNØP09, Theorem 3.28] for the Lévy process case) which states that if a random variable F ∈ L 2 is G T -measurable, then it can be written as

F = E[F ] + T 0 E[D t F |G t ] dB t + T 0 R E[D t,ζ F |G t ] N (dt, dζ),
where D t and D t,ζ denote the generalised Malliavin derivatives at t with respect to B and at (t, ζ) with respect to N , respectively. Applying this to F := U (X φ T ∧τn ) we see that the solution (p n , q n , r n ) to the BSDE (3.2) can be represented as follows, for all t ∈ [0, T ] and ζ ∈ R:

p n t = E U X φ T ∧τn |G t , q n t = E D t U X φ T ∧τn |G t , r n (t, ζ) = E D t,ζ U X φ
T ∧τn |G t . Therefore, in view of Proposition 3.4, we get the following characterization of the optimal terminal wealth X φ * ,n

T ∧τn of the partial information locally optimal portfolio problem.

Corollary 3.5. For any fixed n ∈ N, an element φ ∈ A U F (n) solves Problem 3.1 if and only if the corresponding terminal wealth X φ T ∧τn satisfies the following partial information Malliavin differential equation P-a.s. for a.a.

t ∈ [0, T ∧ τ n ]: E b n t U X φ T ∧τn + σ n t D t U X φ T ∧τn + R γ n (t, ζ) D t,ζ U X φ T ∧τn ν(dζ) F t = 0.
3.2. Partial information equivalent martingale measures (PIEMMs). We now proceed to characterize the density processes of all partial information equivalent martingale measures (PIEMMs), defined below in Definition 3.6. As a preliminary, let us consider a generic probability measure Q ∼ P on (Ω, G T ) and denote by G = (G t ) t∈[0,T ] its density process, i.e., G t := dQ| G t dP| G t for all t ∈ [0, T ], which is a strictly positive G-martingale with E[G T ] = 1. Due to the martingale representation property in the filtration G, there exists a P G -measurable process θ 0 = (θ 0 t ) t∈[0,T ] with T 0 (θ 0 t ) 2 dt < +∞ P-a.s. and a P G ⊗ B(R)-measurable function θ

1 : Ω × [0, T ] × R → (-1, +∞) with T 0 R θ 1 (t, ζ) 2 ν(
dζ)dt < +∞ P-a.s. such that the following holds:

(3.4)      dG t = G t-θ 0 t dB t + R θ 1 (t, ζ) N (dt, dζ) , t ∈ [0, T ], G 0 = 1.
For all t ∈ [0, T ], the SDE (3.4) admits as explicit solution (3.5)

G t = exp t 0 θ 0 s dB s - 1 2 t 0 (θ 0 s ) 2 ds + t 0 R log 1 + θ 1 (s, ζ) N (ds, dζ) + t 0 R log 1 + θ 1 (s, ζ) -θ 1 (s, ζ) ν(dζ) ds .
In the following, we write G θ t := G t , for θ := (θ 0 , θ 1 ), where G t is represented by θ as above. We let Θ denote the family of all P G -measurable processes θ = (θ 0 , θ 1 ) such that the SDE (3.4) has a unique strictly positive martingale solution G θ = (G θ t ) t∈[0,T ] . Similarly, for θ ∈ Θ, we denote by Q θ the measure on (Ω, G T ) defined by dQ θ /dP := G θ T and by E Q θ [•] the corresponding expectation.

Definition 3.6. For n ∈ N, a probability measure Q θ ∼ P on (Ω, G T ) is said to be a partial information equivalent martingale measure (PIEMM) up to τ n if the process S τn is a (Q θ , F)-martingale.

In the next proposition we characterize the density processes of all PIEMMs.

Proposition 3.7. For any fixed n ∈ N, a probability measure Q θ ∼ P on (Ω, G T ) such that (θ 0 , θ 1 ) ∈ L 2 (B) × G 2 ( N ) is a PIEMM up to τ n if and only if the following condition holds:

(3.6) E Q θ b n t + σ n t θ 0 t∧τn + R γ n (t, ζ) θ 1 (t ∧ τ n , ζ)ν(dζ) F t = 0 P-a.s. for a.a. t ∈ [0, T ∧ τ n ].
3.3. Local market viability and PIEMMs. We now combine the results of Sections 3.1-3.2 to obtain our first main result.

Theorem 3.8. For any fixed n ∈ N, the following are equivalent:

(1) the financial market is locally viable up to τ n (in the sense of Definition 3.2) and φ * ∈ A U F (n) solves the partial information locally optimal portfolio problem;

(2) for φ

* ∈ A U F (n), the measure Q φ * ,n ∼ P defined on (Ω, G T ) by (3.7) dQ φ * ,n dP := U X φ * T ∧τn E U X φ * T ∧τn
is a PIEMM up to τ n , in the sense of Definition 3.6.

As we have already mentioned, we did not introduce any no-arbitrage a-priori restriction on the model. The result of Theorem 3.8 can then be interpreted in the following sense: as soon as the financial market is locally viable, in the sense that a portfolio optimization problem admits locally a solution, then there locally exists a partial information equivalent martingale measure. This means that the absence of arbitrage opportunities comes as a direct consequence of local market viability.

Global market viability under partial information

In this section, we adopt a global perspective and aim at characterizing the global viability of the financial market on the full investment horizon [0, T ].

4.1. The case of bounded coefficients. This subsection aims at proving, under a rather strong assumption, the equivalence between the existence of a PIEMM and market viability in a global sense. In the spirit of Definition 3.2, we adopt the following definition of global market viability. Definition 4.1. Let U be a utility function. The financial market is said to be globally viable if there exists an element φ * ∈ A U F such that (4.1) sup

φ∈A U F E U X φ T = E U X φ * T < +∞.
In general, it turns out that the equivalence between the global viability of the financial market and the existence of a PIEMM with density given by the (normalised) marginal utility of the optimal terminal wealth does not hold, as shown by an explicit counterexample in Section 5. However, we can still obtain a direct and global version of Theorem 3.8 if the following assumption is satisfied. Assumption 4.2.

(1) The P G -measurable processes b, σ as well as the P G ⊗ B(R)-measurable function γ are P-a.s. uniformly bounded and the measure ν is finite; (2) the utility function U satisfies the following condition: A useful consequence of part (1) of Assumption 4.2 is that the price process S admits finite moments of every order, as shown in the next lemma. Following the same approach of Section 3.1, we can characterize the solution to the portfolio optimization problem (4.1) via the solution (p, q, r) ∈ M 2 × L 2 (B) × G 2 ( N ) to the associated BSDE As in Section 3.2, the density process G = (G t ) t∈[0,T ] of a probability measure Q ∼ P on (Ω, G T ) admits a representation of the form (3.5), for some P G -measurable process θ 0 and for some P G ⊗B(R)measurable function θ 1 : Ω × [0, T ] × R → (-1, +∞). Let us introduce the following definition, which is a natural extension of Definition 3.6 to the full time interval [0, T ]. Definition 4.5. A probability measure Q θ ∼ P on (Ω, G T ) is said to be a partial information equivalent martingale measure (PIEMM) if the process S is a (Q θ , F)-martingale.

Density processes G θ of PIEMMs can be characterized as follows, similarly to Proposition 3.7. where θ 0 (θ 1 , resp.) is the process (predictable function, resp.) appearing in representation (3.5).

As in Section 3.3, we can now combine Propositions 4.4 and 4.6 in order to obtain the equivalence between global viability (in the sense of Definition 4.1) and the existence of a PIEMM.

Theorem 4.7. Suppose that Assumption 4.2 holds. Then the following are equivalent:

(1) the financial market is globally viable, in the sense of Definition 4.1, and φ * ∈ A U F solves the partial information optimal portfolio problem (4.1);

(2) for φ * ∈ A U F , the measure Q φ * ∼ P on (Ω, G T ) defined by

dQ φ * dP := U X φ * T E U X φ *
T is a PIEMM, in the sense of Definition 3.7.

4.2. The general case. In the present section, we study the issue of global viability in the more general case where b, σ and γ are only F-locally bounded, as in Section 2, without assuming that Assumption 4.2 holds. In this case, as will be shown by the example given in Section 5, we cannot characterize global market viability in terms of PIEMMs and we need to rely on the localization approach described in Section 3, adopting the following definition of global market viability. Definition 4.8 (Global market viability). Let U be a utility function. The financial market is said to be globally viable if, for all n ∈ N, Problem 3.1 admits an optimal solution φ * n ∈ A U F (n).

In general, one cannot hope to obtain a full characterization of global market viability in terms of (partial information) equivalent martingale measures defined on the full interval [0, T ] (not even in terms of partial information equivalent local martingale measures), as will be shown in Section 5. Hence, we need to formulate the following notion, which corresponds to a weaker counterpart to the density process of a PIEMM and extends to the partial information setting the concept of local martingale deflator, already introduced in the previous chapters of this thesis. Definition 4.9. A strictly positive F-local martingale Z = (Z t ) t∈[0,T ] with Z 0 = 1 is said to be a partial information local martingale deflator (PILMD) if ZS is an F-local martingale.

As shown in [START_REF] Kardaras | Market viability via absence of arbitrage of the first kind[END_REF], the existence of a local martingale deflator is equivalent to the no arbitrage of the first kind (NA1) condition. In particular, the NA1 condition can be shown to be strictly weaker than the classical no free lunch with vanishing risk (NFLVR) condition of [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF], the latter being equivalent (in the case of locally bounded processes) to the existence of an equivalent local martingale measure (in Section 5 we will give an example of a globally viable market that does not satisfy NFLVR).

Theorem 4.10. If the financial market is globally viable, in the sense of Definition 4.8, then there exists a PILMD.

Note that the no-arbitrage property of the model (in terms of NA1) comes as a direct consequence of the (global) viability of the financial market. However, the mere existence of a PILMD is not sufficient to ensure the global viability of the financial market. Nevertheless, it turns out that, if the financial market is globally viable, in the sense of Definition 4.8, and if the family of optimal strategies {φ * n } n∈N satisfies the following consistency condition, then global viability is equivalent to the existence of a PILMD which aggregates the density processes of all local PIEMMs. Definition 4.11. If the financial market is globally viable, in the sense of Definition 4.8, we say that the family of optimal strategies {φ * n } n∈N is consistent if the following hold:

(4.4) E U X φ * n T ∧τn |F T ∧τ n-1 E U X φ * n T ∧τn = U X φ * n-1 T ∧τ n-1 E U X φ * n-1 T ∧τ n-1
, for all n ∈ N.

Remark 4.12. As can be readily checked, a general sufficient condition for the validity of (4.4) is given by φ * n-1 = φ * n 1 [ [0,τ n-1 ] ] together with the martingale property of U (X φ * n •∧τn ), for every n ∈ N. In particular, this holds in the following two cases: (a) for a logarithmic utility function in a wide class of jump-diffusion models, for which the reciprocal of the log-optimal portfolio is a local martingale (see, e.g., [START_REF] Korn | The numeraire portfolio in financial markets modeled by a multi-dimensional jump diffusion process[END_REF] as well as the example in Section 5); (b) for an exponential utility function when trading strategies are restricted to a compact set, in a wide class of jump-diffusion models for which the measure ν is finite, as in [START_REF] Morlais | Utility maximization in a jump market model[END_REF] (see also [START_REF] Hu | Utility maximization in incomplete markets[END_REF] for related results in the continuous case).

By relying on Theorem 3.8, we can now formulate the announced equivalence result, which characterizes global market viability, with a consistent family of optimal strategies, under partial information.

Theorem 4.13. The following are equivalent:

(1) the financial market is globally viable, in the sense of Definition 4.8, with a consistent family {φ * n } n∈N of optimal strategies; (2) there exists a family of strategies {φ * n } n∈N , with φ * n ∈ A U F (n), for every n ∈ N, such that the process Z = (Z t ) t∈[0,T ] defined by (4.5)

Z t := 1 {t=0} + +∞ k=1 1 {τ k-1 <t≤τ k } E U X φ * k T ∧τ k |F t E U X φ * k T ∧τ k , for all t ∈ [0, T ], is a PILMD satisfying Z(T ∧ τ n ) = U X φ * n T ∧τn /E U X φ * n
T ∧τn , for all n ∈ N, with τ 0 := 0.

An example

Three-dimensional Bessel processes have been extensively studied in relation with the existence of arbitrage opportunities, see [START_REF] Delbaen | Arbitrage possibilities in Bessel processes and their relations to local martingales[END_REF], [FR13, Section 2] and [KK07, Example 4.6]. We will show that, for a logarithmic utility function, the financial market is viable in the local as well as in the global sense, even though the model allows for arbitrage opportunities.

Let (Ω, G, G, P) be a given filtered probability space, with a standard Brownian motion B and where F := G B ⊆ G is the P-augmented filtration generated by B. We define the (discounted) price process S of a single risky asset as the solution to the following SDE:

(5.1)

   dS t = 1 S t dt + dB t , t ∈ [0, T ]; S 0 = 1.
The solution to the SDE (5.1) is known as the three-dimensional Bessel process (see [RY99, Chapter XI]). There exists a sequence {τ n } n∈N of F-stopping times with τ n +∞ P-a.s. as n → +∞ such that S τn and 1/S τn are P-a.s. uniformly bounded, for every n ∈ N. A simple application of Itô's formula gives that dS -1 t = -S -2 t dB t , thus showing that the stopped process 1/S τn is an F-martingale, for all n ∈ N. Furthermore, for any φ ∈ A U F (n) and n ∈ N:

(5. thus showing that the stopped process (X φ ) τn /S τn is an F-martingale, for every n ∈ N.

Let us consider the logarithmic utility function U (x) = log(x), with an initial endowment of x = 1. Jensen's inequality together with the martingale property of (X φ ) τn /S τn gives that E log X φ T ∧τn /S T ∧τn ≤ log E X φ T ∧τn /S T ∧τn = 0, meaning that E[log(X φ T ∧τn )] ≤ E[log(S T ∧τn )], for any φ ∈ A U F (n). This shows that the optimal strategy is a simple buy-and-hold position in the risky asset itself, i.e., φ * n = 1 ∈ A U F (n) for all n ∈ N. According to Definition 3.2, the financial market is locally viable up to τ n , for every n ∈ N.

We can also verify the local viability of the financial market by applying Theorem 3.8. Indeed, since the stopped process 1/S τn is a strictly positive F-martingale, we can define a probability measure Q n on (Ω, G T ) by letting dQ n /dP := 1/S T ∧τn . Due to Bayes' rule, it is easy to check that Q n is a PIEMM up to τ n , in the sense of Definition 3.6. Theorem 3.8 implies that the financial market is viable up to τ n and that φ * n = 1 ∈ A U F (n) solves Problem 3.1, for every n ∈ N. In this example, Assumption 4.2 fails to hold and we cannot rely on the approach presented in Section 4.1. More precisely, we can prove that the process 1/S cannot be used as the density process of a PIEMM on [0, T ], since 1/S is a strict local martingale, according to the terminology of [ELY99], being a local martingale which fails to be a true martingale, so that E[1/S T ] < 1. Let us explain with some more details this phenomenon. We define the measure Q φ * as follows:

dQ φ * dP := U X φ * T E U X φ * T = 1/S T E 1/S T .
If Q φ * were a PIEMM, then its density process G = (G t ) t∈[0,T ] , with dQ φ * | Ft := G t dP| Ft for all t ∈ [0, T ], would be an F-martingale admitting the following representation, as in (3.5):

G t = exp t 0 θ 0 s dB s - 1 2 t 0 (θ 0 s ) 2 ds , for all t ∈ [0, T ],
for some F-predictable process θ 0 = (θ 0 t ) t∈[0,T ] with T 0 (θ 0 t ) 2 dt < +∞ P-a.s., so that:

( If Q φ * were a PIEMM, then the product GS would be an F-(local) martingale and equation (5.3) would then imply that θ 0 t = -1/S t for a.a. t ∈ [0, T ], meaning that dG t = -G t /S t dB t . But, since G 0 = 1/S 0 = 1, this would in turn imply that G and 1/S solve the same SDE and, hence, one would conclude that G = 1/S, thus contradicting the martingale property of G. This shows that, in the context of the present example, the marginal utility of the optimal terminal wealth cannot be taken as the density of a PIEMM. The failure of the martingale property of 1/S is also linked to the existence of multiple solutions to the BSDE (3.2) beyond the class M 2 × L 2 (B), as discussed in [START_REF] Xing | On backward stochastic differential equations and strict local martingales[END_REF].

We conclude the discussion of this example by showing that the financial market is globally viable with a consistent family of optimal strategies, in the sense of Definitions 4.8 and 4.11. Indeed, we already know that φ * := 1 = φ * n ∈ A U F (n) solves Problem 3.1, for every n ∈ N. Moreover, the consistency condition (4.4) also holds, due to the martingale property of the stopped process 1/S τn (see Remark 4.12). Global viability can also be proven by applying Theorem 4.13. Indeed, take φ = 1 ∈ n∈N A U F (n) and consider the process Z φ defined in (4.5). Since U (x) = 1/x, for every n ∈ N, it is immediate to check that Z φ = 1/S:

Z φ t = 1 {t=0} + +∞ k=1 1 {τ k-1 <t≤τ k } E 1/S T ∧τ k |F t E 1/S T ∧τ k = 1 {t=0} + +∞ k=1 1 {τ k-1 <t≤τ k } 1 S t∧τ k = 1 S t .
Equation (5.2) together with the martingale property of 1/S τn , for all n ∈ N, shows that 1/S is a PILMD and Theorem 4.13 implies then that the financial market is globally viable with a consistent family of optimal strategies.

• Lecture "Financial mathematics", ISIFAR master program, Paris Diderot University.

• Lecture "Econometrics", third year of the BSc program in applied mathematics, Paris Diderot University.

Academic year 2016/2017

•

  , M. Jeanblanc, Q. Li, Information, no-arbitrage and completeness for asset price models with a random change point, Stochastic Processes and their Applications, 124(9): 3009-3030, 2014. (2) C. Fontana, M. Jeanblanc, S. Song, On arbitrages arising with honest times, Finance and Stochastics, 18(3): 515-543, 2014. (3) B. Acciaio, C. Fontana, C. Kardaras, Arbitrage of the first kind and filtration enlargements in semimartingale financial models, Stochastic Processes and their Applications, 126(6): 1761-1784, 2016. (4) C. Fontana, The strong predictable representation property in initially enlarged filtrations under the density hypothesis, Stochastic Processes and their Applications, 128(3): 1007-1033, 2018. (5) C. Fontana, B. Øksendal, A. Sulem, Viability and martingale measures under partial information, Methodology and Computing in Applied Probability, 17(1): 15-39, 2015.

  (a) an informed agent can always realise arbitrage opportunities exactly at time τ , i.e., on the time interval [0, τ ] (see Theorem 4.1); (b) it is never possible to realise arbitrage opportunities strictly before time τ , i.e., on the time interval [0, ], for any G-stopping time with < τ P-a.s. (see Corollary 5.2). Furthermore, in the case of Example 1.1, an informed agent can never realise arbitrage opportunities on the time interval [0, τ ∧ T ], for every T ∈ (0, +∞) (see Theorem 5.1); (c) an informed agent can always realise arbitrage profits which are stronger than arbitrage opportunities (namely, arbitrages of the first kind, see Definition 2.3) by taking a position as soon as time τ has occurred (see Proposition 6.1) as well as arbitrage opportunities after time τ (see Proposition 6.3).

  Proposition 3.1 shows that there always exists at least one local martingale deflator in G on the time horizon [0, τ ], given by the reciprocal of the F-local martingale N appearing in the multiplicative decomposition of the Azéma supermartingale Z associated to the random time τ (see Lemma 2.9).The next lemma describes the general structure of all local martingale deflators in G on the time horizon [0, τ ]. It shows that all local martingale deflators in G are strict G-local martingales in the sense of[START_REF] Delbaen | A simple counterexample to several problems in the theory of asset pricing[END_REF], being G-local martingales which fail to be uniformly integrable G-martingales. Lemma 3.2. Let L = (L t ) t≥0 be a local martingale deflator in G on the time horizon [0, τ ]. Then L admits the following representation:L = E(R) N τwhere R = (R t ) t≥0 is a G-local martingale with R 0 = 0, purely discontinuous on [[0, τ ]] and with {∆R = 0} ⊆ [[τ ]] and ∆R τ > -1 P-a.s. Furthermore, all local martingale deflators in G on the time horizon [0, τ ] fail to be uniformly integrable G-martingales.3.2. Local martingale deflators in G on the time horizon[0, σ ∧ τ ].Let σ be an arbitrary (possibly infinite-valued) F-stopping time. In this section, aiming at characterising the validity of NFLVR on [0, σ ∧ τ ] (see Section 5), we study the martingale property of local martingale deflators in G on the time horizon [0, σ∧τ ]. For any F-stopping time σ, every local martingale deflator L = (L t ) t≥0 in G on the time horizon [0, σ ∧ τ ] admits the following representation when stopped at σ ∧ τ :(3.2)

  +∞ and 1 + k τ > 0 P-a.s. and η is a non-negative G τ -measurable random variable with E[η|G τ -] = 0. Lemma 3.3. Let σ be an F-stopping time and L = (L t ) t≥0 a local martingale deflator in G on the time horizon [0, σ ∧ τ ]. Then the following holds:

3. 3 .

 3 Local martingale deflators in G on the time horizon [0, +∞]. Let us now consider the question of whether there exists a local martingale deflator in G on the global time horizon [0, +∞].

  2) the strategy φ := 1 [ [0,τ ] ] ϕ yields an arbitrage opportunity in M G and, hence, NA fails to hold in M G on the time horizon [0, τ ]; (3) NFLVR fails to hold in M G on the time horizon [0, τ ].

Corollary 5. 2 .

 2 Let be a G-stopping time with < τ P-a.s. Then NFLVR holds in the enlarged financial market M G on the time horizon [0, ].Example 5.3. Due to Theorem 5.1, if σ is an F-stopping time such that P(σ ≥ ν) > 0, then there exist arbitrage opportunities in the enlarged financial market M G on the time horizon [0, σ ∧ τ ]. Let us illustrate this fact by means of a simple example, in the same setting of Example 1.1. Suppose that S 0 = 1 and let us define the F-stopping time τ * := inf{t ≥ 0 : S t = 1/2}, which is P-a.s. finite, and the honest time τ := sup{t ∈ [0, τ * ] : S t = S * t }. Let us also introduce the F-stopping time σ := inf{t ≥ 0 : S t = 3/2}. It can be checked that τ avoids all F-stopping times and we have ν = τ * P-a.s. and P(σ > ν) = P(σ > τ * ) > 0. Hence, due to Theorem 5.1, NFLVR fails to hold in the enlarged financial market M G on [0, σ ∧ τ ]. Indeed, the buy-and-hold strategy 1 [ [0,σ∧τ ] ] provides an arbitrage opportunity on the time horizon [0, σ ∧ τ ], since S σ∧τ -S 0 ≥ 0 P-a.s. and P(S σ∧τ > S 0 ) > 0. In general, it remains an open problem how to explicitly construct arbitrage strategies based on the local martingale N , in the spirit of Theorem 4.1, when P(σ ≥ ν) > 0.

CHAPTER 6

 6 Arbitrage of the first kind and filtration enlargements in semimartingale financial models This chapter is based on B. Acciaio, C. Fontana & C. Kardaras, Arbitrage of the first kind and filtration enlargements in semimartingale financial models, Stoch. Proc. Appl. 126(6): 1761-1784, 2016.

  Whenever Q ∼ P, we use Y(F, S, Q) to denote the class of all strictly positive F-adapted càdlàg processes Y with Y 0 = 1, such that Y and Y S are local martingales on (Ω, F, Q). The elements in Y(F, S, Q) are called strictly positive local martingale deflators (for S on (Ω, F, Q)). When strict positivity is replaced by nonnegativity, we simply talk of local martingale deflators. If Y Q denotes the density process of Q with respect to P, note that Y

3. 1 .

 1 Representation pair associated with τ . The next result is [Kar15, Theorem 1.1].

  , for all φ ∈ A UF and for all ξ ∈ r∈(1,+∞)L r .

Lemma 4. 3 .

 3 If Assumption 4.2-(1) holds, then E sup t∈[0,T ] |S t | r < +∞ for all r ∈ (1, +∞).

  t = q t dB t + R r(t, ζ) N (dt, dζ), t ∈ [0, T ], p T = U X φ T .Proposition 4.4. Suppose that Assumption 4.2 holds. An element φ ∈ A U F solves problem (4.1) if and only if the solution (p, q, r) ∈ M 2 × L 2 (B) × G 2 ( N ) to the corresponding BSDE (4.3) satisfies the following condition P-a.s. for a.a. t ∈ [0, T ]:E b t p t + σ t q t + R γ(t, ζ) r(t, ζ) ν(dζ) F t = 0

••

  Lecture "Insurance mathematics", ISIFAR master program, Paris Diderot University. • Teaching assistant for the course "Stochastic Processes in Finance", master program in Stochastic Modeling, Paris Diderot University. • Lecture and exercise class "Econometrics", third year of the BSc program in applied mathematics, Paris Diderot University. • Lecture "Asset pricing theory", Advanced degree in Quantitative Finance, Politecnico di Milano. Academic year 2015/2016 Teaching assistant for the course "Stochastic Processes in Finance", master program in Stochastic Modeling, Paris Diderot University. • Exercise class for the course "Econometrics", third year of the BSc program in applied mathematics, Paris Diderot University. • Exercise class for the course "Probability", second year of the BSc program in mathematics, Paris Diderot University. • Lecture "Asset pricing theory", Advanced degree in Quantitative Finance, Politecnico di Milano. Academic year 2014/2015 Teaching assistant for the course "Stochastic Processes in Finance", master program in Stochastic Modeling, Paris Diderot University. • Exercise class for the course "Econometrics", third year of the BSc program in applied mathematics, Paris Diderot University. • Exercise class for the course "Probability", second year of the BSc program in mathematics, Paris Diderot University. • Lecture "Asset pricing theory", Advanced degree in Quantitative Finance, Politecnico di Milano. Academic year 2013/2014 • Lecture "Asset pricing theory", Advanced degree in Quantitative Finance, Politecnico di Milano. • Supervision of the PhD seminars at the department of mathematics of the University of Evry. Academic year 2012/2013 • Lecture "Asset pricing theory", Advanced degree in Quantitative Finance, Politecnico di Milano. Academic Year 2011/2012 • Lecture "Asset pricing theory", Advanced degree in Quantitative Finance, Politecnico di Milano. • Exercise class for the course "Mathematical Finance I", third year of the undergraduate program in Mathematical Engineering, Department of Mathematics "F. Brioschi", Politecnico di Milano. • Supervision of the PhD seminars at the department of mathematics of the University of Evry.

  , C. Fontana, Financial Markets Theory: Equilibrium, Efficiency and Information, second edition, 836 pp., Springer, London, 2017. Fontana, J.M.A. Montes, A unified approach to pricing and risk management of equity and credit risk, Journal of Computational and Applied Mathematics, 250: 350-361, 2014.(2) C. Fontana, W.J. Runggaldier, Diffusion-based models for financial markets without martingale measures, in: Biagini, F., Richter, A. & Schlesinger, H. (eds.),

	Risk Measures and
	Attitudes, 45-81, EAA Series, Springer, 2013.
	(3) C. Fontana, M. Schweizer, Simplified mean-variance portfolio optimisation, Mathematics and
	Financial Economics, 6(2): 125-152, 2012.
	(4) C. Fontana, Credit risk and incomplete information: a filtering framework for pricing and
	risk management, in: Perna, C. & Sibillo, M. (eds.), Mathematical and Statistical Methods
	for Actuarial Sciences and Finance, 193-202, Springer, 2012.
	(5) C. Fontana, W.J. Runggaldier, Credit risk and incomplete information: filtering and EM
	parameter estimation, International Journal of Theoretical and Applied Finance, 13(5): 683-
	715, 2010.
	Preprints and articles in preparation
	(1) C. Fontana, M. Pelger, M. Platen, Flash profits via flash strategies and the impossibility of
	predictable jumps, preprint (submitted), 2018.
	(2) H.N. Chau, A. Cosso, C. Fontana, The value of informational arbitrage, in preparation, 2018.
	Theses
	(1) Four Essays in Financial Mathematics, PhD Thesis, University of Padova, 2012.
	(2) Mean-Variance Problems with Applications to Credit Risk Models, MAS Thesis, University
	of Zürich and ETH Zürich, 2010.	-1033,
	2018. (3) Affine Multifactor Credit Risk Models under Incomplete Information: Filtering and Param-
	(4) H.N. Chau, A. Cosso, C. Fontana, O. Mostovyi, Optimal investment with intermediate con-eter Estimation, Master Thesis, University of Padova, 2007.
	sumption under no unbounded profit with bounded risk, Journal of Applied Probability, 54(3):
	710-719, 2017.
	(5) C. Cuchiero, C. Fontana, A. Gnoatto, A general HJM framework for multiple curve modeling,
	Finance and Stochastics, 20(2): 277-320, 2016.
	(6) B. Acciaio, C. Fontana, C. Kardaras, Arbitrage of the first kind and filtration enlargements in
	semimartingale financial models, Stochastic Processes and their Applications, 126(6): 1761-
	1784, 2016.
	(7) C. Fontana, Weak and strong no-arbitrage conditions for continuous financial markets, In-
	ternational Journal of Theoretical and Applied Finance, 18(1): 1550005, 2015.
	(8) C. Fontana, B. Øksendal, A. Sulem, Viability and martingale measures under partial infor-
	mation, Methodology and Computing in Applied Probability, 17(1): 15-39, 2015.
	(9) C. Fontana, M. Jeanblanc, S. Song, On arbitrages arising with honest times, Finance and
	Stochastics, 18(3): 515-543, 2014.
	(10) C. Fontana, Z. Grbac, M. Jeanblanc, Q. Li, Information, no-arbitrage and completeness for
	asset price models with a random change point, Stochastic Processes and their Applications,
	124(9): 3009-3030, 2014.

Peer-reviewed papers and conference proceedings (after the PhD) (1) C. Cuchiero, C. Fontana, A. Gnoatto, Affine multiple yield curve models, Mathematical Finance, forthcoming. (2) C. Fontana, T. Schmidt, General dynamic term structures under default risk, Stochastic Processes and their Applications, forthcoming, 2017. (3) C. Fontana, The strong predictable representation property in initially enlarged filtrations under the density hypothesis, Stochastic Processes and their Applications, 128(3): 1007(11) C. Fontana, A note on arbitrage, approximate arbitrage and the fundamental theorem of asset pricing, Stochastics, 86(6): 922-931, 2014. (12) No-arbitrage conditions and absolutely continuous changes of measure, in: Hillairet, C., Jeanblanc, M. & Jiao, Y. (eds.), Arbitrage, Credit and Informational Risks, 3-18, World Scientific, 2014. Chapter 3. Complete list of publications

Peer-reviewed papers and conference proceedings (related to the PhD thesis)

(1) C.

  NFLVR holds if and only if NA 1 holds and there exists N

	T 0 0 (μ t /V t ) 2 dt < +∞ P-a.s.; T	θ2 t dt < +∞ P-a.s., with the latter condition being equivalent to
	(2)

  s. is said to be a local martingale deflator in H on the time horizon [0, ] if L S is an H-local martingale; (2) a probability measure Q ∼ P on (Ω, F) is said to be an Equivalent Local Martingale Measure in H (ELMM Theorem 2.5. Let H ∈ {F, G and a (possibly infinite-valued) H-stopping time. Then, on the time horizon [0, ], the following hold: (1) NA 1 (or, equivalently, NUPBR) holds in the financial market M H if and only if there exists a local martingale deflator in H; (2) NFLVR holds in the financial market M H if and only if there exists an Equivalent Local Martingale Measure in H. (3) NFLVR holds in the financial market M H if and only if both NA 1 (or, equivalently, NUPBR)

H ) on the time horizon [0, ] if S is an H-local martingale under Q.

Note that the concept of local martingale deflator corresponds to the notion of strict martingale density first introduced by

[START_REF] Schweizer | Martingale densities for general asset prices[END_REF]

. We then have the following fundamental theorem. The first assertion is a partial statement of [Kar10, Theorem 4], while the last two assertions follow from Corollary 1.2 and Corollary 3.4 together with Proposition 3.6, respectively, of

[START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF]

.

  By the Doob-Meyer decomposition of the bounded submartingale (1 {τ n+1 ≤t} ) t∈[0,T ] , it is also a uniformly integrable martingale on (Ω, F, P) (see e.g. [JS03, Theorem I.3.15]). By [HWY92, Corollary 5.31], this implies that (τ n+1 ∧ • -τ n ∧ •) is the dual F-predictable projection of (1 {τ n+1 ≤t} ) t∈[0,T ] , for all n ∈ N, thus proving (2.3).

  Proposition 4.6. Suppose that Assumption 4.2-(1) holds. A probability measure Q θ ∼ P on (Ω, G T ) with dQ θ /dP ∈ L 2 is a PIEMM if and only if the following condition holds:E Q θ b t + σ t θ 0 t + R γ(t, ζ) θ 1 (t,ζ)ν(dζ) F t = 0 P-a.s. for a.a. t ∈ [0, T ]

  5.3) d G t S t = S t dG t + G t dS t + d G, S t = S t dG t + G t dB t + G t

	1 S t	+ θ 0 t dt.

This paper has been awarded the Europlace Institute of Finance prize in

for the best paper in finance.

[START_REF] Jacod | Grossissement initial, hypothèse (H'), et théorème de Girsanov[END_REF] Theorem

2.5], there exists a (BE ⊗ P(F))-measurable version of the map (x, ω, t) → S, q x F t (ω), which is well-defined on the set {q x -> 0}. Hence, S, q x F x=L denotes the F-predictable quadratic variation for x = L.

Acknowledgements

research reported in this thesis benefited from the financial support of the European Union, under a Marie Curie Intra-European fellowship for career development, the Europlace Institute of Finance, and the French Embassy.

I am grateful to Huyên Pham for

Section 3 is devoted to the proof of Theorem 2.3 and Theorem 2.6; several interesting side results are also included there. In Section 2.5, a couple of illustrative examples are given.

2.4. Main results under initial filtration enlargement. We now study the stability of condition NA 1 under an initial enlargement of the filtration F with respect to an A-measurable random variable L taking values in a Lusin space (E, B E ), where B E denotes the Borel σ-field of E. With some abuse of notation, we denote by G = (G t ) t∈R + the right-continuous augmentation of the filtration G 0 = (G 0 t ) t∈R + defined by G 0 t := F t ∨ σ(L), for all t ∈ R + . Let γ : B E → [0, 1] be the law of L (so that γ(B) = P(L ∈ B) holds for all B ∈ B E ). Furthermore, for all t ∈ R + , let γ t : Ω × B E → [0, 1] be a regular version of the F t -conditional law of L, which exists since (E, B E ) is Lusin.

Assumption 2.7. Throughout Section 2.4, we work under the following condition:

(J) for all t ∈ R + , γ t γ holds in the P-a.s. sense.

Assumption 2.7 is the classical density hypothesis introduced in [START_REF] Jacod | Grossissement initial, hypothèse (H'), et théorème de Girsanov[END_REF]. Indeed, as shown in [Jac85, Proposition 1.5] (see also [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF]Theorem VI.11]), condition (J) holds if and only if, for all t ∈ R + there exists a σ-finite measure ν t on (E, B E ) such that γ t ν t holds in the P-a.s. sense. Jacod's density hypothesis plays a prominent role in financial mathematics, notably in relation to the modeling of additional information (see e.g. [AIS98, GP98, GP01, Bau03, GVV06, KH07, KHOL11]).

The next auxiliary result implies the existence of a good version of conditional densities. It essentially corresponds to [Jac85, Lemma 1.8] (see also [START_REF] Amendinger | Martingale representation theorems for initially enlarged filtrations[END_REF]Appendix A.1]). Note that O(F) denotes the F-optional σ-field on Ω × R + .

Lemma 2.8. There exists a (B E ⊗ O(F))-measurable function E × Ω × R + (x, ω, t) → q x t (ω) ∈ [0, +∞), càdlàg in t ∈ R + and such that:

(i) for every t ∈ R + , γ t (dx) = q x t γ(dx) holds P-a.s; (ii) for every x ∈ E, the process q x = (q x t ) t∈R + is a martingale on (Ω, F, P).

For every x ∈ E and n ∈ N, define families of stopping times on (Ω, F) via

(2.4)

For all x ∈ E, it holds that (ζ x n ) n∈N is a nondecreasing sequence, P(lim n→∞ ζ x n = ζ x ) = 1, and [START_REF] Jacod | Grossissement initial, hypothèse (H'), et théorème de Girsanov[END_REF]Lemma 1.8]). Note also that, due to [Jac85, Corollary 1.11], it holds that P(ζ L < +∞) = 0, with ζ L (ω) := ζ L(ω) (ω) for all ω ∈ Ω. For every x ∈ E, we consider the

which is a stopping time on (Ω, F) and represents the time at which q x jumps to zero. We mention that the family of stopping times {η x } x will turn out to be important in establishing a general form of martingale representation in the initially enlarged filtration G, as shown in Chapter 7. Under Assumption 2.7, let us now discuss counterparts to Theorems 2.3 and 2.6 on the validity of NA 1 in initially enlarged filtrations. Note that Assumption 2.7 guarantees that S is a semimartingale on (Ω, G, P), by [Jac85, Theorem 1.1], which is proved by relying on the Bichteler-Dellacherie characterisation of semimartingales. This allows us to define the class X (G, S) and the condition NA 1 (G, S) as done in Section 2.2 with respect to the filtration F. The first result is concerned with stability of condition NA 1 for a fixed semimartingale model. Theorem 2.9. Under Assumption 2.7, suppose further that the space L 1 (Ω, A, P) is separable and

Therefore, if P(η < +∞) > 0, then P(D τ > 0) > 0, hence P(S τ > 1) > 0. This means that NA 1 (G, S τ ) fails, concluding the proof.

Note that, in view of Proposition 3.7, Theorem 2.6 implies that NA 1 is stable for all semimartingale models if and only if the process 1/L τ is a local martingale on (Ω, G, P).

CHAPTER 7

The strong predictable representation property in initially enlarged filtrations under the density hypothesis CHAPTER 8

Market viability and martingale measures under partial information

Introduction

The concepts of no-arbitrage, martingale measure and optimal portfolio can be rightly considered as the cornerstones of modern mathematical finance, starting from the seminal papers [START_REF] Harrison | Martingales and arbitrage in multiperiod securities markets[END_REF][START_REF] Kreps | Arbitrage and equilibrium in economies with infinitely many commodities[END_REF]. Loosely speaking, the no-arbitrage requirement is equivalent to the existence of a martingale measure, which can be used for pricing purposes, and, again loosely speaking, portfolio optimization problems are solvable if and only if arbitrage profits cannot be obtained by trading on the market.

In the context of discrete-time models on a finite probability space (see, e.g., [BF17, PR12, Pli97]), it can actually be shown that the above concepts are equivalent and, furthermore, one can work out explicitly the connections between them. In particular, portfolio optimization problems and solvable if and only if there exists an equivalent martingale measure (EMM) and, moreover, one can obtain an EMM by taking the (normalised) marginal utility of the optimal terminal wealth. This relation also represents a classical and well-known result from the economic literature (see, e.g., [BF17, Section 4.4]). In the case of discrete-time models on a general probability space, the validity of this equivalence has been studied in [START_REF] Rasony | On utility maximization in discrete-time financial market models[END_REF][START_REF] Schäl | Price systems constructed by optimal dynamic portfolios[END_REF][START_REF] Schäl | Portfolio optimization and martingale measures[END_REF].

When one moves from discrete-time to continuous-time financial models, then things become quickly more complicated and the equivalences discussed so far do not hold any more in full generality. For instance, in order to recover the equivalence between EMMs and no-arbitrage, one has to replace the notion of martingale with the notion of local martingale (or sigma-martingale) and the condition of no-arbitrage with the no free lunch with vanishing risk (NFLVR) criterion adopted in [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF][START_REF] Delbaen | The fundamental theorem of asset pricing for unbounded stochastic processes[END_REF] (see also [START_REF] Frittelli | Some remarks on arbitrage and preferences in securities market models[END_REF] for an equivalent characterization of NFLVR in terms of no market free lunch, a noarbitrage criterion based on the structure of investors' preferences). Furthermore, the marginal utility of the optimal terminal wealth does not necessarily yield the density of an equivalent (local/sigma)martingale measure but only the terminal value of a supermartingale deflator.

In the present chapter, we consider a general jump-diffusion model with locally bounded coefficients and study the issue of the viability of the financial market, defined as the ability to solve a portfolio optimization problem. Our main goal consists in characterizing the notion of viability in terms of martingale measures, in a sense to be made precise in the following, studying under which conditions the marginal utility of terminal wealth gives rise to a martingale measure. We refrain from a-priori imposing no-arbitrage restrictions on the model, tackling instead directly the solvability of portfolio optimization problems. Furthermore, we suppose that market participants have only access to a partial information flow, which does not reveal the full information of the stochastic basis. In order to solve portfolio optimization problems under partial information, we shall employ necessary and sufficient maximum principles for stochastic control problems under partial information, as discussed in [START_REF] Baghery | A maximum principle for stochastic control with partial information[END_REF] (see also [ØS12] for related results in the complete information case). This approach allows us to characterize the optimal solution via an associated BSDE, which in turn requires a good control on the The chapter is structured as follows. Section 2 presents the modeling framework. In Section 3, we prove the equivalence between local market viability and the existence of a PIEMM up to a stopping time. Section 4 deals with the issue of the global viability of the financial market. Section 5 closes the chapter by illustrating some of the main concepts and results in the context of a simple example.

Setting

On a given probability space (Ω, G, P), let us consider a Brownian motion B = (B t ) t≥0 and a homogeneous Poisson random measure N on R + × R, in the sense of Definition II.1.20 of [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF], independent of B. Let G = (G t ) t≥0 be the filtration generated by B and N , assumed to satisfy the usual conditions of right-continuity and P-completeness, and denote by P G the predictable sigma-field of G. We denote by m(dt, dζ) := ν(dζ) udt the compensator of the random measure N (dt, dζ), where ν is a σ-finite measure on (R, B(R)), and by N (dt, dζ) := N (dt, dζ) -ν(dζ) dt the corresponding compensated random measure. Finally, we let T ∈ (0, +∞) represent a fixed investment horizon.

As mentioned in the introduction to this chapter, we are interested in financial models where agents do not have access to the full information filtration G. To this effect, we introduce a filtration F = (F t ) t≥0 , which represents the partial information actually available. We assume that the filtration F satisfies the usual conditions and that F t ⊆ G t , for all t ≥ 0, and we denote by P F the predictable sigma-field of F.

We consider an abstract financial market with two investment possibilities (all the results of the present chapter can be generalised to multi-dimensional models in a straightforward way):

(1) a risk-free asset with unit price S 0 t = 1, for all t ∈ [0, T ]; (2) a risky asset, with unit (discounted) price S t given by the solution to the SDE (2.1)

where the two processes b = (b t ) t∈[0,T ] and σ = (σ t ) t∈[0,T ] are P G -measurable and γ : Ω×[0, T ]×R → R is a predictable function in the sense of [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF], i.e., P G ⊗B(R)-measurable, and integrable with respect to N . We also impose the following assumption on b, σ and γ and on the sub-filtration F: Assumption 2.1.

(1) The P G -measurable processes b and σ as well as the

where

Part (1) of Assumption 2.1 implies that there exists a common sequence of F-stopping times {τ n } n∈N with τ n +∞ P-a.s. as n → +∞ such that the stopped processes S τn , b τn , σ τn and γ(• ∧ τ n , •) are P-a.s. uniformly bounded, for every n ∈ N. In particular, part (1) of Assumption 2.1 is always satisfied for the processes b and σ if they are P F -measurable and left-continuous or rightcontinuous with limits from the left (see, e.g., [HWY92, Theorem 7.7]). This is for instance the case if b t and σ t are given as continuous functions of S t-. Part (2) of Assumption 2.1 implies that every market participant can observe the evolution of the (discounted) price of the risky asset S. Note, however, that the filtration F S is in general strictly smaller than G, since the observation of the price process S does not suffice to unveil the two sources of randomness B and N .

We say that a function U : (-∞, +∞] → [-∞, +∞) of class C 1 on (-∞, +∞) is a utility function if it is concave and strictly increasing on (-∞, +∞) and we denote by U its first derivative (marginal utility). Aiming at describing the activity of trading in the financial market on the basis of the partial information represented by the sub-filtration F and according to the preference structure represented