domain: math qfin

text:
Claudio Fontana
email: fontana@math.unipd.it

Zorana Grbac
email: grbac@math.univ-paris-diderot.fr

Sandrine Gümbel
email: sandrine.guembel@stochastik.uni-freiburg.de

Thorsten Schmidt
email: thorsten.schmidt@stochastik.uni-freiburg.de

AND Sandrine G Ümbel

Term structure modelling

Keywords: . 2010 Mathematics Subject Classification. 60G44, 60G48, 60G57, 91B70, 91G20, 91G30. JEL classification: C02, C60, E43, G12 HJM model, semimartingale, affine process, NAFLVR, large financial market, multiple yield curves, stochastic discontinuities, forward rate agreement, market models, Libor rate

come

Introduction

This work aims at providing a general analysis of interest rate markets in the postcrisis environment. These markets exhibit two key characteristics. The first one is the presence of stochastic discontinuities, meaning jumps occurring at predetermined dates. Indeed, a view on historical data of European reference interest rates (see Figure 1) shows surprisingly regular jumps: many of the jumps occur in correspondence of monetary policy meetings of the European Central Bank (ECB), and the latter take place at pre-scheduled dates. This important feature, present in interest rate markets even before the crisis, has been surprisingly neglected by existing stochastic models.

The second key characteristic is the co-existence of different yield curves associated to different tenors. This phenomenon originated with the 2007 -2009 financial crisis, when the spreads between different yield curves reached their peak beyond 200 basis points. Since then the spreads have remained on a non-negligible level, as shown in Figure 2. This was accompanied by a rapid development of interest rate models, treating multiple yield curves at different levels of generality and following different modeling paradigms. The most important curves to be considered in the current economic environment are the overnight indexed swap (OIS) rates and the interbank offered rates (abbreviated as Ibor, such as Libor rates from the London interbank market) of various tenors. In the European market these are respectively the Eonia-based OIS rates and the Euribor rates.

It is our aim to propose a general treatment of markets with multiple yield curves in the light of stochastic discontinuities, meanwhile unifying the existing multiple curve modeling approaches. The building blocks of this study are OIS zero-coupon bonds and forward rate agreements (FRAs), which constitute the basic traded assets of a multiple curve financial market. While OIS bonds are bonds bootstrapped from quoted OIS rates, a FRA is an over-the-counter derivative consisting of an exchange of a payment based on a floating rate against a payment based on a fixed rate. FRAs can be regarded as the fundamental components of all interest rate derivatives written on Ibor rates.

The main contributions of the present paper can be outlined as follows:

' A general forward rate setup for the term structure of FRAs and OIS bond prices inspired by the seminal Heath-Jarrow-Morton (HJM) approach of [START_REF] Heath | Bond pricing and the term structure of interest rates[END_REF], suitably extended to allow for stochastic discontinuities. We derive a set of necessary and sufficient conditions characterizing risk-neutral measures with respect to a general numéraire process (Theorem 3.7). This framework unifies and generalizes the existing approaches in the literature. ' We study market models in general and, on the basis of minimal assumptions, derive necessary and sufficient drift conditions in the presence of stochastic discontinuities (Theorem 4.1). This approach covers modeling under forward measures as a special case. Moreover, the generality of our forward rate formulation with stochastic discontinuities enables us to directly embed market models. ' We propose a new class of model specifications, based on affine semimartingales as recently introduced in Keller- [START_REF] Keller-Ressel | Affine processes beyond stochastic continuity[END_REF], going beyond the classical requirement of stochastic continuity. We illustrate the potential for practical applications by means of some simple examples. ' Finally, we provide a general description of a multiple curve financial market under minimal assumptions and a characterization of absence of arbitrage. We prove the equivalence between the notion of no asymptotic free lunch with vanishing risk (NAFLVR) and the existence of an equivalent separating measure (Theorem 6.3).

To this effect, we rely on the theory of large financial markets and we extend to multiple curves and to an infinite time horizon the main result of [START_REF] Cuchiero | A new perspective on the fundamental theorem of asset pricing for large financial markets[END_REF]. To the best of our knowledge, this represents the first rigorous formulation of an FTAP in the context of multiple curve financial markets.

1.1. The modeling framework. We briefly illustrate the ingredients of our modeling framework, referring to the sections in the sequel for full details. First, forward rate agreements (FRAs) are quoted in terms of forward rates. More precisely, the forward Ibor rate Lpt, T, δq at time t ď T with tenor δ and settlement date T is given as the unique value of the fixed rate which assigns the FRA value zero at inception t. This leads to the fundamental representation of FRA prices Π FRA pt, T, δ, Kq " δ `Lpt, T, δq ´K˘P pt, T `δq, (1.1)

where P pt, T `δq is the price at time t of an OIS zero-coupon bond with maturity T `δ and K is a fixed rate. Formula (1.1) implicitly defines the yield curves T Þ Ñ Lpt, T, δq for different tenors δ, thus explaining the terminology multiple yield curves. In the following, we will simply call the associated markets multiple curve financial markets (compare with Definition 2.2 below).

The forward rate formulation makes some additional assumptions on the yield curves. More specifically, it postulates that the right-hand side of (1.1) admits the representation Π FRA pt, T, δ, Kq " S δ t e ´şpt,T s f pt,u,δqηpduq ´e´ş pt,T `δs f pt,uqηpduq p1 `δKq.

(1.2)

Here, f pt, T q denotes the OIS forward rate, so that P pt, T q " e ´şpt,T s f pt,uqηpduq , while f pt, T, δq is the δ-tenor forward rate and S δ is a multiplicative spread. We extend the usual HJM formulation by considering a measure η containing atoms which by no-arbitrage will be precisely related to the set of stochastic discontinuities in the dynamics of forward rates and multiplicative spreads.

Representations (1.1) and (1.2) constitute two seemingly different starting points for multiple curve modeling: market models and HJM models, respectively. In the following, we shall derive no-arbitrage drift restrictions for both classes. Moreover, we will show that the two classes can be analyzed in a unified setting (see Appendix B).

1.2. Stochastic discontinuities in interest rate markets. One of the main novelties of our approach consists in the presence of stochastic discontinuities, representing events occurring at announced dates but with a possibly unanticipated informational content. The importance of jumps at predetermined times is widely acknowledged in the financial literature, see for example [START_REF] Merton | On the pricing of corporate debt: the risk structure of interest rates[END_REF], [START_REF] Piazzesi | An econometric model of the yield curve with macroeconomic jump effects. Piazzesi, M[END_REF][START_REF] Piazzesi | An econometric model of the yield curve with macroeconomic jump effects. Piazzesi, M[END_REF][START_REF] Piazzesi | Affine term structure models[END_REF], [START_REF] Kim | Jumps in bond yields at known times[END_REF], [START_REF] Lando | Term structures of credit spreads with incomplete accounting information[END_REF] (see also the introductory section of [START_REF] Keller-Ressel | Affine processes beyond stochastic continuity[END_REF]). However, to the best of our knowledge, stochastic discontinuities have never been taken explicitly into account in stochastic models for the term structure of interest rates. This feature is extremely relevant in financial markets. For instance, the Governing Council (GC) of the European Central Bank (ECB) holds its monetary policy meetings on a regular basis at predetermined dates, which are publicly known for about two years ahead. At such dates the GC takes its monetary policy decisions and determines whether the main ECB interest rates will change. In turn, these key interest rates are principal determinants of the Eonia rate, as illustrated by Figure 1.

A closer inspection of Figure 1 reveals the presence of two different types of stochastic discontinuities in the Eonia rate. On the one hand, there are structural jumps in correspondence to monetary policy decisions. This type of discontinuity is evidenced by a step-like jump of the Eonia rate in correspondence to a new level of the ECB lending rate (see Figure 3, right panel). On the other hand, there are spiky jumps which are unrelated to the monetary policy and occur at the end of the maintenance periods of banks' deposits. Indeed, in the Eurosystem banks are required to hold deposits on accounts with their national central bank over fixed maintenance periods. Banks who fail to keep sufficient reserves during the period need to borrow in the interbank market before the close of the maintenance period, thereby generating a temporary liquidity pressure in interbank lending which leads to a jump in the Eonia rate (see e.g. [START_REF] Beirne | The EONIA spread before and during the crisis of 2007-2009: The role of liquidity and credit risk[END_REF] and [START_REF] Hernandis | The information content of Eonia swap rates before and during the financial crisis[END_REF]). This second type of stochastic discontinuity is evidenced by the spikes in the left panel of Figure 3. More formally, we distinguish these two kinds of stochastic discontinuities as follows: jumps of type I are step-like jumps to a new level and jumps of type II are upward/downward jumps followed by a fast continuous decay/ascent to the pre-jump level.

Our framework allows for the possibility of both type I and type II stochastic discontinuities. In addition, by relaxing the classical assumption that the term structure of bond prices is absolutely continuous (see equation (1.2)), we also allow for discontinuities in time-to-maturity at predetermined dates. In a credit risky setting, term structures with stochastic discontinuities have been recently studied in [START_REF] Gehmlich | Dynamic defaultable term structure modelling beyond the intensity paradigm[END_REF] and [START_REF] Fontana | General dynamic term structures under default risk[END_REF]. Finally, besides stochastic discontinuities as described above, we also allow for totally inaccessible jumps, representing events occurring as a surprise to the market and generated by a general random measure with absolutely continuous compensator. Such jumps have been already considered in several multiple curve models (see e.g. [START_REF] Crépey | A multiple-curve HJM model of interbank risk[END_REF] and [START_REF] Cuchiero | A general HJM framework for multiple yield curve modeling[END_REF]).

1.3. Overview of the literature. The literature on multiple curve models has witnessed a tremendous growth over the last years. Therefore, we only give an overview of the contributions that are the most related to the present paper, referring to the volume of Bianchetti and Morini (2013) and the monographs by [START_REF] Henrard | Interest Rate Modelling in the Multi-curve Framework[END_REF] and [START_REF] Grbac | Interest Rate Modeling: Post-Crisis Challenges and Approaches[END_REF] for further references and a guide to post-crisis interest rate markets. Multiplicative spreads for modeling multiple curves have been first considered in [START_REF] Henrard | The irony in the derivatives discounting[END_REF].

Adopting a short rate approach, an insightful empirical analysis has been conducted by [START_REF] Filipović | The term structure of interbank risk[END_REF], showing that spreads can be decomposed into credit and liquidity components. The extended HJM approach developed in Section 3 generalizes the framework of [START_REF] Cuchiero | A general HJM framework for multiple yield curve modeling[END_REF], who consider Itô semimartingales as driving processes and, therefore, do not allow for stochastic discontinuities (see Remark 3.12 for a detailed comparison). HJM models taking into account multiple curves have been proposed in [START_REF] Crépey | A Lévy HJM multiple-curve model with application to CVA computation[END_REF] with Lévy processes as drivers and in [START_REF] Moreni | Parsimonious HJM modelling for multiple yieldcurve dynamics[END_REF] in a Gaussian framework. In the market model setup, the extension to multiple curves was pioneered by [START_REF] Mercurio | Modern LIBOR market models: using different curves for projecting rates and for discounting[END_REF] and further developed in [START_REF] Mercurio | The basis goes stochastic[END_REF]. More recently, Grbac et al. (2015) have developed an affine market model in a forward rate setting, further generalized by [START_REF] Cuchiero | Affine multiple yield curve models[END_REF]. All these models, both HJM and market models, can be easily embedded in the general framework proposed in this paper.

1.4. Outline of the paper. In Section 2, we introduce the basic assets in a multiple curve financial market. The general multi-curve framework inspired by the HJM philosophy, extended to allow for stochastic discontinuities, is developed and fully characterized in Section 3. In Section 4, we introduce and analyze general market models with multiple curves. In Section 5, we propose a flexible class of models based on affine semimartingales, in a setup which allows for stochastic discontinuities. In Section 6, we prove a version of the fundamental theorem of asset pricing for multiple curve financial markets, by relying on the theory of large financial markets. Finally, the two appendices contain some technical results and a result on the embedding of market models into the extended HJM framework.

A general analysis of multiple curve financial markets

In this section, we provide a general description of a multiple curve market under minimal assumptions. We assume that the interbank offered rates (Ibor) are quoted for a finite set of tenors D :" tδ 1 , . . . , δ m u, with 0 ă δ 1 ă . . . ă δ m . Typically, about seven tenors, ranging from 1 day to 12 months, are available in the market. For a tenor δ P D, the Ibor rate for the time interval rT, T `δs fixed at time T is denoted by LpT, T, δq. For 0 ď t ď T ă `8, we denote by P pt, T q the price at date t of an OIS zero-coupon bond with maturity T . Definition 2.1. A forward rate agreement (FRA) with tenor δ, settlement date T , strike K and unitary notional amount, is a contract in which a payment based on the Ibor rate LpT, T, δq is exchanged against a payment based on the fixed rate K at maturity T `δ. The price of a FRA contract at date t ď T `δ is denoted by Π FRA pt, T, δ, Kq and the payoff at maturity T `δ is given by Π FRA pT `δ, T, δ, Kq " δLpT, T, δq ´δK.

(2.1)

The two addends in (2.1) are typically referred to as floating leg and fixed leg, respectively. We define the multiple curve financial market as follows.

Definition 2.2. The multiple curve financial market is the financial market containing the following two sets of traded assets:

(i) OIS zero-coupon bonds, for all maturities T ě 0; (ii) FRAs, for all tenors δ P D, all settlement dates T ě 0 and all strikes K P R.

The assets included in Definition 2.2 represent the quantities that we assume to be tradable in the financial market. We emphasize that, in the post-crisis environment, FRA contracts have to be considered on top of OIS bonds as they cannot be perfectly replicated by the latter, due to the risks implicit in interbank transactions.

We work under the standing assumption that FRA prices are determined by a linear valuation functional. This assumption is standard in interest rate modeling and is also coherent with the fact that we consider clean prices, i.e., prices which do not model explicitly counterparty and liquidity risk (the counterparty and liquidity risk of the interbank market as a whole is of course present in Ibor rates, recall Figure 2). Clean prices are fundamental quantities in interest rate derivative valuation and they also form the basis for the computation of XVA adjustments, see Section 1.2.3 in [START_REF] Grbac | Interest Rate Modeling: Post-Crisis Challenges and Approaches[END_REF] and [START_REF] Brigo | Risk-neutral valuation under differential funding costs, defaults and collateralization[END_REF].

Recalling (2.1), the value of the fixed leg of a FRA at time t ď T `δ is given by δKP pt, T `δq. Hence, we obtain that Π FRA pt, T, δ, Kq is an affine function of K.

Definition 2.3. The forward Ibor rate Lpt, T, δq at t P r0, T s for tenor δ P D and maturity T ą 0 is the unique value K satisfying Π FRA pt, T, δ, Kq " 0.

Due to the affine property of FRA prices combined with the above definition, the fundamental representation Π FRA pt, T, δ, Kq " δ `Lpt, T, δq ´K˘P pt, T `δq, follows immediately for t ď T , while for t P rT, T `δs we have of course Π FRA pt, T, δ, Kq " δpLpT, T, δq ´KqP pt, T `δq.

Starting from this expression, under no additional assumptions, we can decompose the value of the floating leg of the FRA into a multiplicative spread and a tenor-dependent discount factor. Indeed, setting Kpδq :" 1 `δK, we can write Π FRA pt, T, δ, Kq " `1 `δLpt, T, δq ˘P pt, T `δq ´KpδqP pt, T `δq ": S δ t P pt, T, δq ´KpδqP pt, T `δq, (2.2)

where S δ t represents a multiplicative spread and P pt, T, δq a discount factor satisfying P pT, T, δq " 1, for all T ě 0 and δ P D. More precisely, it holds that S δ t " P pt, t `δq `1 `δLpt, t, δq

˘" 1 `δLpt, t, δq 1 `δF pt, t, δq ,
where F pt, t, δq denotes the simply compounded OIS rate at date t for the period rt, t `δs.

The discount factor P pt, T, δq is therefore given by P pt, T, δq " P pt, T `δq P pt, t `δq 1 `δLpt, T, δq 1 `δLpt, t, δq .

We shall sometimes refer to P p¨, T, δq as δ-tenor bonds. These bonds essentially span the term structure, while S δ accounts for the counterparty and liquidity risks in the interbank market, which do not vanish as t Ñ T .

Remark 2.4. In the classical pre-crisis single curve setup, the FRA price is given by the textbook formula Π FRA pt, T, δ, Kq " P pt, T q ´P pt, T `δq Kpδq.

The single curve setting can be recovered from our approach by setting S δ " 1 and P pt, T, δq :" P pt, T q, for all δ P D and 0 ď t ď T ă `8. This also highlights that, in a single curve setup, FRA prices are fully determined by OIS bond prices.

Remark 2.5. Representation (2.2) allows for a natural interpretation via a foreign exchange analogy, following some ideas going back to [START_REF] Bianchetti | Two curves, one price[END_REF]. Indeed, Ibor rates can be thought of as simply compounded rates in a foreign economy, with the currency risk playing the role of the counterparty and liquidity risks of interbank transactions. In this perspective, P pt, T, δq represents the price at date t (in units of the foreign currency) of a foreign zero-coupon bond with maturity T , while S δ t represents the spot exchange rate between the foreign and the domestic currencies. The quantity S δ t P pt, T, δq appearing in (2.2) corresponds to the value at date t (in units of the domestic currency) of a payment of one unit of the foreign currency at maturity T . In view of Remark 2.4, the pre-crisis scenario assumes the absence of currency risk, in which case S δ t P pt, T, δq " P pt, T q. Related foreign exchange interpretations of multiplicative spreads have been discussed in [START_REF] Cuchiero | A general HJM framework for multiple yield curve modeling[END_REF], [START_REF] Macrina | Consistent valuation across curves using pricing kernels[END_REF] and [START_REF] Nguyen | The multi-curve potential model[END_REF].

With the additional assumption that OIS and δ-tenor bond prices are of HJM form, we obtain our second fundamental representation (1.2). In the following, we will show that such a representation allows for a precise characterization of arbitrage-free multiple curve markets and leads to interesting specifications by means of affine semimartingales.

3. An extended HJM approach to term structure modeling

In this section, we present a general framework for modeling the term structures of OIS bonds and FRA contracts, inspired by the seminal work by [START_REF] Heath | Bond pricing and the term structure of interest rates[END_REF]. We work in an infinite time horizon (models with a finite time horizon T ă `8 can be treated by stopping the relevant processes at T ). As mentioned in the introduction, a key feature of the proposed framework is that we allow for the presence of stochastic discontinuities, occurring in correspondence of a countable set of predetermined dates pT n q nPN , with T n`1 ą T n , for every n P N, and lim nÑ`8 T n " `8.

We assume that the stochastic basis pΩ, F , F, Qq supports a d-dimensional Brownian motion W " pW t q tě0 together with an integer-valued random measure µpdt, dxq on R `ˆE, with compensator νpdt, dxq " λ t pdxqdt, where λ t pdxq is a kernel from pΩ ˆR`, Pq into pE, B E q, with P denoting the predictable sigma-field on Ω ˆR`a nd pE, B E q a Polish space with its Borel sigma-field. We refer to [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] for all unexplained notions related to stochastic calculus.

As a first ingredient, we assume the existence of a general numéraire process X 0 " pX 0 t q tě0 , given by a strictly positive semimartingale admitting the representation

X 0 " E `B `H ¨W `L ˚pµ ´νq ˘, (3.1)
where H " pH t q tě0 is an R d -valued progressively measurable process so that ş T 0 }H s } 2 ds ă `8 a.s. for all T ą 0 and L : Ω ˆR`ˆE Ñ p´1, `8q is a P b B E -measurable function satisfying ş T 0 ş E pL 2 pt, xq ^|Lpt, xq|qλ t pdxqdt ă `8 a.s. for all T ą 0. Note that, in view of (Jacod and Shiryaev, 2003, Theorem II.1.33), the last condition is necessary and sufficient for the well-posedness of the stochastic integral L ˚pµ ´νq. The process B " pB t q tě0 is assumed to be a finite variation process of the form

B t " ż t 0 r s ds `ÿ nPN ∆B Tn 1 tTnďtu , for all t ě 0, (3.2)
where r " pr t q tě0 is an adapted process satisfying ş T 0 |r s |ds ă `8 a.s. for all T ą 0 and ∆B Tn is an F Tn -measurable random variable taking values in p´1, `8q, for each n P N. Note that this specification of X 0 explicitly allows for jumps at times pT n q nPN , the stochastic discontinuity points of X 0 . The assumption that lim nÑ`8 T n " `8 ensures that the summation in (3.2) involves only a finite number of terms, for every t ě 0.

Remark 3.1. Requiring minimal assumptions on X 0 enables us to unify different modeling approaches. Usually, it is postulated that X 0 " expp ş 0 r OIS s dsq, with r OIS representing the OIS short rate. In the setting considered here, X 0 can also be generated by a sequence of OIS bonds rolled over at dates pT n q nPN , compare (Klein et al., 2016, Definition 5) for a precise notion. This allows to avoid the unnecessary assumption of existence of a bank account. In market models, the usual choice for X 0 is the OIS-bond with the longest available maturity, see Remark 4.2. Moreover, it is also possible to choose Q as the physical probability measure and X 0 as the growth-optimal portfolio. By this, we cover the benchmark approach to term structure modeling (see [START_REF] Bruti-Liberati | Real-world jumpdiffusion term structure models[END_REF] and [START_REF] Platen | A Benchmark Approach to Quantitative Finance[END_REF]). While these examples refer to situations where the numéraire X 0 is tradable, we do not necessarily assume that X 0 represents the price process of a traded asset or portfolio (with the exception of Section 6). This generality yields additional flexibility, since X 0 may also represent a state-price density or pricing kernel in the spirit of [START_REF] Constantinides | A theory of the nominal term structure of interest rates[END_REF], embedding a choice of the discounting asset and a probability change into a single process (compare also with Remark 3.11). As explained below, the focus of Sections 3-5 will be on deriving necessary and sufficient conditions for the local martingale property of X 0 -denominated prices under Q.

The reference probability measure Q is said to be a risk-neutral measure for the multiple curve financial market with respect to X 0 if the X 0 -denominated price process of every asset included in Definition 2.2 is a Q-local martingale. One of our main goals consists in deriving necessary and sufficient conditions for Q to be a risk-neutral measure. In Section 6, under the additional assumption that the numéraire X 0 is tradable, we will prove a fundamental theorem characterizing absence of arbitrage in the sense of NAFLVR, for which the existence of a risk-neutral measure is a sufficient condition (see Remark 6.4).

In view of representation (2.2), modeling a multiple curve financial market requires the specification of multiplicative spreads S δ and δ-tenor bond prices, for δ P D. The multiplicative spread process S δ " pS δ t q tě0 is assumed to be a strictly positive semimartingale, for each δ P D. Similarly as in (3.1), we assume that S δ admits the representation

S δ " S δ 0 E `Aδ `Hδ ¨W `Lδ ˚pµ ´νq ˘, (3.3)
for every δ P D, where A δ , H δ and L δ satisfy the same requirements of the processes B, H and L, respectively, appearing in (3.1). In line with (3.2), we furthermore assume that

A δ t " ż t 0 α δ s ds `ÿ nPN ∆A δ Tn 1 tTnďtu , for all t ě 0, (3.4)
where pα δ t q tě0 is an adapted process satisfying ş T 0 |α δ s |ds ă `8 a.s., for all δ P D and T ą 0, and ∆A δ

Tn is an F Tn -measurable random variable taking values in p´1, `8q, for each n P N and δ P D.

We let P pt, T, 0q :" P pt, T q, for all 0 ď t ď T ă `8. We assume that, for every T P R ànd δ P D 0 :" D Ť t0u, the δ-tenor bond price process pP pt, T, δqq 0ďtďT is of the form

P pt, T, δq " exp ˜´ż pt,T s f pt, u, δqηpduq ¸, for all 0 ď t ď T, (3.5)
where ηpduq " du `ÿ nPN δ Tn pduq.

(3.6)

Note that ηpr0, T sq ă `8, for all T ą 0. We adopt the convention ş pT,T s f pT, u, δqηpduq " 0, for all T P R `and δ P D 0 . For every T P R `and δ P D 0 , we assume that the forward rate process pf pt, T, δqq 0ďtďT satisfies

f pt, T, δq " f p0, T, δq `ż t 0 aps, T, δqds `V pt, T, δq `ż t 0 bps, T, δqdW s `ż t 0 ż E gps, x, T, δq `µpds, dxq ´νpds, dxq ˘, (3.7)
for all 0 ď t ď T , where V p¨, T, δq " V pt, T, δq 0ďtďT is a pure jump adapted process of the form V pt, T, δq "

ÿ nPN ∆V pT n , T, δq1 tTnďtu , for all 0 ď t ď T,
with ∆V pt, T, δq " 0 for all 0 ď T ă t ă `8. Moreover, for all n P N, T P R `and δ P D 0 , we also assume that ş T 0 |∆V pT n , u, δq|du ă `8. Remark 3.2.

(1) The above framework allows for a general modeling of type I and type II stochastic discontinuities (see Section 1.2), as we illustrate by means of explicit examples in Section 5. Moreover, the dependence on δ in equations (3.3)-(3.7) allows the discontinuities to have a different impact on different yield curves. This is consistent with the typical market behavior, which shows a dampening of the discontinuities over longer tenors.

(2) The discontinuity dates pT n q nPN play two distinct but equally important roles.

On the one hand, they represent stochastic discontinuities in the dynamics of all relevant processes. On the other hand, they represent discontinuity points in maturity of bond prices (see equation (3.5)). As shown in Theorem 3.7 below, absence of arbitrage will imply a precise relation between these two aspects.

Assumption 3.3. The following conditions hold a.s. for every δ P D 0 :

(i) the initial forward curve T Þ Ñ f p0, T, δq is F 0 b BpR `q-measurable, real-valued and satisfies ş T 0 |f p0, u, δq|du ă `8, for all T P R `; (ii) the drift process ap¨, ¨, δq : Ω ˆR2
`Ñ R is a real-valued, progressively measurable process, in the sense that the restriction ap¨, ¨, δq| r0,ts : Ω ˆr0, ts ˆR`Ñ R is F t b Bpr0, tsq b BpR `q-measurable, for every t P R `. Moreover, it satisfies for all 0 ď T ă t ă `8 that apt, T, δq " 0, and

ż T 0 ż u 0 |aps, u, δq|ds ηpduq ă `8,
for all T ą 0;

(iii) the volatility process bp¨, ¨, δq : ΩˆR 2 `Ñ R d is an R d -valued progressively measurable process, in the sense that the restriction bp¨, ¨, δq| r0,ts : Ω ˆr0, ts ˆR`Ñ R d is F t b Bpr0, tsq b BpR `q-measurable, for every t P R `. Moreover, it satisfies for all 0 ď T ă t ă `8 that bpt, T, δq " 0, and

d ÿ i"1 ż T 0 ˆż u 0 |b i ps, u, δq| 2 ds ˙1{2 ηpduq ă `8, for all T ą 0; (iv) the jump function gp¨, ¨, ¨, δq : Ω ˆR`ˆE ˆR`Ñ R is a P b B E b BpR `q-measurable
real-valued function satisfying gpt, x, T, δq " 0 for all x P E and 0 ď T ă t ă `8. Moreover, it satisfies

ż T 0 ż E ż T 0 |gps, x, u, δq| 2 ηpduqνpds, dxq ă `8, for all T ą 0.
Assumption 3.3 implies that the integrals appearing in the forward rate equation (3.7) are well-defined for η-a.e. T P R `. Moreover, the integrability requirements appearing in conditions (ii)-(iv) of Assumption 3.3 ensure that we can apply ordinary and stochastic Fubini theorems, in the versions of [START_REF] Veraar | The stochastic Fubini theorem revisited[END_REF] for the Brownian motion and Proposition A.2 in [START_REF] Björk | Towards a general theory of bond markets[END_REF] for the compensated random measure. The mild measurability requirement in conditions (ii)-(iii) holds if ap¨, ¨, δq and bp¨, ¨, δq are P rog bBpR `q-measurable, for every δ P D 0 , with P rog denoting the progressive sigma-algebra on Ω ˆR`, see (Veraar, 2012, Remark 2.1).

Remark 3.4. There is no loss of generality in taking a single measure η instead of different measures η δ for each tenor δ P D 0 . Indeed, dependence on the tenor can be embedded in our framework by suitably specifying each forward rate f pt, T, δq in (3.7) and using a common measure η " ř δPD 0 η δ . For all 0 ď t ď T ă `8, δ P D 0 and x P E, we set āpt, T, δq :"

ż rt,T s apt, u, δqηpduq, bpt, T, δq :" ż rt,T s bpt, u, δqηpduq, V pt, T, δq :" ż rt,T s ∆V pt, u, δqηpduq, ḡpt, x, T, δq :" ż rt,T s gpt, x, u, δqηpduq.
As a first result, the following lemma (whose proof is postponed to Appendix A) gives a semimartingale representation of the process P p¨, T, δq.

Lemma 3.5. Suppose that Assumption 3.3 holds. Then, for every T P R `and δ P D 0 , the process pP pt, T, δqq 0ďtďT is a semimartingale and admits the representation

P pt, T, δq " exp ˆ´ż T 0 f p0, u, δqηpduq ´ż t 0 āps, T, δqds ´ÿ nPN V pT n , T, δq1 tTnďtu ´ż t 0 bps, T, δqdW s ´ż t 0 ż E ḡps, x, T, δq `µpds, dxq ´νpds, dxq ż t 0 f pu, u, δqηpduq ˙, for all 0 ď t ď T. (3.8)
The δ-tenor bond price process pP pt, T, δqq 0ďtďT admits an equivalent representation as a stochastic exponential, which will be used in the following. The following corollary is a direct consequence of Lemma 3.5 and (Jacod and Shiryaev, 2003, Theorem II.8.10), using the fact that µptT n u ˆEq " 0 a.s., for all n P N.

Corollary 3.6. Suppose that Assumption 3.3 holds. Then, for every T P R `and δ P D 0 , the process P p¨, T, δq " pP pt, T, δqq 0ďtďT admits the representation ˙.

P p¨, T, δq " E ˆ´ż T 0 f p0, u, δqηpduq ´ż 0 āps, T, δqds `1 2 ż 0 } bps, T,
We are now in a position to state the central result of this section, which provides necessary and sufficient conditions for the reference probability measure Q to be a riskneutral measure with respect to the numéraire X 0 . We recall that a random variable ξ on pΩ, F , Qq is said to be sigma-integrable with respect to a sigma-field G Ď F if there exists a sequence of measurable sets pΩ n q nPN Ď G increasing to Ω such that ξ 1 Ωn P L 1 pQq for every n P N, see Definition 1.15 in [START_REF] He | Semimartingale Theory and Stochastic Calculus[END_REF]. A random variable ξ is sigma-finite with respect to G if and only if the generalized conditional expectation E Q rξ|G s is a.s. finite. For convenience of notation, let α 0 t :" 0, H 0 t :" 0, L 0 pt, xq :" 0 and ∆A 0 Tn :" 0 for all n P N, t P R `and x P E, so that S 0 :" EpA 0 `H0 ¨W `L0 ˚pµ ´νqq " 1. Let (iii) for every n P N, it holds that

E Q « 1 `∆A δ Tn 1 `∆B Tn

ˇˇˇˇF

Tn´ff " e ´f pTn´,Tn,δq ;

(iv) for every n P N and T ě T n , it holds that (3.12)

E Q « 1 `∆A
The conditions of Theorem 3.7 together with Remark 3.8 admit the following interpretation. First, for δ " 0 condition (i) requires that the drift rate r t of the numéraire process X 0 equals the short end of the instantaneous yield f pt, t, 0q on OIS bonds, plus two additional terms accounting for the volatility of X 0 itself.1 For δ ‰ 0, condition (i) requires that, at the short end, the instantaneous yield α δ t `f pt, t, δq on the floating leg of a FRA equals the instantaneous yield f pt, t, 0q plus a risk premium determined by the covariation between the numéraire process X 0 and the multiplicative spread process S δ .

Second, condition (ii) is a generalization of the well-known HJM drift condition. In particular, if D " H and the process X 0 does not have local martingale components, then condition (ii) reduces to the drift restriction established in Proposition 5.3 of [START_REF] Björk | Towards a general theory of bond markets[END_REF] for single-curve jump-diffusion models.

Finally, conditions (iii) and (iv) are new and specific to our setting with stochastic discontinuities. Together, they correspond to excluding the possibility that, at some predetermined date T n , prices of X 0 -denominated assets exhibit jumps whose size can be predicted on the basis of the information contained in F Tn´. Indeed, such a possibility would violate absence of arbitrage (compare with Fontana et al. ( 2019)).

Proof. of Theorem 3.7 Recall that P pt, T, 0q " P pt, T q, 0 ď t ď T ă `8. By definition, Q is a risk-neutral measure with respect to X 0 if and only if the processes P p¨, T q{X 0 and Π FRA p¨, T, δ, Kq{X 0 are Q-local martingales, for every T P R `, δ P D and K P R. In view of (2.2) and using the notational convention introduced above, this holds if and only if the process S δ P p¨, T, δq{X 0 is a Q-local martingale, for every T P R `and δ P D 0 . An application of Corollary A.1 together with Corollary 3.6 and equations (3.1)-(3.4) yields S δ P p¨, T, δq X 0 "S δ 0 P p0, T, δq Note that the set t∆K p1q pT, δq ‰ 0u Ş t∆K p2q pT, δq ‰ 0u is evanescent for every T P R ànd δ P D 0 , as a consequence of the fact that µptT n u ˆEq " 0 a.s. for all n P N. Suppose that S δ P p¨, T, δq{X 0 is a Q-local martingale, for every T P R `and δ P D 0 . In this case, (3.13) implies that the finite variation process ş 0 k s pT, δqds `Kp1q pT, δq Kp2q pT, δq is also a Q-local martingale. By means of (Jacod and Shiryaev, 2003, Lemma I.3.11), this implies that the pure jump finite variation process K p1q pT, δq `Kp2q pT, δq is of locally integrable variation. Since the two processes K p1q pT, δq and K p2q pT, δq do not have common jumps, it holds that |∆K piq pT, δq| ď |∆K p1q pT, δq `∆K p2q pT, δq|, for i " 1, 2.

¨E
As a consequence of this fact, both processes K p1q pT, δq and K p2q pT, δq are of locally integrable variation. Noting that

K p2q pT, δq " ÿ nPN ∆K p2q
Tn pT, δq1 rrTn,`8rr , Theorem 5.29 of [START_REF] He | Semimartingale Theory and Stochastic Calculus[END_REF] implies that the random variable ∆K p2q Tn pT, δq is sigmaintegrable with respect to F Tn´, for every n P N. This is equivalent to the sigmaintegrability of 1 `∆A δ Tn 1 `∆B Tn e ´şpTn,T s ∆V pTn,u,δqηpduq`f pTn´,Tn,δq (3.14) with respect to F Tn´. Since f pT n ´, T n , δq is F Tn´-measurable, the sigma-integrability of (3.14) with respect to F Tn´c an be equivalently stated as the sigma-integrability of (3.10) with respect to F Tn´. Moreover, the fact that K p1q pT, δq is of locally integrable variation is equivalent to the a.s. finiteness of the integral

ż T 0 ż E ˇˇΛps, x, T, δq ˇˇλ s pdxqds,
thus proving the integrability conditions (3.9), (3.10). Having established that the two processes K p1q pT, δq and K p2q pT, δq are of locally integrable variation, we can take their compensators (dual predictable projections), see (Jacod and Shiryaev, 2003, Theorem I.3.18). This leads to Considering then the pure jump part, the condition ∆ p K p2q Tn pT, δq " 0 a.s., for all n P N, leads to

S δ P p¨, T, δq X 0 " S δ 0 P p0, T,
E Q « 1 `∆A δ
Tn 1 `∆B Tn e ´şpTn,T s ∆V pTn,u,δqηpduq ˇˇˇˇF Tn´ff " e ´f pTn´,Tn,δq (3.17) a.s. for all n P N. Condition (iii) in the statement of the theorem is obtained by taking T " T n , while condition (iv) follows by inserting condition (iii) into (3.17).

Conversely, if the integrability conditions (3.9), (3.10) are satisfied then the finite variation processes K p1q pT, δq and K p2q pT, δq appearing in (3.13) are of locally integrable variation. One can therefore take their compensators and obtain representation (3.15). It is then easy to verify that, if the four conditions (i)-(iv) hold, then the processes kpT, δq and p K p2q pT, δq appearing in (3.15) are null, up to an evanescent set. This proves the local martingale property of S δ P p¨, T, δq{X 0 , for every T P R `and δ P D 0 .

Remark 3.9. The foreign exchange analogy introduced in Remark 2.5 carries over to the conditions established in Theorem 3.7. In particular, in the special case where H t " Lpt, xq " 0, for all pt, xq P R `ˆE, it can be easily verified that conditions (i)-(ii) reduce exactly to the HJM conditions established in [START_REF] Koval | Time-inhomogeneous Lévy processes in Cross-Currency Market Models[END_REF] in the context of multi-currency HJM semimartingale models.

3.1. The OIS bank account as numéraire. In HJM models, the numéraire is usually chosen as the OIS bank account expp ş 0 r OIS s dsq, with r OIS denoting the OIS short rate. In this context, an application of Theorem 3.7 enables us to characterize all equivalent local martingale measures (ELMMs, see Section 6) with respect to the OIS bank account numéraire. To this effect, let Q 1 be a probability measure on pΩ, F q equivalent to Q and denote by Z 1 its density process, i.e., Z 1 t " dQ 1 | Ft {dQ| Ft , for all t ě 0. We denote the expectation with respect to Q 1 by E Q 1 and assume that

Z 1 " E ˆ´θ ¨W ´ψ ˚pµ ´νq ´ÿ nPN Y n 1 rrTn,`8rr ˙, (3.18)
for an R d -valued progressively measurable process θ " pθ t q tě0 satisfying the integrability condition ş T 0 }θ s } 2 ds ă `8 a.s. for all T ą 0, a P b B E -measurable function ψ : Ω ˆR`Ê Ñ p´8, `1q satisfying the integrability condition ş T 0 ş E p|ψps, xq| ^ψ2 ps, xqqλ s pdxqds ă `8 a.s. for all T ą 0, and a family pY n q nPN of random variables taking values in p´8, `1q such that Y n is F Tn -measurable and E Q rY n |F Tn´s " 0, for all n P N. Denote Λ ˚ps, x, T, δq " `1 ´ψps, xq ˘`p1 `Lδ ps, xqqe ´ḡps,x,T,δq ´1˘´L δ ps, xq `ḡps, x, T, δq.

Corollary 3.10. Suppose that Assumption 3.3 holds. Let Q 1 be a probability measure on pΩ, F q equivalent to Q, with density process Z 1 given in (3.18 (ii) for every T P R `and for a.e. t P r0, T s, it holds that āpt, T, δq "

1 2 } bpt, T, δq} 2 `bpt, T, δq J `θt ´Hδ t żE
´`1 ´ψpt, xq ˘`1 `Lδ pt, xq ˘`e ´ḡpt,x,T,δq ´1˘`ḡ pt, x, T, δq ¯λt pdxq;

(iii) for every n P N, it holds that

E Q 1 " ∆A δ
Tn ˇˇF Tn´‰ " e ´f pTn´,Tn,δq ´1;

(iv) for every n P N and T ě T n , it holds that

E Q 1 " p1 `∆A δ Tn q
´e´ş pTn,T s ∆V pTn,u,δqηpduq ´1¯ˇˇˇF Tn´ı " 0.

Proof. By means of Bayes' formula, Q 1 is an ELMM if and only if Z 1 S δ P p¨, T, δqe a.s., for all T ą 0, it can be easily checked that condition (3.19) is equivalent to (3.9). The corollary then follows from Theorem 3.7 noting that, for any F Tn -measurable random variable ξ which is sigma-integrable under Q 1 with respect to F Tn´, it holds that

E Q 1 rξ|F Tn´s " E Q rZ 1 Tn ξ|F Tn´s Z 1
Tn´"

E Q " p1 ´Yn qξ ˇˇF Tn´‰ " E Q " ξ 1 `∆B Tn ˇˇˇF Tn´ ,
where we have used the fact that Z 1 Tn " Z 1 Tn´p 1 ´Yn q, for every n P N.

Remark 3.11. The proof of Corollary 3.10 permits to obtain a characterization of all equivalent local martingale deflators for the multiple curve financial market, i.e., all strictly positive Q-local martingales Z of the form (3.18) such that ZS δ P p¨, T, δqe ´ş¨0 r OIS s ds is a Q-local martingale, for every T P R `and δ P D 0 .

Remark 3.12. The HJM framework of [START_REF] Cuchiero | A general HJM framework for multiple yield curve modeling[END_REF] can be recovered as a special case with no stochastic discontinuities, setting ηpduq " du in (3.6), taking the OIS bank account as numéraire and a jump measure µ generated by a given Itô semimartingale. [START_REF] Cuchiero | A general HJM framework for multiple yield curve modeling[END_REF] show that most of the existing multiple curve models are covered by their framework, which a fortiori implies that they can be easily embedded in our framework.

General market models

In this section, we consider market models and develop a general arbitrage-free framework for modeling Ibor rates. As shown in Appendix B, market models can be embedded into the extended HJM framework considered in Section 3, in the spirit of [START_REF] Brace | The market model of interest rate dynamics[END_REF]. This is possible due to the fact that the measure ηpduq in the term structure equation (3.5) may contain atoms. However, it turns out to be simpler to directly study market models as follows.

In the spirit of market models, and differently from Definition 2.2, in this section we assume that only finitely many assets are traded. For each δ P D, let T δ " tT δ 0 , . . . , T δ N δ u be the set of settlement dates of traded FRA contracts associated to tenor δ, with T δ 0 " T 0 and T δ N δ " T ˚, for 0 ď T 0 ă T ˚ă `8. We consider an equidistant tenor structure, i.e. T δ i T δ i´1 " δ, for all i " 1, . . . , N δ and δ P D. Let us also define T :" Ť δPD T δ , corresponding to the set of all traded FRAs. The starting point of our approach is representation (1.1), Π FRA pt, T, δ, Kq " δ `Lpt, T, δq ´K˘P pt, T `δq, (4.1)

for δ P D, T P T δ , t P r0, T s and K P R. The financial market contains OIS zero-coupon bonds for all maturities T P T 0 :" T Ť tT ˚`δ i : i " 1, . . . , mu 2 as well as FRA contracts for all δ P D, T P T δ and K P R.

Let pΩ, F , F, Qq be a filtered probability space supporting a d-dimensional Brownian motion W and a random measure µ, as described in Section 3. We assume that, for every tenor δ P D and maturity T P T δ , the forward Ibor rate Lp¨, T, δq " pLpt, T, δqq 0ďtďT satisfies Lpt, T, δq " Lp0, T, δq

`ż t 0 a L ps, T, δqds `ÿ nPN ∆LpT n , T, δq1 tTnďtu `ż t 0 b L ps, T, δqdW s `ż t 0 ż E g L ps, x, T, δq `µpds, dxq ´νpds, dxq ˘. (4.2)
In the above equation, a L p¨, T, δq " pa L pt, T, δqq 0ďtďT is a real-valued adapted process satisfying ş T 0 |a L ps, T, δq|ds ă `8 a.s., b L p¨, T, δq " pb L pt, T, δqq 0ďtďT is a progressively measurable R d -valued process satisfying the integrability condition ş T 0 }b L ps, T, δq} 2 ds ă `8 a.s., p∆LpT n , T, δqq nPN is a family of random variables such that ∆LpT n , T, δq is F Tnmeasurable, for each n P N, and g L p¨, ¨, T, δq : Ω ˆr0, T s ˆE Ñ R is a P b B E -measurable function that satisfies

ż T 0 ż E
´`g L ps, x, T, δq ˘2 ^|g L ps, x, T, δq| ¯λs pdxqds ă `8 a.s.

The dates pT n q nPN represent the stochastic discontinuities occurring in the market. We assume that OIS bond prices are of the form (3.5) for δ " 0, for all T P T 0 , with the associated forward rates f pt, T, 0q being as in (3.7).

The main goal of this section consists in deriving necessary and sufficient conditions for a reference probability measure Q to be a risk-neutral measure with respect to a general numéraire X 0 of the form (3.1) for the financial market where FRA contracts and OIS zero-coupon bonds are traded, and FRA prices are modeled via (4.1) and (4.2) for the discrete set T of settlement dates. We recall that bpt, T `δ, 0q " ż rt,T `δs bpt, u, 0qηpduq, ḡpt, x, T `δ, 0q " ż rt,T `δs gpt, x, u, 0qηpduq.

2 Note that we need to consider an extended set of maturities for OIS bonds since the payoff of a FRA contract with settlement date T and tenor δ takes place at date T `δ.

Theorem 4.1. Suppose that Assumption 3.3 holds for δ " 0 and for all maturities T P T 0 . Then Q is a risk-neutral measure with respect to X 0 if and only if all the conditions of Theorem 3.7 are satisfied for δ " 0 and for all T P T 0 , and, for every δ P D,

ż T 0 ż E ˇˇg L ps, x, T, δq ˜e´ḡps,x,T `δ,0q
1 `Lps, xq ´1¸ˇˇˇλ s pdxqds ă `8 (4.3) a.s. for all T P T δ , and, for each n P N and T δ Q T ě T n , the random variable ∆LpT n , T, δq 1 `∆B Tn e ´şpTn,T `δs ∆V pTn,u,0qηpduq (4.4)

is sigma-integrable with respect to F Tn´, and the following two conditions hold a.s.:

(i) for all T P T δ and a.e. t P r0, T s, it holds that a L pt, T, δq " b L pt, T, δq J `Ht `bpt, T `δ, 0q żE g L pt, x, T, δq ˜e´ḡpt,x,T `δ,0q

1 `Lpt, xq ´1¸λ t pdxq;

(ii) for all n P N and T δ Q T ě T n , it holds that E Q " ∆LpT n , T, δq 1 `∆B Tn e ´şpTn,T `δs ∆V pTn,u,0qηpduq ˇˇˇF Tn´ " 0.

Condition (i) of Theorem 4.1 is a drift restriction for the Ibor rate process. In the context of a continuum of traded maturities, as in Theorem 3.7, this condition can be separated into a condition on the short end and an HJM-type drift restriction (see conditions (i) and (ii) in Theorem 3.7). Condition (ii), similarly to conditions (iii), (iv) of Theorem 3.7, corresponds to requiring that, for each n P N, the size of the jumps occurring at date T n in FRA prices cannot be predicted on the basis of the information contained in F Tn´.

Proof. In view of representation (4.1), Q is a risk-neutral measure with respect to X 0 if and only if P p¨, T q{X 0 is a Q-local martingale, for every T P T 0 , and Lp¨, T, δqP p¨, T `δq{X 0 is a Q-local martingale, for every δ P D and T P T δ . Considering first the OIS bonds, Theorem 3.7 implies that P p¨, T q{X 0 is a Q-local martingale, for every T P T 0 , if and only if conditions (3.9), (3.10) as well as conditions (i)-(iv) of Theorem 3.7 are satisfied for δ " 0 and for all T P T 0 . Under these conditions, equation (3.15) for δ " 0 gives that P p¨, T q X 0 " P p0, T q E `M 1 pT, 0q ˘, (4.5)

for every T P T 0 , where the local martingale M 1 pT, 0q is given by ∆LpT n , T, δq 1 `∆B Tn e ´şpTn,T `δs ∆V pTn,u,0qηpduq`f pTn´,Tn,0q .

M 1 pT, 0q " K p2q pT
If Lp¨, T, δqP p¨, T `δq{X 0 is a local martingale, for every δ P D and T P T δ , then (4.6) implies that the processes J p1q pT, δq and J p2q pT, δq are of locally integrable variation. Similarly as in the proof of Theorem 3.7, this implies the validity of conditions (4.3) and (4.4), due to Theorem 5.29 in [START_REF] He | Semimartingale Theory and Stochastic Calculus[END_REF]. Let us denote by p J piq pT, δq the compensator of J piq pT, δq, for i P t1, 2u, δ P D and T P T δ . We have that p J p1q pT, δq "

ż 0 ż E g L ps, x, T, δq ˜e´ḡps,x,T `δ,0q 1 `Lps, xq ´1¸λ s pdxqds, p J p2q pT, δq " ÿ nPN ˆEQ "
∆LpT n , T, δq 1 `∆B Tn e ´şpTn,T `δs ∆V pTn,u,0qηpduq ˇˇˇF Tn´ e f pTn´,Tn,0q 1 rrTn,`8rr

˙.

The local martingale property of Lp¨, T, δqP p¨, T `δq{X 0 together with equation (4.6) implies that the predictable finite variation process ż 0 j s pT, δqds `p J p1q pT, δq `p J p2q pT, δq (4.7) is null (up to an evanescent set), for every δ P D and T P T δ . Considering separately the absolutely continuous and discontinuous parts, this implies the validity of conditions (i), (ii) in the statement of the theorem. Conversely, by Theorem 3.7, if conditions (3.9), (3.10) as well as conditions (i)-(iv) of Theorem 3.7 are satisfied for δ " 0 and for all T P T 0 , then P p¨, T q{X 0 is a Qlocal martingale, for all T P T 0 . Furthermore, if conditions (4.3), (4.4) are satisfied and conditions (i), (ii) of the theorem hold, then the process given in (4.7) is null. In turn, by equation (4.6), this implies that Lp¨, T, δqP p¨, T `δq{X 0 is a Q-local martingale, for every δ P D and T P T δ , thus proving that Q is a risk-neutral measure with respect to X 0 . Remark 4.2. In market models, the numéraire is usually chosen as the OIS zero-coupon bond with the longest available maturity T ˚(terminal bond). In addition, the reference probability measure Q is the associated T ˚-forward measure, see Section 12.4 in [START_REF] Musiela | Martingale Methods in Financial Modelling[END_REF]. Exploiting the generality of the process X 0 , this setting can be easily accommodated within our framework. Indeed, if ş T 0 ş E |e ´ḡps,x,T ˚,0q ´1 gps, x, T ˚, 0q|λ s pdxqds ă `8 a.s., Corollary 3.6 shows that X 0 " P p¨, T ˚q{P p0, T ˚q holds as long as the processes appearing in (3.1) and (3.2) are specified as

H t " ´bpt, T ˚, 0q,
Lpt, xq " e ´ḡpt,x,T ˚,0q ´1, ∆B Tn " e ´şpTn,T ˚s ∆V pTn,u,0qηpduq`f pTn´,Tn,0q ´1,

r t " f pt, t, 0q ´āpt, T ˚, 0q `1 2 } bpt, T ˚, 0q} 2
`żE `e´ḡpt,x,T ˚,0q ´1 `ḡpt, x, T ˚, 0q ˘λt pdxq.

Under this specification, a direct application of Theorem 4.1 yields necessary and sufficient conditions for Q to be a risk-neutral measure with respect to the terminal OIS bond as numéraire.

4.1. Martingale modeling. Typically, market models start directly from the assumption that each Ibor rate Lp¨, T, δq is a martingale under the pT `δq-forward measure Q T `δ associated to the numéraire P p¨, T `δq. In our context, this assumption is generalized into a local martingale requirement under the pT `δq-forward measure, whenever the latter is well-defined. More specifically, suppose that P p¨, T `δq{X 0 is a true martingale and define the pT `δq-forward measure by dQ T `δ | F T `δ :" pP p0, T `δqX 0 T `δ q ´1dQ| F T `δ . As a consequence of Girsanov's theorem (see (Jacod and Shiryaev, 2003, Theorem III.3.24)) and equation (4.5), the forward Ibor rate Lp¨, T, δq satisfies under the measure Q T `δ Lpt, T, δq " Lp0, T, δq

`ż t 0 a L,T `δ ps, T, δqds `ÿ nPN ∆LpT n , T, δq1 tTnďtu `ż t 0 b L ps, T, δqdW T `δ s `ż t 0 ż E g L ps,
x, T, δq `µpds, dxq ´νT `δ pds, dxq ˘, (4.8)

for some adapted real-valued process a L,T `δ p¨, T, δq, where the process W T `δ is a Q T `δ -Brownian motion defined by W T `δ :" W `ş¨0 pH s `bps, T `δ, 0qqds and the compensator ν T `δ pds, dxq of the random measure µpds, dxq under Q T `δ is given by ν T `δ pds, dxq " e ´ḡps,x,T `δ,0q 1 `Lps, xq λ s pdxqds.

In this context, Theorem 4.1 leads to the following proposition, which provides a characterization of the local martingale property of forward Ibor rates under forward measures.

Proposition 4.3. Suppose that Assumption 3.3 holds for δ " 0 and for all T P T 0 . Assume furthermore that P p¨, T q{X 0 is a true Q-martingale, for every T P T 0 . Then the following are equivalent:

(i) Q is a risk-neutral measure;

(ii) Lp¨, T, δq is a local martingale under Q T `δ , for every δ P D and T P T δ ; (iii) for every δ P D and T P T δ , it holds that a L,T `δ pt, T, δq " 0, outside a subset of Ω ˆr0, T s of pQ b dtq-measure zero, and, for every n P N and T δ Q T ě T n , the random variable ∆LpT n , T, δq satisfies

E Q T `δ r∆LpT n , T, δq|F Tn´s " 0 a.s.
Proof. Under these assumptions, Q is a risk-neutral measure if and only if Lp¨, T, δqP p0, T δq{X 0 is a local martingale under Q, for every δ P D and T P T δ . The equivalence piq ô piiq then follows from the conditional version of Bayes' rule (see (Jacod and Shiryaev, 2003, Proposition III.3.8)), while the equivalence piiq ô piiiq is a direct consequence of equation (4.8) together with (He et al., 1992, Theorem 5.29).

Affine specifications

One of the most successful classes of processes in term-structure modeling is the class of affine processes. This class combines a great flexibility in capturing the important features of interest rate markets with a remarkable analytical tractability, see e.g. [START_REF] Kan | A yield-factor model of interest rates[END_REF], [START_REF] Duffie | Affine processes and applications in finance[END_REF], as well as [START_REF] Filipović | Term Structure Models: A Graduate Course[END_REF] for a textbook account. In the literature, affine processes are by definition stochastically continuous and, therefore, do not allow for jumps at predetermined dates. In view of our modeling objectives, we need a suitable generalization of the notion of affine process. To this effect, [START_REF] Keller-Ressel | Affine processes beyond stochastic continuity[END_REF] have recently introduced affine semimartingales by dropping the requirement of stochastic continuity. Related results on affine processes with stochastic discontinuities in credit risk may be found in [START_REF] Gehmlich | Dynamic defaultable term structure modelling beyond the intensity paradigm[END_REF]. In the present section, we aim at showing how the class of affine semimartingales leads to flexible and tractable multiple curve models with stochastic discontinuities.

We consider a countable set T " tT n : n P Nu of discontinuity dates, with T n`1 ą T n , for every n P N, and lim nÑ`8 T n " `8. We assume that the filtered probability space pΩ, F , F, Qq supports a d-dimensional special semimartingale X " pX t q tě0 which is further assumed to be an affine semimartingale in the sense of [START_REF] Keller-Ressel | Affine processes beyond stochastic continuity[END_REF] and to admit the canonical decomposition

X " X 0 `BX `Xc `x ˚`µ X ´νX ˘,
where B X is a finite variation predictable process, X c is a continuous local martingale with quadratic variation C X and µ X ´νX is the compensated jump measure of X. Let B X,c be the continuous part of B X and ν X,c the continuous part of the random measure ν X , in the sense of (Jacod and Shiryaev, 2003, § II.1.23). In view of (Keller-Ressel et al., 2018, Theorem 3.2), under weak additional assumptions it holds that B X,c t pωq "

ż t 0 ´β0 psq `d ÿ i"1 X i s´p ωqβ i psq ¯ds, C X t pωq " ż t 0 ´α0 psq `d ÿ i"1 X i s´p ωqα i psq ¯ds, ν X,c pω, dt, dxq " ´µ0 pt, dxq `d ÿ i"1 X i t´p ωqµ i pt, dxq ¯dt, ż R d `exu,xy ´1˘ν X pω, ttu, dxq " ˜exp ´γ0 pt, uq `d ÿ i"1 xX i t´p ωq, γ i pt, uqy ¯´1 ¸.
(5.1)

In (5.1), we have that β i : R `Ñ R d and α i : R `Ñ R dˆd , for i " 0, 1, . . . , d, γ 0 : R `ˆC d Ñ C ´and γ i : R `ˆC d Ñ C d , for i " 1, . . . , d. µ i pt, dxq is a Borel measure on R d zt0u for all i " 0, 1, . . . , d, such that for all t P R `, ş R d zt0u p1 `|x| 2 qµ i pt, dxq ă `8. Finally, we assume that ν X pttu ˆRd q vanishes a.s. outside the set of stochastic discontinuities pT n q nPN . We use the affine semimartingale X as the driving process of a multiple curve model, as presented in Section 3. In particular, we focus here on modeling the δ-tenor bond prices P pt, T, δq and the multiplicative spreads S δ t in such a way that the resulting model is affine in the sense of the following definition, which extends the approach of (Keller-Ressel et al., 2018, Section 5.3)

ˆż T 0 |ϕ i p¨, u, δq| 2 ηpduq ˙1{2 P LpX i q and ż T 0 ż T 0 |ϕpt, u, δq|ηpduq|dB X,c t | ă `8 a.s.,
with LpXq denoting the set of R d -valued predictable processes which are integrable with respect to X in the semimartingale sense, and similarly for LpX i q. The measure η is specified as in equation (3.6).

For all 0 ď t ď T ă `8 and δ P D 0 , let us also define φpt, T, δq :" ż rt,T s ϕpt, u, δqηpduq.

We furthermore assume that ş T 0 e pψ δ t q J x 1 tpψ δ t q J xą1u ν X,c pdt, dxq ă `8 a.s., for all T P R `, which ensures that S δ is a special semimartingale (see (Jacod and Shiryaev, 2003, Proposition II.8.26)). To complete the specification of the model, we suppose that X 0 takes the form (5.4) where pr t q tě0 is an adapted real-valued process satisfying ş T 0 |r t |dt ă `8 a.s., for all T P R `, and ψ Tn is a d-dimensional F Tn´-measurable random vector, for all n P N.

X 0 t " exp ´ż t 0 r s ds `ÿ nPN ψ J Tn ∆X Tn 1 tTnďtu ¯, for all t ě 0,
We aim at characterizing when Q is a risk-neutral measure for an affine multiple curve model. By Remark 3.8, we see that a necessary condition is that r t " f pt, t, 0q, for a.e. t ě 0.

(5.5)

Under the present assumptions and in the spirit of Theorem 3.7, the following proposition provides sufficient conditions for Q to be a risk-neutral measure for the affine multiple curve model introduced above. For convenience of notation we let ψ 0 t :" 0 for all t P R ànd S 0 0 :" 1, so that S 0 :" S 0 0 expp ş 0 ψ 0 s dX s q " 1.

Proposition 5.2. Consider an affine multiple curve model as in Definition 5.1 and satisfying (5.5). Assume furthermore that

ż T 0 ż R d zt0u
ˇˇe pψ δ s q J x `e´φps,T,δq J x ´1˘`φ ps, T, δq J x ˇˇν X,c pds, dxq ă `8 a.s.

(5.6)

for every δ P D 0 and T P R `. Then Q is a risk-neutral measure with respect to X 0 given as in (5.4) if the following three conditions hold a.s. for every δ P D 0 :

(i) for a.e. t P R `, it holds that r t ´f pt, t, δq " pψ δ t q J ˜β0 ptq `d ÿ

i"1

X i t´β i ptq ¸`1 2 pψ δ t q J ˜α0 ptq `d ÿ i"1 X i t´α i ptq ¸ψδ t `żR d zt0u ´epψ δ t q J x ´1 ´pψ δ t q J x ¯˜µ 0 pt, dxq `d ÿ i"1 X i t´µ i pt, dxq ¸;
(ii) for every T P R `, a.e. t P r0, T s and for every i " 0, 1, . . . , d, it holds that φpt, T, δq

J β i ptq " φpt, T, δq J α i ptq ˆ1 2 φpt, T, δq ´ψδ t żR d zt0u
´epψ δ t q J x ´e´φpt,T,δq J x ´1¯`φ pt, T, δq J x ¯µi pt, dxq; (5.7)

(iii) for every n P N and T ě T n , it holds that ´f pT n ´, T n , δq " γ 0 ´Tn , ψ δ Tn ´ψTn ´żpTn,T s ϕpT n , u, δqηpduq

d ÿ i"1 A X i Tn´, γ i ´Tn , ψ δ Tn ´ψTn ´żpTn,T s ϕpT n , u, δqηpduq ¯E.
Proof. For all δ P D 0 , the present integrability assumptions ensure that ψ δ ¨X and S δ are special semimartingales. Hence, (Jacod and Shiryaev, 2003, Theorem II.8.10) implies that S δ admits a stochastic exponential representation of the form (3.3), (3.4), with

α δ t " pψ δ t q J ˜β0 ptq `d ÿ i"1 X i t´β i ptq ¸`1 2 pψ δ t q J ˜α0 ptq `d ÿ i"1 X i t´α i ptq ¸ψδ t `żR d zt0u ´epψ δ t q J x ´1 ´pψ δ t q J x ¯˜µ 0 pt, dxq `d ÿ i"1 X i t´µ i pt, dxq ¸,

∆A δ

Tn " e pψ δ Tn q J ∆X Tn ´1, for all n P N, and L δ pt, xq " pe pψ δ t q J x ´1q1 J c ptq, for all pt, xq P R `ˆR d zt0u, where we define the set J c :" R `zT. Due to (5.4), condition (i) of Theorem 3.7 reduces to a δ t " f pt, t, 0q´f pt, t, δq, for a.e. t P R `and δ P D (see also equation (3.12) in Remark 3.8), from which condition (i) directly follows. The integrability conditions appearing in Definition 5.1 enable us to apply the stochastic Fubini theorem in the version of Theorem IV.65 of [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF] and, moreover, ensure that ϕp¨, T, δq ¨X is a special semimartingale, for every δ P D 0 and T P R `. This permits to obtain a representation of P pt, T, δq as in Lemma 3.5, namely

P pt, T, δq " exp ˆ´ż T 0 f p0, u, δqηpduq ´ż t 0 φps, T, δqdB X,c s ´ÿ nPN φpT n , T, δq J ∆X Tn 1 tTnďtu ´ż t 0 φps, T, δqdX c s ´ż t 0 ż R d zt0u φps, T, δq J x1 J c psq `µX pds, dxq ´νX pds, dxq ż t 0 f pu, u, δqηpduq ˙.
In view of the affine structure (5.1) and comparing with (3.8), it holds that āpt, T, δq " φpt, T, δq J ´β0 ptq `d ÿ

i"1

X i t´β i ptq ¯, } bpt, T, δq} 2 " φpt, T, δq J ´α0 ptq `d ÿ i"1 X i t´α i ptq ¯φpt, T, δq, bpt, T, δq J H δ t " φpt, T, δq J ´α0 ptq `d ÿ i"1 X i t´α i ptq ¯ψδ t ,
and ḡpt, x, T, δq " φpt, T, δq J x1 J c ptq, for all 0 ď t ď T ă `8, δ P D 0 and x P R d zt0u. In the present setting condition (ii) of Theorem 3.7 takes the form φpt, T, δq J ˆβ0 ptq `d ÿ

i"1

X i t´β i ptq ˙" φpt, T, δq J ˆα0 ptq `d ÿ i"1 X i t´α i ptq ˙ˆ1 2 φpt, T, δq ´ψδ t żR d zt0u ´epψ δ t q J x `e´φpt,T,δq J x ´1˘`φ pt, T, δq J x ¯´µ 0 pt, dxq `d ÿ i"1 X i t´µ i pt, dxq ¯. (5.8)
Clearly, condition (ii) of the proposition is sufficient for (5.8) to hold, for every T P R ànd a.e. t P r0, T s. In the present setting, conditions (iii), (iv) of Theorem 3.7 can be together rewritten as follows, for every δ P D 0 , n P N and T ě T n , e ´f pTn´,Tn,δq " E Q « 1 `∆A δ Tn 1 `∆B Tn e ´şpTn,T s ϕpTn,u,δq J ∆X Tn ηpduq ˇˇˇF Tn´ff

" E Q « exp ˜ˆψ δ Tn ´ψTn ´żpTn,T s ϕpT n , u, δqηpduq ˙J∆X Tn ¸ˇˇˇF Tn´ff ,
from which condition (iii) of the proposition follows by making use of (5.1). Finally, in the present setting the integrability condition (3.9) appearing in Theorem 3.7 reduces to condition (5.6). In view of Theorem 3.7, we can conclude that Q is a risk-neutral with respect to X 0 .

Remark 5.3. Condition (ii) is only sufficient for the necessary condition (5.8). Only if the coordinates of X i are linearly independent, then this condition is also necessary.

The following examples illustrate the conditions of Proposition 5.2.

Example 5.4 (A single-curve Vasiček specification). As first example we study a classical single-curve (i.e., D " H) model without jumps, driven by a one-dimensional Gaussian Ornstein-Uhlenbeck process. Let ξ be the solution of

dξ t " κpθ ´ξt qdt `σdW t ,
where W is a Brownian motion and κ, θ, σ are positive constants. As driving process in (5.2) we choose the three-dimensional affine process

X t " ˆt, ż t 0 ξ s ds, ξ t ˙J , t ě 0.
The coefficients in the affine semimartingale representation (5.1) are time-homogeneous, i.e. α i ptq " α i and β i ptq " β i , i " 0, . . . , 3, given by

β 0 " ¨1 0 κθ ', β 1 " ¨0 0 0 ', β 2 " ¨0 0 0 ', β 3 " ¨0 1 ´κ ', α 0 " ¨0 0 0 0 0 0 0 0 σ 2 ',
and α 1 " α 2 " α 3 " 0. The drift condition (5.7) implies φ1 pt, T, 0q " σ 2 2 `φ 3 pt, T, 0q ˘2 ´κθ φ3 pt, T, 0q, φ2 pt, T, 0q " κ φ3 pt, T, 0q.

We are free to specify ϕ 3 pt, T, 0q and choose φ3 pt, T, 0q "

1 κ ´1 ´e´κpT ´tq ¯.
This in turn implies that ϕ 1 pt, T, 0q " σ 2 κ ´e´κpT ´tq ´e´2κpT ´tq ¯´κθe ´κpT ´tq , ϕ 2 pt, T, 0q " κe ´κpT ´tq , ϕ 3 pt, T, 0q " e ´κpT ´tq .

It can be easily verified that this corresponds to the Vasiček model, see Section 10.3.2.1 in [START_REF] Filipović | Term Structure Models: A Graduate Course[END_REF]. Note that this also implies f pt, t, 0q " ξ t . Choosing r t " f pt, t, 0q leads to the numéraire X 0 " expp ş 0 f ps, s, 0qdsq. Hence, all conditions in Proposition 5.2 are satisfied and the model is free of arbitrage. An extension to the multi-curve setting is presented in Example 5.6.

Example 5.5 (A single-curve Vasiček specification with discontinuity). As next step, we extend the previous example by introducing a discontinuity at time 1. Our goal is to provide a simple, illustrative example with jump size depending on the driving process ξ and we therefore remain in the single-curve framework.

We assume that there is a multiplicative jump in the numéraire at time T 1 " 1 depending on exppaξ 1 ` q, where a P R and " N p0, b 2 q is an independent normally distributed random variable with variance b 2 . As driving process in (5.2) we consider the five-dimensional affine process

X t " ˆż t 0 ηpdsq, ż t 0 ξ s ds, ξ t , 1 ttě1u ξ 1 , 1 ttě1u ˙J ,
where ηpdsq " ds `δ1 pdsq. The size of the jump in X 0 is specified by

ψ J t ∆X t " 1 tt"1u paξ 1 ` q,
which can be achieved by ψ J 1 " p0, 0, 0, a, 1q. The coefficients in the affine semimartingale representation (5.1) α i , β i , i " 0, . . . , 3, are as in Example 5.4, with zeros in the additional rows and columns. In addition we have β 4 " β 5 " 0 and α 4 " α 5 " 0. Moreover,

ż e xu,xy ν X pttu, dxq " 1 tt"1u exp ˆu1 `u4 X 3 1 `u2 5 b 2 2 ˙, u P R 5 .
Finally, we choose for t ď T ϕ 3 pt, T, 0q "

$ ' ' & ' ' % 0 for t " 1 ď T,
ae ´κp1´tq for t ă 1 " T, e ´κpT ´tq otherwise, ϕ 1 p1, 1, 0q " b 2 {2, ϕ 4 pt, T, 0q " p1 ´aq1 tt"T "1u , and ϕ 5 pt, T, 0q " 0. ϕ 1 pt, T, 0q for pt, T q ‰ p1, 1q and ϕ 2 pt, T, 0q for t ď T can be derived from ϕ 3 pt, T, 0q as in the previous example by means of the drift condition (5.7). Condition (iii) is the interesting condition for this example. This condition is equivalent to

aX 3 1 ´b2 2 " f p1´, 1, 0q, (5.9)
which can be satisfied by choosing f p0, 1, 0q " ´b2 {2. Equation (5.9), together with the specification of ϕ i pt, T, 0q for i " 1, . . . , 5 ensures that f pt, t, 0q " ξ t . Choosing r t " f pt, t, 0q we obtain that the model is free of arbitrage and the term structure is fully specified: indeed, we recover for 1 ď t ď T and 0 ď t ď T ă 1 the bond pricing formula from the previous example P pt, T, 0q " exp ´´ApT ´t, 0q ´BpT ´t, 0qX Example 5.6 (A simple multi-curve Vasiček specification). We extend Example 5.4 to the multi-curve setting and consider D " tδu. For simplicity, we choose as driving diffusive part a two-dimensional Gaussian Ornstein-Uhlenbeck process:

dξ i t " κ i pθ i ´ξi t qdt `σi dW i t , i " 1, 2,
where pW 1 , W 2 q J is a two-dimensional Brownian motion with correlation ρ. The driving process X in (5.2) is specified as

X t " ˆt, ż t 0 ξ 1 s ds, ξ 1 t , ż t 0 ξ 2 s ds, ξ 2 t ˙J .
The coefficients α i and β i , i " 0, . . . , 5 are time-homogeneous and obtained similarly as in Example 5.4 from (5.1). Note that

α 0 " ¨0 0 0 0 0 0 0 0 0 0 0 0 σ 2 1 0 ρσ 1 σ 2 0 0 0 0 0 0 0 ρσ 1 σ 2 0 σ 2 2 ‹ ‹ ‹ ‹ ‹ '
.

The coefficients ϕ 1 pt, T, 0q, . . . , ϕ 3 pt, T, 0q are chosen as in Example 5.4, while ϕ 4 pt, T, 0q " ϕ 5 pt, T, 0q " 0. We note that f pt, t, 0q " ξ 1 t and set r t " f pt, t, 0q. Moreover, we choose ϕ 2 pt, T, δq " ϕ 3 pt, T, δq " 0 and φ5 pt, T, δq " 1 κ 2 ´1 ´e´κ 2 pT ´tq ¯.

Now, choose pψ δ t q J " p0, 1, 0, ´1, 0q, so that ϕ 1 pt, T, δq and ϕ 4 pt, T, δq can be calculated from φ5 pt, T, δq by means of the drift condition (5.7). At this stage, the model is fully specified. It is not difficult to verify that we are in the affine framework computed in detail in Section 4.2 of [START_REF] Brigo | Interest Rate Models -Theory and Practice[END_REF], where explicit expressions for bond prices may be found. Moreover, we obtain f pt, t, δq " ξ 2 t " X 5 t and condition (ii) (and (iii), trivially) from Proposition 5.2 is satisfied. Condition (i) also holds: in this regard, note that

pψ δ t q J ˜β0 `5 ÿ i"1 X i t β i ¸" pψ δ t q J ¨1 X 3 t κ 1 θ 1 ´κ1 X 3 t X 5 t κ 2 θ 2 ´κ2 X 5 t ‹ ‹ ‹ ‹ ‹ ' " f pt, t, 0q ´f pt, t, δq.
Since all conditions of Proposition 5.2 are now satisfied, we can conclude that the model is free of arbitrage.

Example 5.7 (A multi-curve Vasiček specification with discontinuities). We extend the previous example by allowing for discontinuities, which can be of type I as well as of type II (see Section 1.2) and can have a different impact on the OIS and on the Ibor curves.

As in Example 5.6, we consider a two-dimensional Gaussian Ornstein-Uhlenbeck process:

dξ i t " κ i pθ i ´ξi t qdt `σi dW i t , i " 1, 2.
The driving process X in (5.2) is enlarged as follows:

X t " ˆż t 0 ηpdsq, ż t 0 ξ 1 s ds, ξ 1 t , ż t 0 ξ 2 s ds, ξ 2 t , ż t 0 J s ds, J t ˙J ,
where the process J is defined as

J t " ÿ T i ďt i e ´κ3 pt´T i q , t ě 0,
for some κ 3 ě 0. A large value of κ 3 corresponds to a high speed of mean-reversion in J and generates a spiky behavior, corresponding to discontinuities of type II (recall Figure 3). On the contrary, a small value of κ 3 generates long-lasting jumps, which are in line with discontinuities of type I. For simplicity, the random variables p i q iě1 are i.i.d. standard normal, independent of ξ 1 and ξ 2 . The set of stochastic discontinuities is described by the time points pT n q nPN and the measure ηpduq is defined as in (3.6). The coefficients α i and β i are time-homogeneous and

β 0 " ¨1 0 κ 1 θ 1 0 κ 2 θ 2 0 0 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' , β 3 " ¨0 1 ´κ1 0 0 0 0 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' , β 5 " ¨0 0 0 1 ´κ2 0 0 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' , β 7 " ¨0 0 0 0 0 1 ´κ3 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' , β 1 " β 2 " β 4 " β 6 " 0, α 0 " ¨0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 σ 2 1 0 ρσ 1 σ 2 0 0 0 0 0 0 0 0 0 0 0 ρσ 1 σ 2 0 σ 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ '
, and α i " 0 for i " 1, ..., 7. Moreover, ż R 7 e xu,xy ν X pttu, dxq "

ÿ nPN 1 tt"Tnu exp ˆu1 `u2 7 2 ˙, u P R 7 , so that γ 0 pT n , uq " u 1 `u2 7 2 , u P R 7
and γ j pT n , uq " 0 for all j " 1, . . . , 7 and n P N.

We assume that jumps in X 0 and in the spread occur at the stochastic discontinuities pT n q nPN and are specified by

ψ J t ∆X t " ÿ nPN 1 tt"Tnu c n , pψ δ t q J ∆X t " ÿ nPN 1 tt"Tnu a n , κ 2 ´e´2κ 2 pT ´tq ´e´κ 2 pT ´tq ρσ 1 σ 2 κ 1 κ 2
´´κ 1 e ´κ1 pT ´tq ´κ2 e ´κ2 pT ´tq `pκ 1 `κ2 qe ´pκ 1 `κ2 qpT ´tq ¯, for t, T R T, p1 `aκ 3 q κ 3 ´p1 `cκ 3 qe ´κ3 pT ´tq ´p1 `aκ 3 qe ´2κ 3 pT ´tq ¯, for t P T S T, is satisfied for all n P N and T ě T n . We can conclude that the term structure is fully specified and, by Proposition 5.2, the model is free of arbitrage.

6. An FTAP for multiple curve financial markets

In this section, we characterize absence of arbitrage in a multiple curve financial market. At the present level of generality, this represents the first rigorous analysis of absence of arbitrage in post-crisis fixed-income markets.

As introduced in Definition 2.2, a multiple curve financial market is a large financial market containing uncountably many securities. An economically convincing notion of no-arbitrage for large financial markets has been introduced in Cuchiero, [START_REF] Klein | No arbitrage theory for bond markets[END_REF] under the name of no asymptotic free lunch with vanishing risk (NAFLVR), generalizing the classic requirement of NFLVR for finite-dimensional markets (see [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF] and [START_REF] Cuchiero | A convergence result for the Emery topology and a variant of the proof of the fundamental theorem of asset pricing[END_REF]). In this section, we extend the main result of [START_REF] Cuchiero | A new perspective on the fundamental theorem of asset pricing for large financial markets[END_REF] to an infinite time horizon and apply it to a general multiple curve financial market.

Since each element X P X is a semimartingale up to infinity, the limit X 8 exists pathwise and is finite. We can therefore define K 0 :" tX 8 : X P X u and C :" pK 0 ´L0 `q Ş L 8 , the convex cone of bounded claims super-replicable with zero initial capital. Definition 6.2. We say that the multiple curve financial market satisfies no asymptotic free lunch with vanishing risk (NAFLVR) if

C

Ş L 8 `" t0u, where C denotes the norm closure in L 8 of the set C.

The following result provides a general formulation of the fundamental theorem of asset pricing for multiple curve financial markets.

Theorem 6.3. The multiple curve financial market satisfies NAFLVR if and only if there exists an equivalent separating measure Q, i.e., a probability measure Q " P on pΩ, F q such that E Q rX 8 s ď 0 for all X P X .

Proof. We divide the proof into several steps, with the goal of reducing our general multiple curve financial market to the setting considered in Cuchiero, [START_REF] Klein | No arbitrage theory for bond markets[END_REF].

1) In view of Remark 6.1, it suffices to consider FRA contracts with fixed strike K " 0, for all tenors δ P D and settlement dates T P R `. Consequently, the parameter space I " R `ˆD 0 ˆR can be reduced to I 1 :" R `ˆt0, 1, . . . , mu, which can be further transformed into a subset of R `via I 1 Q pT, iq Þ Ñ i `T {p1 `T q P r0, m `1q ": J .

2) Without loss of generality, we can assume that pX 0 q ´1Π FRA p¨, T, δ, 0q is a semimartingale up to infinity, for every T P R `and δ P D 0 . Indeed, let n P N and A P J n . Similarly as in the proof of (Cherny and Shiryaev, 2005, Theorem 5.5), for each i " 1, . . . , n, there exists a deterministic function K i ą 0 such that pK i q ´1 P LpS i q and Y i :" pK i q ´1 ¨Si P S. Setting Y A " pY 1 , . . . , Y n q, the associativity of the stochastic integral together with (Cherny and Shiryaev, 2005, Theorem 4.2) allows to prove that X A 1 " φ ¨Y A : φ P L 8 pY A q, φ 0 " 0 and pφ ¨Y A q t ě ´1 a.s. for all t ě 0 ( .

Henceforth, we shall assume that S A P S, for all A P J n and n P N.

3) For t P r0, 1q and u P r0, `8q, let αptq :" t{p1´tq and βpuq :" u{p1`uq. The functions α and β are two inverse isomorphisms between r0, 1q and r0, `8q and can be extended to r0, 1s and r0, `8s. For A P J n , n P N, let us define the process S A " pS A t q tPr0,1s by S A t :" S A αptq , for all t P r0, 1s. Since S A P S, the process S A is a semimartingale on pΩ, F, Pq. Let θ P L 8 pS A q. We define the process θ " pθ t q tPr0,1s by θ t :" θ αptq , for all t ă 1, and θ 1 :" 0. As in the proof of (Cherny and Shiryaev, 2005, Theorem 4.2), it holds that θ P LpS A q. Moreover, it can be shown that pθ ¨SA q t " pθ ¨SA q αptq , (6.1) for all t P r0, 1s. Conversely, if θ P LpS A q, the process θ " pθ t q tě0 defined by θ t :" θ βptq , for t ě 0, belongs to L 8 pS A q and it holds that pθ ¨SA q t " pθ ¨SA q βptq , for all t ě 0. Furthermore, pθ ¨SA q 8 " pθ ¨SA q 1 holds if θ 1 " 0. 4) In view of step 3), we can consider an equivalent financial market indexed over r0, 1s in the filtration F. To this effect, for each A P J n , n P N, let us define X A 1 :" θ ¨SA : θ P LpS A q, θ 0 " θ 1 " 0 and pθ ¨SA q t ě ´1 a.s. @t P r0, 1s ( and the sets

X n 1 :" ď API n X A 1 , X 1 :" ď nPN X n 1 S , X :" ď λą0 λX 1
and K 0 :" tX 1 : X P X u, where the closure in the definition of X 1 is taken in the semimartingale topology on the filtration F. Let pX k q kPN Ď Ť nPN X n 1 be a sequence converging to X in the topology of S (on the filtration F). By definition, for each k P N, there exists a set A k such that X k " θ k ¨SA k for some 1-admissible strategy θ k P L 8 pS A k q. In view of (6.1), it holds that

X k αptq " pθ k ¨SA k q t ": X k t ,
for all t P r0, 1s. Since the topology of S is stable with respect to changes of time (see Proposition 1.3 in [START_REF] Stricker | Quelques remarques sur la topologie des semimartingales[END_REF]), the sequence pX k q kPN converges in the semimartingale topology (on the filtration F) to X " X αp¨q P X 1 . This implies that K 0 Ď K 0 . An analogous argument allows to show the converse inclusion, thus proving that K 0 " K 0 . In view of Definition 6.2, this implies that NAFLVR holds for the original financial market if and only if it holds for the equivalent financial market indexed over r0, 1s on the filtration F.

5) It remains to show that, for every A P J n , n P N, the set X A 1 satisfies the requirements of [START_REF] Cuchiero | A new perspective on the fundamental theorem of asset pricing for large financial markets[END_REF], Definition 2.1). First, X A 1 is convex and, by definition, each element X P X A 1 starts at 0 and is uniformly bounded from below by ´1. Second, let X 1 , X 2 P X A 1 and two bounded F-predictable processes H 1 , H 2 ě 0 such that H 1 H 2 " 0. By definition, there exist processes θ 1 and θ 2 such that X

i " θ i ¨SA , for i " 1, 2. If Z :" H 1 ¨X1 `H2 ¨X2 ě ´1, then Z " pH 1 θ 1 `H2 θ 2 q ¨SA P X A 1 ,
so that the required concatenation property holds. Moreover,

X A 1 Ă X A 2 if A 1 Ă A 2 .
The theorem finally follows from (Cuchiero, Klein and Teichmann, 2016, Theorem 3.2). Remark 6.4. An equivalent local martingale measure (ELMM) is a probability measure Q " P on pΩ, F q such that pX 0 q ´1Π FRA p¨, T, δ, Kq is a Q-local martingale, for all T P R `, δ P D 0 and K P R. Under additional conditions (namely of locally bounded discounted price processes, see (Cuchiero, Klein and Teichmann, 2016, Section 3.3)), it can be shown that NAFLVR is equivalent to the existence of an ELMM. In general, one cannot replace in Theorem 6.3 a separating measure with an ELMM, as shown by an explicit counterexample in Cuchiero, [START_REF] Klein | No arbitrage theory for bond markets[END_REF]. However, as a consequence of Fatou's lemma, the existence of an ELMM always represents a sufficient condition for NAFLVR. Assuming that the numéraire X 0 is tradable, an ELMM corresponds to a risk-neutral measure (see Section 3), which has been precisely characterized in the previous sections of the paper. Remark 6.5. Absence of arbitrage in large financial markets has also been studied by [START_REF] Kabanov | Asymptotic arbitrage in large financial markets[END_REF] in the sense of no asymptotic arbitrage of the first kind (NAA1), which is a weaker requirement than NAFLVR, see (Cuchiero, Klein and Teichmann, 2016, Section 4). Differently from [START_REF] Kabanov | Asymptotic arbitrage in large financial markets[END_REF], we work on a fixed filtered probability space pΩ, F , F, Pq and not on a sequence of probability spaces. On the other hand, we allow for uncountably many traded assets (see Definition 2.2).

Conclusions

The aim of this paper has been to introduce stochastic discontinuities into term structure modeling in a multi-curve setup. Stochastic discontinuities are a key feature in interest rate markets and we introduced two types for the classification of these jumps. To this end, we provided a general analysis of post-crisis multiple curve markets under minimal assumptions.

Three key results have been developed in our work: first, we provide a characterization of absence of arbitrage in an extended HJM setting. Second, we provide a similar characterization for market models. Both results rely on a fundamental theorem of asset pricing for multiple curve financial markets. Third, we provide a flexible class of multi-curve models based on affine semimartingales, a setup allowing for stochastic discontinuities.

While the focus of our analysis is a fundamental treatment of pricing in multiple curve markets, it is worth emphasizing that this framework has a large potential for many other applications such as risk management, requiring further studies. In particular for the latter, a proper modeling of the market price of risk and taking macro-economic variables into account are equally important.

Appendix A. Technical results

The following technical result on ratios and products of stochastic exponentials easily follows from Yor's formula, see (Jacod and Shiryaev, 2003, § II.8.19).

Corollary A.1. For any semimartingales X, Y and Z with ∆Z ą ´1, it holds that

EpXqEpY q

EpZq " E ˜X `Y ´Z `xX c , Y c y ´xY c , Z c y ´xX c , Z c y `xZ c , Z c y `ÿ 0ăsď¨ˆ∆ Z s p´∆X s ´∆Y s `∆Z s q `∆X s ∆Y s 1 `∆Z s ˙¸.

Proof. of Lemma 3.5 Due to Assumption 3.3 it can be verified by means of Minkowski's integral inequality and Hölder's inequality that the stochastic integrals appearing in (3.8) are well-defined, for every T P R `and δ P D 0 . Let F pt, T, δq :" ş pt,T s f pt, u, δqηpduq, for all 0 ď t ď T ă `8. For t ă T , equation (3.7) implies that In (A.1), the finiteness of ş 0 f pu, u, δqηpduq follows by Assumption 3.3 together with an analogous application of ordinary and stochastic Fubini theorems.

To complete the proof, it remains to establish (3.8) for t " T P R `. To this effect, it suffices to show that ∆GpT, T, δq " ∆F pT, T, δq for all T P R `, where ∆GpT, T, δq :" GpT, T, δq ´GpT ´, T, δq, and similarly for ∆F pT, T, δq. By (Jacod and Shiryaev, 2003, Proposition II.1.17), νptT uˆEq " 0 implies that, for every T P R `, QrµptT uˆEq ‰ 0s " 0. Therefore, it holds that Qr∆GpT, T, δq ‰ 0s ą 0 only if T " T n , for some n P N. For T " T 1 , equations (A.1) and (3.7) together imply that ∆GpT 1 , T 1 , δq " V pT 1 , T 1 , δq ´f pT 1 , T 1 , δq " ´f pT 1 ´, T 1 , δq " ´F pT 1 ´, T 1 , δq " ∆F pT 1 , T 1 , δq, where the last equality follows from the convention F pT 1 , T 1 , δq " 0. By induction over n, the same reasoning yields that ∆GpT n , T n , δq " ∆F pT n , T n , δq, for all n P N. Finally, the semimartingale property of δ-tenor bond prices pP pt, T, δqq 0ďtďT follows from (A.1).

∆A δ

Tn "

ˆ1 `δLpT n , T n , δq 1 `δLpT n ´, T n , δq ˙efpTn´,Tn,0q´fpTn´,Tn,δq´∆V pTn,T n`1 ,0q ´1.

Moreover, the resulting HJM model satisfies all the conditions of Theorem 3.7.

Proof. Since the proof involves rather lengthy computations, we shall only provide a sketch.

For T P T δ , by means of Theorem 4.1 and the assumption Lpt, T, δq ą ´1{δ a.s. for all t P r0, T s, the process p1 `δLp¨, T, δqqP p¨, T `δq{X 0 is a strictly positive Q-local martingale, so that Lpt´, T, δq ą ´1{δ a.s. for all t P r0, T s and T P T δ . Let us define the process Y pT, δq " pY t pT, δqq 0ďtďT by Y t pT, δq :" S δ t P pt, T, δq{P pt, T `δq. An application of Corollary A.1, together with equation (3.3) and Corollary 3.6, yields a stochastic exponential representation and a semimartingale decomposition of the process Y pT, δq.

For the spread process S δ given in (3.3), we start by imposing H δ " 0 and L δ " 0. We then proceed to determine the processes describing the forward rates tf p¨, T, δq : T P T δ u satisfying (3.7). In view of (B.1), for each T P T δ , we determine the process bp¨, T, δq by matching the Brownian part of Y pT, δq with the Brownian part of δLp¨, T, δq, while the jump function gp¨, ¨, T, δq is obtained in a similar way by matching the totally inaccessible jumps of Y pT, δq with the totally inaccessible jumps of δLp¨, T, δq. The drift process ap¨, T, δq is then univocally determined by imposing condition (ii) of Theorem 3.7. As a next step, for each n " 1, . . . , N , the random variable ∆A δ Tn appearing in (3.3), (3.4) is determined by requiring that ∆Y Tn pT n , δq " δ∆LpT n , T n , δq.

(B.3) Then, for each n " 1, . . . , N ´1 and T P tT n`1 , . . . , T N u, the random variable ∆V pT n , T, δq is determined by requiring that ∆Y Tn pT, δq " δ∆LpT n , T, δq, (B.4) while ∆V pT n , T, δq :" 0 for T ď T n . Note that ∆V pT n , T N `1, δq " 0 for δ ‰ 0 and n " 1, . . . , N `1. At this stage, the forward rates tf p¨, T, δq : T P T δ u are completely specified.

With this specification, it can be verified that conditions (4.3) and (4.4) respectively imply that conditions (3.9) and (3.10) of Theorem 3.7 are satisfied, using the fact that Assumption 3.3 as well as conditions (3.9), (3.10) are satisfied for δ " 0 and T P T 0 by assumption. Moreover, it can be checked that, if condition (ii) of Theorem 4.1 is satisfied, then the random variables ∆A δ Tn and ∆V pT n , T, δq resulting from (B.3), (B.4) satisfy conditions (iii), (iv) of Theorem 3.7, for every n P N and T P T δ . It remains to specify the process α δ appearing in (3.4). To this effect, an inspection of Lemma 3.5 and Corollary 3.6 reveals that, since the measure η is purely atomic, the terms f pt, t, δq and f pt, t, 0q do not appear in condition (i) of Theorem 3.7 and in condition (3.11), respectively. Since (3.11) holds by assumption, α δ " 0 follows by imposing condition (i) of Theorem 3.7. We have thus obtained that the two processes `1 `δLp¨, T, δq ˘P p¨, T `δq X 0 and S δ P p¨, T, δq X 0 are two local martingales starting from the same initial values, with the same continuous local martingale parts and with identical jumps. By means of (Jacod and Shiryaev, 2003, Theorem I.4.18 and Corollary I.4.19), we conclude that (B.1) holds for all 0 ď t ď T P T δ .

We want to point out that the specification described in Proposition B.1 is not the unique HJM model which allows embedding a given market model tLp¨, T, δq : T P T δ u. Indeed, bpt, T iptq , δq and H δ t can be arbitrarily specified as long as they satisfy bpt, T iptq , δq ´Hδ t " bpt, T iptq , 0q `bpt, T iptq`1 , 0q ´δ b L pt, T iptq , δq 1 `δLpt´, T iptq , δq , together with suitable integrability requirements. An analogous degree of freedom exists concerning the specification of the functions gpt, x, T iptq , δq and L δ pt, xq. Note also that the random variable ∆A δ Tn given in Proposition B.1 can be equivalently expressed as ∆A δ Tn "

1 `δLpT n , T n , δq 1 `δLpT n´1 , T n´1 , δq P pT n , T n`1 q P pT n´1 , T n q ´1, for n " 1, . . . , N.

Figure 1 .

 1 Figure 1. Historical series of the Eonia rate, the ECB deposit facility rate, the ECB marginal lending facility rate and the ECB main refinancing operations rate from January 1999 -June 2019. Source: European Central Bank.

Figure 2 .

 2 Figure 2. Euribor -Eonia OIS Spread for different maturities (1 month to 12 months) from January 2005 -September 2018. Source: Bloomberg and European Central Bank.

Figure 3 .

 3 Figure 3. Eonia and ECB rates from January 2010 -December 2010 (left panel) and July 2015 -June 2016 (right panel). The exposed discontinuities on the left panel are of type II, while the exposed discontinuity on the right panel is of type I. Source: European Central Bank.

F

  , u, δq `µpds, dxq ´νpds, dxq ˘˙ηpduq " , u, δq `µpds, dxq ´νpds, dxq ˘ηpduq.Due to Assumption 3.3, we can apply ordinary and stochastic Fubini theorems, in the versions of Theorem 2.2 in[START_REF] Veraar | The stochastic Fubini theorem revisited[END_REF] for the stochastic integral with respect to W and in the version of Proposition A.2 in[START_REF] Björk | Towards a general theory of bond markets[END_REF] for the stochastic integral with respect to the compensated random measure µ ´ν. We therefore obtain F pt, T, δq " , T, δq `µpds, dxq ´νpds, dxq ˘´ż t 0 f pu, u, δqηpduq ": Gpt, T, δq.(A.1)

  δq} 2 ds ´ż 0 bps, T, δqdW s ´ż 0 ż

E

ḡps, x, T, δq `µpds, dxq ´νpds, dxq ż 0 ż E `e´ḡps,x,T,δq ´1 `ḡps, x, T, δq ˘µpds, dxq `ż 0 f pu, u, δqdu `ÿ nPN `e´V pTn,T,δq`f pTn,Tn,δq ´1˘1 rrTn,`8rr

  Suppose that Assumption 3.3 holds. Then Q is a risk-neutral measure with respect to X 0 if and only if, for every δ P D 0 ,

	Λps, x, T, δq :"	1 `Lδ ps, xq 1 `Lps, xq	e ´ḡps,x,T,δq `Lps, xq ´Lδ ps, xq `ḡps, x, T, δq ´1.
	Theorem 3.7. ż T	ż
		0

E

ˇˇΛps, x, T, δq ˇˇλ s pdxqds ă `8

(3.9) a.s. for every T P R `and, for every n P N and T ě T n , the random variable

1 `∆A δ Tn 1 `∆B Tn e ´şpTn,T s ∆V pTn,u,δqηpduq (3.10)

is sigma-integrable with respect to F Tn´, and the following four conditions hold a.s.:

(i) for a.e. t P R `, it holds that r t ´αδ t " f pt, t, δq ´HJ t H δ t `}H t } 2 `żE Lpt, xq 1 `Lpt, xq `Lpt, xq ´Lδ pt, xq ˘λt pdxq; (ii) for every T P R `and for a.e. t P r0, T s, it holds that āpt, T, δq " 1 2 } bpt, T, δq} 2 `bpt, T, δq J `Ht ´Hδ t żE ˆ1 `Lδ pt, xq 1 `Lpt, xq `e´ḡpt,x,T,δq ´1˘`ḡ pt, x, T, δq ˙λt pdxq;

  Remark 3.8. By considering separately the cases δ " 0 and δ P D, we can obtain a more explicit version of condition (i) of Theorem 3.7, which is equivalent to the validity of the two conditions, for every δ P D and a.e. t P R `,

δ Tn 1 `∆B Tn ´e´ş pTn,T s ∆V pTn,u,δqηpduq ´1¯ˇˇˇˇF Tn´ff " 0. r t " f pt, t, 0q `}H t } 2 `żE L 2 pt, xq 1 `Lpt, xq λ t pdxq; (3.11) α δ t " f pt, t, 0q ´f pt, t, δq `HJ t H δ t `żE L δ pt, xqLpt, xq 1 `Lpt, xq λ t pdxq.

  Tn e ´şpTn,T s ∆V pTn,u,δqηpduq`f pTn´,Tn,δq ´1¸, where in the last equality we made use of (3.7) together with the definition of the process V . The process M pT, δq " pM t pT, δqq 0ďtďT appearing in (3.13) is the local martingale M t pT, δq :"

	and pK t pT, δqq 0ďtďT is a pure jump finite variation process given by p2q
	K t pT, δq :" p2q	ÿ nPN		1 tTnďtu	˜∆A δ Tn 1 `∆B Tn `1 1 `∆B Tn	´e´V pTn,T,δq`f pTn,Tn,δq ´1∆
		B Tn 1 `∆B Tn	`∆A δ Tn 1 `∆B Tn	´e´V pTn,T,δq`f pTn,Tn,δq ´1¯"
	ÿ nPN 1 `∆B ż t ˜1 `∆A δ Tn 1 tTnďtu
		`Hδ s ´Hs ´bps, T, δq ˘dW s
		0
	`ż t	ż
		0
	ż t	ż	´e´ḡps,x,T,δq ´1 `ḡps, x, T, δq ¯µpds, dxq
	0	E
	`ż 0 ż E	Lps, xq 1 `Lps, xq	´´L δ ps, xq ´`e ´ḡps,x,T,δq ´1˘`L ps, xq ¯µpds, dxq
	`ż 0 ż E	L δ ps, xq 1 `Lps, xq	´e´ḡps,x,T,δq ´1¯µ	pds, dxq
	ż t	ż
	"		Λps, x, T, δqµpds, dxq,
	0	E

ˆż 0 k s pT, δqds `Kp1q pT, δq `Kp2q pT, δq `M pT, δq ˙, (3.13) where pk t pT, δqq 0ďtďT is an adapted process given by k t pT, δq :" α δ t ´rt ´āpt, T, δq`1 2 } bpt, T, δq} 2 `f pt, t, δq`bpt, T, δq J `Ht ´Hδ t ˘´H J t H δ t `}H t } 2 , pK p1q t pT, δqq 0ďtďT is a pure jump finite variation process given by K p1q t pT, δq :" E `Lδ ps, xq ´Lps, xq ´ḡps, x, T, δq ˘`µpds, dxq ´νpds, dxq ˘.

  δq E ˆż 0 ks pT, δqds `p K p2q pT, δq `M 1 pT, δq ˙, (3.15)where M 1 pT, δq :" M pT, δq`K p1q pT, δq`K p2q pT, δq´ż 0 `k s pT, δq´k s pT, δq ˘ds´p K p2q pT, δq (3.16) is a local martingale, p kt pT, δqq 0ďtďT is an adapted process given by kt pT, δq " k t pT, δq p2q pT, δq must be null (up to an evanescent set), being a predictable local martingale of finite variation, see(Jacod and Shiryaev, 2003, Corollary I.3.16). In particular, analyzing separately its absolutely continuous and discontinuous parts, this holds if and only if kt pT, δq " 0 outside of a set of pQbdtq-measure zero and ∆ p K

	`żE	Λpt, x, T, δq λ t pdxq
	and, in view of (He et al., 1992, Theorem 5.29), p K p2q pT, δq "	ř	nPN ∆ p K Tn pT, δq1 rrTn,`8rr p2q
	is a pure jump finite variation predictable process with
	∆ p K

p2q

Tn pT, δq " e f pTn´,Tn,δq E Q « 1 `∆A δ Tn 1 `∆B Tn e ´şpTn,T s ∆V pTn,u,δqηpduq ˇˇˇˇF Tn´ff ´1, for all n P N. If S δ P p¨, T, δq{X 0 is a Q-local martingale, then by equation (3.15) the process ş 0 ks pT, δqds`p K p2q Tn pT, δq " 0 a.s. for every n P N. Let us first consider the absolutely continuous part 0 " kt pT, δq " α δ t ´rt ´āpt, T, δq `1 2 } bpt, T, δq} 2 `f pt, t, δq `bpt, T, δq J `Ht ´Hδ t ˘´H J t H δ t `}H t } 2 `żE Λpt, x, T, δq λ t pdxq. The integral appearing in the last line is a.s. finite for a.e. t P r0, T s as a consequence of (3.9). Taking T " t leads to the requirement r t ´αδ t " f pt, t, δq ´HJ t H δ t `}H t } 2 `żE Lpt, xq 1 `Lpt, xq `Lpt, xq ´Lδ pt, xq ˘λt pdxq, for a.e. t P R `, which gives condition (i) in the statement of the theorem. In turn, inserting this last condition into the equation kt pT, δq " 0 directly leads to condition (ii).

  ). Assume furthermore that ş T Then, Q 1 is an ELMM with respect to the numéraire expp ş 0 r OIS s dsq if and only if, for every δ P D 0 ,

	ż T	ż
	0

0 ş tψps,xqPr0,1su ψ 2 ps, xq{p1 ´ψps, xqqλ s pdxqds ă `8 a.s. for all T ą 0. E ˇˇΛ ˚ps, x, T, δq ˇˇλ s pdxqds ă `8 (3.19) a.s. for every T P R `and, for every n P N and T ě T n , the random variable `1 `∆A δ Tn ˘e´ş pTn,T s ∆V pTn,u,δqηpduq is sigma-integrable under Q 1 with respect to F Tn´, and the following conditions hold a.s.: (i) for a.e. t P R `, it holds that r OIS t " f pt, t, 0q, α δ t " f pt, t, 0q ´f pt, t, δq `θJ t H δ t `żE ψpt, xqL δ pt, xqλ t pdxq;

  martingale under Q, for every T P R `and δ P D 0 . The result therefore follows by applying Theorem 3.7 with respect to X 0 :" e ,1su ψ 2 ps, xq{p1 ´ψps, xqqλ s pdxqds ă `8 a.s. together with the elementary inequality x 2 {p1 ´xq ď |x| ^x2 , for x ď 0. The process X 0 is of the form (3.1), (3.2) with

							´ş¨0 r OIS s	ds
	is a local ş 0 r OIS s	ds {Z 1 . By applying Lemma A.1,
	we obtain that
	X 0 " E	ˆż 0 ´rOIS s	`}θ s } 2 `żE	ψ 2 ps, xq 1 ´ψps, xq	λ s pdxq ¯ds `θ	¨W
			`ψ 1 ´ψ ˚pµ ´νq	`ÿ nPN	1 ´Yn Y n	1 rrTn,`8rr	˙.
	Note that sumption that ş T 0 ş	E ψ 2 ps, xq{p1 ´ψps, xqqλ s pdxqds ă `8 a.s., as a consequence of the as-ş T ş 0 tψps,xqPr0r t " r OIS t `}θ t } 2 `żE ψ 2 pt, xq 1 ´ψpt, xq λ t pdxq,
			ż T 0	ż E	ψ 2 ps, xq 1 ´ψps, xq	λ s pdxqds ă	`8

H " θ, L " ψ{p1 ´ψq and ∆B Tn " Y n {p1 ´Yn q. Since

.

  Definition 5.1. The multiple curve model is said to be affine if ˆR`ˆD Ñ R d are predictable processes such that, for every i " 1, . . . , d and T P R `,

	f pt, T, δq " f p0, T, δq	`ż t	ϕps, T, δqdX s ,	for all δ P D 0 ,	(5.2)
					0
	S δ t " S δ 0 exp	ˆż t	ψ δ s dX s ˙,	for all δ P D,	(5.3)
				0
	ψ δ P LpXq	and		ż T	|ψ δ t ||dB X,c
					0

for all 0 ď t ď T ă `8, where ϕ : Ω ˆR2

`ˆD 0 Ñ R d and ψ δ : Ω t | ă `8 a.s., for all δ P D, and, for all δ P D 0 and T P R `,

  The coefficients Apτ, uq and Bpτ, uq are the well-known solutions of the Riccati equations, such that

	E Q	" e ´şτ 0 ξsds`uξτ	ı	" e ´Apτ,uq´Bpτ,uqξ 0 ,	for τ ě 0,
	see Section 10.3.2.1 and Corollary 10.2 in Filipović (2009) for details and explicit formulae.
	The example presented here extends Example 6.15 of Keller-Ressel et al. (2018) to a fully
	specified term-structure model.
						3	¯,
						t
	while, for 0 ď t ă 1 ď T ,
	P pt, T, 0q " exp ´´ApT ´1, 0q	´A`1 ´t, ´BpT ´1, 0q	´aB
				`1 ´t, ´BpT ´1, 0q	t ´a˘X 3	`b2 2	¯.

  Therefore, condition (i) of Proposition 5.2 is satisfied by setting r t " ξ 1 t `Jt . By choosing f p0, T n , 0q " ´c2 {2 and f p0, T n , δq " ´1 2 pa ´cq 2 and calculating

			ϕ 6 pt, T, δq "	#	κ 3 p1 `aκ 3 qe ´κ3 pT ´tq , for t, T R T, 0, otherwise,
			ϕ 7 pt, T, δq "	#	p1 `aκ 3 qe ´κ3 pT ´tq , for T R T, 0, otherwise,
	and ϕ 4 pt, T, δq " κ 2 ϕ 5 pt, T, δq. With this specification, it can be checked that condition
	(ii) of Proposition 5.2 is satisfied. Furthermore, it can be verified that
	f pt, t, 0q " ξ 1 t `Jt				and	f pt, t, δq " ξ 1 t	´ξ2 t `p1 `aκ 3 qJ t .
	ż
	pTn,T s	ϕ 1 pT n , u, 0qηpduq "	´c κ 3	`e´κ 3 pT ´Tnq ´1˘`1 2κ 2 3	`e´κ 3 pT ´Tnq ´1˘2 ,
	ż
	pTn,T s	ϕ 7 pT n , u, 0qηpduq "	´1 κ 3	`e´κ 3 pT ´Tnq ´1˘,
	ż pTn,T s	ϕ 1 pT n , u, δqηpduq "	pa ´cqp1 `aκ 3 q κ 3	`e´κ 3 pT ´Tnq ´1p
							1 `aκ 3 q 2 3 2κ 2	`e´κ 3 pT ´Tnq ´1˘2 ,
	ż pTn,T s	ϕ 7 pT n , u, δqηpduq "	´p1 `aκ 3 q κ 3	`e´κ 3 pT ´Tnq ´1˘,
	we can see that condition (iii)
	´f pT n ´, T n , 0q "	´żpTn,T s	ϕ 1 pT n , u, 0qηpduq	`1 2	˜´c	´żpTn,T s	ϕ 7 pT n , u, 0qηpduq ¸2 ,
	´f pT n ´, T n , δq "	´żpTn,T s	ϕ 1 pT n , u, δqηpduq	`1 2	˜a ´c	´żpTn,T s	ϕ 7 pT n , u, δqηpduq ¸2 ,
		1 2 pa ´cq 2 ,					for t " T P T,
		0,						otherwise,
			ϕ 5 pt, T, δq "	#	´e´κ 2 pT ´tq , for t, T R T, 0, otherwise,

Note that, at the present level of generality, the rate rt does not represent a riskless rate of return.

anonymous referees, an Associate Editor and the Editor for valuable remarks that helped improving the paper. The financial support from the Europlace Institute of Finance and the DFG project No. SCHM 2160/9-1 is gratefully acknowledged.

which can be achieved by choosing ψ J t " p0, 0, 0, 0, 0, 0, cq and pψ δ t q J " p0, 0, 0, 1, 0, 0, aq.

From this specification, it follows that the spread is given by S δ t " S δ 0 exp ˆż t 0 ξ 2 s ds `aJ t ˙.

In line with Remark 3.2, the parameters c and a control the different impact of the stochastic discontinuities on the numéraire (and, hence, on the OIS curve) and on the spread (and, hence, on the Ibor curve). The functions ϕ i pt, T, 0q, for i " 1, ..., 7 and t ď T , are chosen as

´θ1 κ 1 e ´κ1 pT ´tq ´σ2 1 κ 1 pe ´2κ 1 pT ´tq ´e´κ 1 pT ´tq q, for t, T R T, ce ´κ3 pT ´tq ´1 κ 3 pe ´2κ 3 pT ´tq ´e´κ 3 pT ´tq q, for t P T S T,

ϕ 2 pt, T, 0q " ϕ 2 pt, T, δq " κ 1 ϕ 3 pt, T, 0q and ϕ 4 pt, T, 0q " ϕ 5 pt, T, 0q " 0. For ϕpt, T, δq we choose

κ 2 e ´κ2 pT ´tq ´σ2

Let pΩ, F , F, Pq be a filtered probability space satisfying the usual conditions of rightcontinuity and P-completeness, with F :" Ž tě0 F t . Let us recall that a process Z " pZ t q tě0 is said to be a semimartingale up to infinity if there exists a process Z " pZ t q tPr0,1s satisfying Z t " Z t{p1´tq , for all t ă 1, and such that Z is a semimartingale with respect to the filtration F " pF t q tPr0,1s defined by [START_REF] Cherny | On stochastic integrals up to infinity and predictable criteria for integrability[END_REF]. We denote by S the space of real-valued semimartingales up to infinity equipped with the Emery topology, see [START_REF] Stricker | Quelques remarques sur la topologie des semimartingales[END_REF]. For a set C Ă S, we denote by C S its closure with respect to the Emery topology.

We denote by I :" R `ˆD 0 ˆR the parameter space characterizing the traded assets included in Definition 2.2. We furthermore assume the existence of a tradable numéraire with strictly positive adapted price process X 0 . For notational convenience, we represent OIS zero-coupon bonds by setting Π FRA pt, T, 0, Kq :" P pt ^T, T q, for all pt, T q P R 2 `and K P R. We also set Π FRA pt, T, δ, Kq " Π FRA pT `δ, T, δ, Kq for all δ P D, K P R and t ě T `δ.

For n P N, we denote by I n the family of all subsets A Ă I containing n elements. For each A " ppT 1 , δ 1 , K 1 q, . . . , pT n , δ n , K n qq P I n , we define the collection of X 0 -discounted prices S A " pS 1 , . . . , S n q by S i :" pX 0 q ´1Π FRA p¨, T i , δ i , K i q, for i " 1, . . . , n.

For each A P I n , n P N, we assume that S A is a semimartingale on pΩ, F, Pq and we denote by L 8 pS A q the set of all R |A| -valued, predictable processes θ " pθ 1 , . . . , θ |A| q which are integrable up to infinity with respect to S A , in the sense of Definition 4.1 in [START_REF] Cherny | On stochastic integrals up to infinity and predictable criteria for integrability[END_REF]. We assume that trading occurs in a self-financing way and we say that a process θ P L 8 pS A q is a 1-admissible trading strategy if θ 0 " 0 and pθ ¨SA q t ě ´1 a.s. for all t ě 0. The set X A 1 of wealth processes generated by 1-admissible trading strategies with respect to S A is defined as

1 :" θ ¨SA : θ P L 8 pS A q and θ is 1-admissible

The set of wealth processes generated by trading in at most n arbitrary assets is given by

By allowing to trade in arbitrary finitely many assets and letting the number of assets increase to infinity, we arrive at generalized portfolio wealth processes. The corresponding set of 1-admissible wealth processes is given by X 1 :"

so that all admissible generalized portfolio wealth processes in the multiple curve financial market are finally given by X :"

Remark 6.1. The set X can be equivalently described as the set of all admissible generalized portfolio wealth processes which can be constructed in the financial market consisting of the following two subsets of assets:

(i) OIS zero-coupon bonds, for all maturities T P R `, (ii) FRAs, for all tenors δ P D, all settlement dates T P R `and strike K 1 , for some fixed arbitrary strike K 1 P R. This follows from our standing assumption of linear valuation of FRAs together with the associativity of the stochastic integral.

Appendix B. Embedding of market models into the HJM framework

The general market model considered in Section 4, as specified by equation (4.2), can be embedded into the extended HJM framework of Section 3. For simplicity of presentation, let us consider a market model for a single tenor (i.e., D " tδu) and suppose that the forward Ibor rate Lp¨, T, δq is given by (4.2), for all T P T δ " tT 1 , . . . , T N u, with T i`1 ´Ti " δ for all i " 1, . . . , N ´1. Always for simplicity, let us assume that there is a fixed number N `1 of discontinuity dates, coinciding with the set of dates T 0 :" T δ Ť tT N `1u, with T N `1 :" T N `δ. We say that tLp¨, T, δq : T P T δ u can be embedded into an extended HJM model if there exists a sigma-finite measure η on R `, a spread process S δ and a family of forward rates tf p¨, T, δq : T P T δ u such that Lpt, T, δq " 1 δ ˆSδ t P pt, T, δq P pt, T `δq ´1˙, for all 0 ď t ď T P T δ , (B.1)

where P pt, T, δq is given by (3.5), for all 0 ď t ď T P T δ . In other words, in view of equation (2.2), the HJM model generates the same forward Ibor rates as the original market model, for every date T P T δ . We remark that, since a market model involves OIS bonds only for maturities T 0 " tT 1 , . . . , T N `1u, there is no loss of generality in taking the measure η in (3.5) as a purely atomic measure of the form ηpduq "

More specifically, if OIS bonds for maturities T 0 are defined through (3.5) via a generic measure of the form (3.6), then there always exists a measure η as in (B.2) generating the same bond prices, up to a suitable specification of the forward rate process.

The following proposition explicitly shows how a general market model can be embedded into an HJM model. For t P r0, T N s, we define iptq :" mintj P t1, . . . , N u : T j ě tu, so that T iptq is the smallest T P T δ such that T ě t.

Proposition B.1. Suppose that all the conditions of Theorem 4.1 are satisfied, with respect to the measure η given in (B.2), and assume furthermore that Lpt, T, δq ą ´1{δ a.s. for all t P r0, T s and T P T δ . Then, under the above assumptions, the market model tLp¨, T, δq : T P T δ u can be embedded into an HJM model by choosing (i) a family of forward rates tf p¨, T, δq :

and satisfying equation (3.7) where, for all i " 1, . . . , N , the volatility process bp¨, T i , δq, the jump function gp¨, ¨, T i , δq and p∆V pT n , T i , δqq n"1,...,N are respectively given by bpt, T i , δq "

´log ˆ1 `δg L pt, x, T i , δq 1 `δLpt´, T i , δq ˙, if i " iptq, gpt, x, T i`1 , 0q ´log ˜1`δ g L pt,x,T i ,δq 1`δLpt´,T i ,δq 1`δ g L pt,x,T i´1 ,δq 1`δLpt´,T i´1 ,δq ¸, if i ą iptq, ∆V pT n , T i , δq " ∆V pT n , T i`1 , 0q ´log ¨1`δLpTn,T i ,δq 1`δLpTn´,T i ,δq 1`δLpTn,T i´1 ,δq 1`δLpTn´,T i´1 ,δq ', for i ě n `1, and the process ap¨, T i , δq is determined by condition (ii) of Theorem 3.7; (ii) a spread process S δ with initial value S δ 0 " `1 `δLp0, 0, δq ˘P p0, δq and satisfying (3.3), (3.4), where the processes α δ , H δ , the function L δ and the random variables p∆A δ Tn q n"1,...,N are respectively given by α δ t " 0, H δ t " 0, L δ pt, xq " 0,