Tefan Sarkadi
email: stefan.sarkadi@inria.fr

Andrea G B Tettamanzi
email: andrea.tettamanzi@unice.fr

Fabien Gandon
email: fabien.gandon@inria.fr

INTEROPERABLE AI: EVOLUTIONARY RACE TOWARDS SUSTAINABLE KNOWLEDGE SHARING *

The advancement and deployment of artificial intelligent agents brought numerous benefits in knowledge and data gathering and processing. However, one of the key challenges in deploying such agents in an open environment like the Web is their interoperability as they currently mostly run in silos. In this paper we report on a simulation and evaluation based on evolutionary agent-based modelling to empirically test how sustainable different strategies are for knowledge sharing in open multi-agent systems (MAS). Our results show the importance of translation-based approaches and the need for incentives to support these.

Related Work & Motivation

Early work on interoperability in MAS has mostly focused on the standardisation of Agent Communication Languages, like FIPA and KQML.

. These multi-agent approaches are very different from ours both conceptually and implementationwise, as they do not aim to study the economics of knowledge-sharing under evolutionary constraints.

Introduction

We are still waiting for a large scale deployment of intelligent agents on the World Wild Web [START_REF] Hendler | Where are all the intelligent agents?[END_REF]. One of the key challenges in deploying such agents in an open environment like the Web is their interoperability.

To ensure interoperability, approaches in distributed AI (MAS) range from enforcing a global schema to allowing as many schemas as existing components, as well as every possible hybrid solution in between these extremes. Smith [START_REF] Smith | Chapter 11: Ontology"Blackwell Guide to the Philosophy of Computing and Information[END_REF] hypothesized that a universal ontology is unlikely to be realizable. Inversely, as soon as more than one schema exists, the question of mapping and translating between these schemas arises, with an associated cost.

Our objective in this paper is to understand, from the perspective of economics of deployment, the advantages of choosing one approach over another according to the costs and benefits of each to reach interoperable MAS. Each approach is treated as a knowledge-sharing strategy, and more precisely a strategy for adopting one vs a strategy to adopt several ontologies. The simulation remains at the level of strategies and does not consider specific ontologies. We define knowledge-sharing games powered by an evolutionary mechanism for social learning that selects agents of a population to interact over long periods of time.

All these approaches share a similar conceptual foundation for ensuring interoperability, namely that two interacting agents share or produce a common representation of the world [START_REF] Gruber | A translation approach to portable ontology specifications[END_REF]. Some argue that if a common ontology had been established for artificial agents to interact on the Web, then we would have seen those seamless interactions. But, perhaps Smith [START_REF] Smith | Chapter 11: Ontology"Blackwell Guide to the Philosophy of Computing and Information[END_REF] was right suggesting that such a universal ontology is unlikely to be realized. Alternatively, it might be the case that mediation services could instead be more sustainable to overcome the challenge of interoperability.

To test Smith's intuition, we do not engineer a specific MAS. Instead, we derive our approach from the evolutionary agent-based modelling of cooperation, communication, and knowledge-sharing in complex social systems as the ones in [START_REF] Nowak | The evolutionary language game[END_REF][START_REF] Sarkadi | The evolution of deception[END_REF][START_REF] Čače | Agent based modelling of communication costs: Why information can be free[END_REF]. None of these previous studies have looked at the question of shared language vs. translation strategies in the context of ensuring sustainable communication. This is exactly the research question we set for this paper.

Research Questions & Method

We know that the diversity of resources in modern infospheres like the Web calls for artificial ecosystems with a diversity of interacting agents ranging from reactive to deliberative paradigms, maintaining the information ecology [START_REF] Gandon | Combining reactive and deliberative agents for complete ecosystems in infospheres[END_REF] and supporting their hybrid communities in sustainable ways. We also know that all approaches in AI come with associated costs of resources and maintenance, from classical knowledge representation methods such as ontologies [START_REF] Gómez-Pérez | Ontological Engineering: with examples from the areas of Knowledge Management, e-Commerce and the Semantic Web[END_REF] to latest machine learning ones.

In particular, alternative agent communication strategies can differ in terms of software maintenance and in terms of costs incurred to store and share knowledge.

At the two extremes, the burden could be put either on the agents to pay the price of having different ontologies when they need to interact, or on a shared ontology provider to ensure a lingua franca. Each extreme has its costs and benefits and there is a continuum of hybrid options in-between. For instance, letting agents have their own ontologies means they will pay the cost of whatever transformation they have to perform when they communicate, whereas maintaining a shared ontology comes with the price of achieving consensus, commitment and versioning every time it needs to be changed. But how sustainable would these different strategies be from an economy of knowledge perspective?

In terms of economy of knowledge sharing, the Web can be seen as an open MAS, where knowledge is mostly public but can also be private or restricted to a group or even a couple of users or agents.We can also say that Web agents that contribute to Wikipedia are building a public good. The over-arching goals of Web agents would be to share as much knowledge as necessary to ensure they achieve their goals without disclosing their entire knowledge base (there might be some knowledge that needs to remain private). From an evolutionary perspective, this search for interoperability makes Web agents compete with each other, and assuming that the knowledge shared is relevant, the agents with the fittest strategies are able to share the most knowledge.

One way to identify more 'sustainable' knowledge-sharing is to allow agents to optimise between the strategies that they choose to share knowledge and the cost of doing so in various contexts. If these agents were able to choose between different knowledge sharing strategies, and able to learn from their peers the costs and benefits of these strategies, then how would such a society of agents behave in the long-run? What kind of strategy would they choose, from an evolutionary perspective? If they had the option to choose a common universal ontology as a knowledge sharing strategy, would these 'sustainability-aware' agents choose it over other strategies? As a result, We have formulated a set of research questions to test the different alternatives:

1. Does the choice of a common universal ontology emerge as an evolutionary stable strategy in populations of self-interested knowledge-sharing agents that learn from each other through imitation?

2. Do voluntary interactions between agents of a population have an impact on the choice of strategy?

3. Does the number of agents actively engaged in knowledge-sharing between themselves affect the choice of strategy?

4. Do mediating agents/services (that translate knowledge between agents) play a role in the sustainability of such systems?

We answer these questions empirically by performing extensive simulations with evolutionary agent-based models.

Agent-Based Modelling of Knowledge Sharing

We use the word strategy, to refer to a strategy for sharing knowledge the context of an evolutionary game. Agents are able to change their strategies by switching to other existing strategies through social learning. We consider that strategies for knowledge sharing correspond to strategies for choosing one or more languages, or one or more ontologies for knowledge-sharing in MAS. We will use language and ontology interchangeably in this context of the knowledge sharing games and we will not be considering a specific language or ontology, only global strategies.

Evolutionary Knowledge-Sharing Games

When sharing knowledge publicly, e.g. on the Wild Web, it is reasonable to assume that self-interested intelligent agents would do so rationally and strategically. In particular, they take into account the existing incentives for sharing knowledge as well as what knowledge to keep private to preserve some of their advantages. The Knowledge-sharing Game (KSG) is a race between a set of knowledge sharing strategies that pushes agents to pick the best one. Notably, KSG optimises cooperation. What the agents in the KSG are actually doing is to compete with each other to find the best cooperative strategy, and even though KSG is a competitive game, it is actually a race between agents towards selecting the best mechanism for cooperative knowledge-sharing.

We define a KSG as a tuple ⟨S, N, M, c, r, θ, s, µ, α, β⟩, whose parameters are defined in Table 1.

P Value Definition S ̸ = ∅
a non-empty set of strategies N 100 the number of agents in a population to play a KSG N Si ≤ N the number of agents using strategy S i ∈ S M 2 or 5 the number of agents selected to play the KSG at a given iteration from a population of N k 10 the entire amount of knowledge of any agent q 2 the amount of knowledge every agent decides to keep private c = k-q 8 the amount of knowledge that every agent contributes to a KSG r 1.5 a multiplication factor that acts as an incentive for sharing a particular amount of knowledge c θ ∈ [0, 1] 0.5 the compatibility rate between any two or more agents that interact s ≥ 0 10 3 the social learning (imitation) strength µ ≥ 0 0.001 the exploration rate i.e. the inclination of agents to randomly adopt another strategy ϵ 2 the cost of using one own's ontology/language α 2 the cost of translating knowledge to another ontology/language β 2 the cost for an agent of adopting a common ontology/language T 10 4 number of iterations of a KSG Table 1: KSG parameters.

For each strategy, a payoff is computed that takes into account the costs and benefits of choosing the respective strategy. We explore the changes in frequencies over time of agents who are able to learn strategies for knowledge sharing from other agents. For each KSG, we perform explicit computations of what payoffs the agents will receive given a sub-population that is selected to play the game at each iteration. The relative differences between the payoffs obtained by the agents with different strategies in the sub-population determine the probability that an agent will adopt a different strategy.

Social learning s represents the tendency of an agent that is selected to adopt a strategy that, compared to its current strategy, maximises the agent's payoff. The higher the value of s, the stronger the tendency of adopting the better strategy. If s = 0, this gives a random outcome, while if s ∞ -→ this will always make the agent select the stronger strategy. The exploration rate µ represents random mistakes in actions as well as purposeful exploration regardless of relative payoffs. While µ represents the probability of an agent to randomly adopt any other strategy, the probability of an agent i with a payoff P i to switch to another agent's strategy j with the payoff P j is given by

P ij = 1 1+exp[-s(Pj -Pi)] .
This stochastic approach allows us to dynamically represent how the frequencies of the different types of agents evolve over time.

We assume that the incompatibility between two or more agents that interact can vary, hence the compatibility label between M agents that interact θ ∈ [0, 1] and the compatibility difference given by 1 -θ. For some strategies, the

Strategy

Payout & Cost Own Language O: The agents share knowledge in their own language only.

Payout: c × r × θ -c Cost: ϵ Common Language CL: The agents communicate in a common language shared by all the agents in a population. The common language has an exponential cost of maintenance.

Payout: c × r -c Cost: β N Common Language v2 CL 2 : idem but the cost of maintaining the common language is proportional to the number of agents in the system that actually choose to use it.

Payout: c × r -c Cost: β × N CL2
Translation T R: The agents choose to pay a cost for translating the knowledge that they wish to share with other agents.

Payout: c × r -c Cost: α × c
Translation Per Knowledge T R 2 : idem but agents only pay for the translation of knowledge that they do not understand i.e. for the exact pieces of knowledge for which they do not share exactly the same ontologies.

Payout: c × r -c Cost: (1 -θ) × α × c
Hybrid H: Whenever possible, agents communicate in a common language the knowledge that they wish to share. For other pieces of knowledge they pay the cost for translation.

Payout: c × r -c Cost: α × c × (1 -θ) + θ × β N CL +N H Hybrid 2 H 2 :
The agents that communicate in a partially known common language that is maintained for agents that actually choose to use it. For knowledge that they cannot express in the common language, they pay the cost of translating that knowledge, similarly to T R 2 .

Payout: c × r -c Cost: α×c×(1-θ)+θ×β×(N CL2 +N H2 )
*Loner L: The agents that choose not to interact in the KSG. We call the agents that choose no interaction a Loner agent. The role of the Loner is to give a chance to other strategies to invade the population.

The payout and costs of the Loner always amount to a fixed payoff L = σ Table 2: Different strategies for knowledge sharing with their payout and cost. All parameters were defined in Table 1. compatibility between agents determines how much knowledge can be exchanged without additional cost. For an agent using a translation service the cost α of communicating knowledge is multiplied by c, which measures the amount of knowledge it intends to share while β is the cost for the adoption and maintenance of a common language (ontology).

Strategies for Knowledge Sharing

In Table 2 we define 7 strategies that agents can use to play a KSG, and one strategy (Loner) for agents that choose not to play the KSG in voluntary games. These strategies are selected through mutation or social learning at every iteration of a KSG, as described in the previous section.

Additionally, KSGs can be played under two mutually-exclusive conditions regarding interactions. The first condition is that interactions between agents of a population are assumed to be compulsory. KSGs played under this condition are called Non-Voluntary. Each agent that plays the game receives a reward (a payout) which is multiplied by (M -1) (the number of other sampled agents playing the same game), whereas the costs are proportional to the fraction of other sampled agents playing the game from the entire population M -1 N -1 . The second condition relaxes this assumption and agents are allowed not to participate in KSGs. Hence the latter KSGs are called Voluntary. In this paper we simulate KSGs under both of these conditions. To do this, we have defined two different payoffs for each strategy, found in Table 4-Appendix.

While the payoffs for non-voluntary KSGs follow straight from the strategies defined in Table 2, voluntary KSGs require an additional strategy. A voluntary KSG means that there is a probability that the agents are not willing to exchange knowledge with others in the population. To model this phenomenon, we introduce an additional strategy to the KSG, namely, the Loner strategy L.

To compute the payoffs of the other strategies in voluntary KSGs, we need to take into account the probability that all other M -1 sampled individuals are Loners, namely:

Pσ = NL!(N -M )! (NL -M + 1)!(N -1)! = N L M -1 N -1 M -1 (1)
Table 3: Experimental Setups. These interactions happen under different conditions identified by the prefix "P" if they are public and/or suffix "-V" if they are voluntary (agents have the possibility to choose not to interact at all).

Type: KSG1 Type: KSG2 Type: KSG*

KSG1, KSG1-V, PKSG1, PKSG1- V KSG2, KSG2-V, PKSG2, PKSG2- V KSG*, KSG*-V,
T R > CL, CL > T R T R 2 > CL 2 , CL 2 > T R 2 CL 2 > CL, CL > CL 2 , T R 2 > T R, T R > T R 2 , T R 2 > CL 2 , CL 2 > T R 2
Verified Hypotheses: Verified Hypotheses: Verified Hypotheses:

T R > CL T R 2 > CL 2 CL 2 > CL, T R 2 > T R, T R 2 > CL 2
where N L is the number of Loners in the population.

The other 7 strategy payoffs are adjusted in order to take into account the likelihood of sampling a Loner from the agent population pool when two or more agents interact in knowledge sharing games. This is not the case in the non-voluntary model, where there are no potential non-participants to the KSG.

Experimental Design

The name of each experiment encodes its configuration. First, we separate two sets of experiments corresponding to two types of KSGs designed to test the impact of private vs. public knowledge exchange as these two settings exist on the Web. The first set (prefixed by "KSG") explores the direct competition between 2 agents that share knowledge between themselves in a private way. The second set (prefixed by "PKSG") explores the idea of knowledge exchanged publicly.We have set the number of agents that interact in these games M = 5 following the study presented in [START_REF] Sarkadi | The evolution of deception[END_REF], which uses similar evolutionary agent-based modelling techniques to ours, based on [START_REF] Nowak | The evolutionary language game[END_REF][START_REF] Sigmund | Social learning promotes institutions for governing the commons[END_REF]. If we consider that knowledge is shared publicly (e.g., on the Web) in domains where agents interact in groups, a first reasonable setup is to adopt the same value for number of agents engaged in these interactions.

The KSG contribution c was given the same arbitrary integer value for all strategies in order not to affect the strategy payoffs, and r was given the value of 1.5 to represent a 50% return on the contribution. The number of iterations T = 10 4 was set to show the effect of a selecting a strategy over great periods of time, whereas the population size N = 100 was selected to represent a relatively large and open MAS, but for which we also perform a sensitivity analysis to show its effect of even larger MAS in the Appendix. We adopt the fixed intermediate value for social learning, s = 10 3 , which was picked with the purpose to represent a relatively strong capability of agents to adopt other strategies without assuming that agents are always successful at it. Contrastingly the value of 0.001 was picked for the exploration rate µ to represent a low tendency to explore. For both s and µ we perform a sensitivity analysis in the Appendix. θ was given the value 0.5 to represent a 50% chance (coin flip) of sharing knowledge successfully with another agent when both agents use their own language. ϵ, α, β were given the same fixed integer value of 2 to isolate their effect on the payoffs, but for which we also provide a sensitivity analysis in the Appendix. Loner payoff σ was adopted from [START_REF] Sigmund | Social learning promotes institutions for governing the commons[END_REF] with the value 1.

Both pairwise and public interactions are experimented in non-voluntary and voluntary games, the latter being identified by the "-V" suffix. Moreover, there are two groups of assumptions about Common Language and Translation that we wish to explore. These assumptions correspond to the underlying language strategy assumptions that correspond to KSG1 (exponentially costly common ontology) and KSG2 (where a common ontology has a cost proportional to the number of agents that adopt it), respectively. KSG* represents an additional experiment designed to compare the two groups of underlying assumptions by putting them in the same competition.

Table 3 summarizes the different combinations of strategies and experimental conditions. In total, we have 12 experiments summarized and, for instance, the name "PKSG1-V" identifies a Knowledge Sharing Game where interactions are voluntary, public and based on the KSG1 assumptions. The games, strategies and hypotheses we will now introduce and test are summarised in Table 3 and the parameters are in Table 1.

Hypotheses

In order to test the sustainability of each strategy we formulate sets of hypotheses for each type of KSG, in pairs of the form X > Y that means that the long-run avg. frequency of strategy X is greater than that of strategy Y . What interpret this as X is more sustainable than Y (see Table 3).

Results

For each type of KSG we present the statistical results of long-run average frequencies for 100 independent runs of 10 4 iterations. The first run starts from a population of agents that use the Own Language (O) strategy. Each subsequent run starts from the population distribution generated at the end of the previous run. All the results and the sensitivity analysis for PKSG*-V are included in the Appendix (Supplementary Material). They are presented as means, standard deviations, t-tests, and One-way ANOVA results in Tables. Significant results where p < 0.01 are represented by *. The results are also visually presented in figures, where the average frequencies of strategies are presented with variance bars (that represent one standard deviation from the mean). Additionally, we plot the dynamics of a single individual-based simulation run for each KSG, where each coloured line represents the frequency of a given strategy at a given time t.

We report on each test we performed and its results in the rest of this section.

CL 2 > CL, T R 2 > T R, T R 2 > CL 2 .)
What have we shown? Generally, mediation strategies in open systems in which either voluntary and non-voluntary interactions take place, are more sustainable than the implementation of common ontologies for knowledge sharing in pairwise interactions between rational social learning agents.

CL 2 > CL, T R 2 > T R, T R 2 > CL 2 .)
What have we shown? Similarly to the experimental conditions from pairwise interactions, mediation strategies in open systems in which either voluntary and non-voluntary interactions take place, are more sustainable than the implementation of common ontologies for knowledge sharing in public interactions between rational social learning agents.

Discussion

In this paper we have studied the sustainability of different knowledge-sharing strategies from an evolutionary social learning perspective by running extensive individual agent-based simulations. To do so, we have defined a set of hypotheses about the sustainability of different strategies for knowledge sharing and we have designed 12 knowledgesharing games to test these hypotheses. The previous section summarized the results of the games vs our hypotheses.

From these results we showed that under the assumption of cooperation and the possibility of decentralisation, translation strategies in open MAS in which either voluntary or non-voluntary interactions take place, are more sustainable than shared ontology strategies for both public and private knowledge sharing between rational agents. We also showed that Translation 2 (where agents pay only for the items of knowledge that need mediating) is more sustainable than Translation (where agents pay for always using mediation services) in all configurations. In other words, we observed that the translation strategies outperform common language strategies in terms of sustainable knowledge sharing. We also observe some niche cases, however unlikely these might be, where common-language strategies with non-exponential costs are more sustainable than translation strategies. Such niche cases can arise for both voluntary and non-voluntary interaction conditions (illustrated in KSG* Fig. 4c and KSG*-V Fig. 4d), as well as in cases of group interactions (illustrated in PKSG* Fig. 7c and PKSG*-V 7d). This could suggest the possibility of having different organizations in a MAS on the Web with different strategies (centralised and decentralised), for instance at different scales of social structures, e.g. a very small local sub group with a shared ontology (e.g. a factory room) vs a large and distributed group (e.g. a B2C or B2B network). Another observation we can make is that a distributed system of strategies emerges (CL 2 + T R 2 ) instead of a single hybrid strategy (H 2 ). Intelligent agents seem to prefer separate systems of communication to run in parallel within the same population.

We must note that our results are only valid for cooperative systems, where the goal of the agents is to optimise knowledge-sharing with other agents -to select the best strategy for cooperation. In specific cases, centralisation CL 2 can be the optimal strategy, for instance in an IoT factory where a single agent dictates the language that must be used for inter-operation. For such a group of agents that use the same strategy but in a larger MAS, they can be collapsed into a single agent. Another example from the real world is when a certain entity, e.g. Google or Apple dictate what kind of software code must be implemented by independent software developers in order for their apps to run on iOS or 'play nicely' with Google services. But in its current state, Web 2.0 is not a good example of a decentralised open MAS because it is a collection of silo-ed and centralised systems.

When we designed the KSGs, we had in mind the possibility of agents to self-organise as truly decentralised distributed systems with independent autonomous agents that interact and self-organise in order to share knowledge. As future work, our KSG model opens the path towards studying the interoperbility of autonomous and self-organising AI agents on existing graphs-based structures of the Web.

In a nutshell, assuming cooperation in knowledge-sharing and the possibility of a fully decentralized Web of resources and services, our results provide reasons for adopting translation strategies. However we still have a number of challenges to address even just for interoperability. We are in a chicken-and-egg problem where we don't have translation services because there is no demand, no existing agent in the wild, and inversely we have no agents in the wild because, among other problems, they have no support to interoperate. By extension, finding incentives for translation services to be provided and maintained is problematic. The question includes the study of business models for translators or opportunities for saving costs in IT systems, e.g., by supporting the outsourcing of some IT services. Classic business models on the Web include advertisement, but this kind of funding model doesn't apply to interactions between software agents. Clearly, to us, the question of incentive and business models for translation-based interoperability appears to be on the critical path of achieving MAS on the Web.

A Supplementary Material -Results & Data Analysis

Payoff in Non-Voluntary Model Payoff in Voluntary Model

O = (M -1) (c × θ × r -c) -ϵ M -1 N -1 (2) O * = Pσ × σ + (1 -Pσ)(N -NL -1) (c × θ × r -c) -ϵ M -1 N -1 (3)
CL = (M -1) (c × r -c) -β N M -1 N -1 (4) CL * = Pσ × σ + (1 -Pσ)(N -NL -1) (c × r -c) -β N M -1 N -1 (5)
CL2 = (M -1) (c × r -c) -β × NCL 2 M -1 N -1 (6) CL * 2 = Pσ × σ + (1 -Pσ)(N -NL -1) (c × r -c) -β × NCL 2 M -1 N -1 (7)
T R = (M -1) (c × r -c) -α × c M -1 N -1 (8) T R * = Pσ × σ + (1 -Pσ)(N -NL -1) (c × r -c) -α × c M -1 N -1 (9)
T R2 = (M -1) (c × r -c) -(1 -θ) × α × c M -1 N -1 (10)
T R * 2 = Pσ × σ + (1 -Pσ)(N -NL -1) (c × r -c) -(1 -θ) × α × c M -1 N -1 (11)
H = (M -1) (c × r -c) -α × c × (1 -θ) + θ × β (N CL +N H ) M -1 N -1 (12)
H * = Pσ × σ + (1 -Pσ)(N -NL -1) (c × r -c) -α × c × (1 -θ) + θ × β (N CL +N H ) M -1 N -1 (13) H2 = (M -1) (c × r -c) -(α × c × (1 -θ) +θ × β × (NCL 2 + NH 2 ))) M -1 N -1 (14)
H * 2 = Pσ × σ + (1 -Pσ)(N -NL -1) (c × r -c) -(α×c×(1-θ) + θ×β × (NCL 2 + NH 2 )) M -1 N -1 (15)
Table 4: Different payoffs in non-voluntary and voluntary models combining costs and payouts from Table 2 and parameters defined in Table 1. On the left-hand side of the table we can find the payoffs for Non-Voluntary KSGs, while on the right-hand side we find the payoffs for the Voluntary KSGs.

6. 1

 1 Pairwise Interactions two at-a-time M=2 KSG1 and KSG2 ( Data: Tab. 5a and Tab. 5b; Fig.2c and Fig.2d -Confirmed Hypotheses: T R > CL, T R 2 > CL 2 .) KSG1-V and KSG2-V (Data: Tab. 6a and Tab. 6b; Fig.3c and Fig.3d -Confirmed Hypotheses: T R > CL, T R 2 > CL 2 .) KSG* and KSG*-V (Data: Tab. 7a and Tab. 7b; Fig.4c and Fig.4d -Confirmed Hypotheses:

6. 2

 2 Public Interactions five at-a-time M=5 PKSG1 and PKSG2 (Data: Tab. 8a and Tab. 8b; Fig.5c and Fig.5d -Confirmed Hypotheses: T R > CL, T R 2 > CL 2 .) PKSG1-V and PKSG2-V (Data: Tab. 9a and Tab. 9b; Fig.6c and Fig.6d --Confirmed Hypotheses: T R > CL, T R 2 > CL 2 .) PKSG* and PKSG*-V (Data: Tab. 10a and Tab. 10b; Fig.7c and Fig.7d -Confirmed Hypotheses:

  (a) Long-run average frequencies for PKSG1-V. (b) Long-run average frequencies for PKSG2-V. (c) Long-run average frequencies for PKSG*-V.

Figure 1 :

 1 Figure 1: Long-run average frequencies for a sample of 100 individual-based runs in public and voluntary interactions.

  (a) Individual-based simulation run Dynamics for KSG1-V. (b) Individual-based simulation run Dynamics for KSG2-V. (c) Long-run average frequencies for KSG1-V. (d) Long-run average frequencies for KSG2-V.

Figure 3 :

 3 Figure 3: Population dynamics for voluntary interactions in KSG1-V fig.3a vs KSG2-V fig. 3b. Long-run average frequencies for a sample of 100 individual-based runs are also reported for KSG1-V in fig. 3c and for KSG2-V in fig. 3d.

Figure 4 :

 4 Figure 4: Population dynamics for interactions in KSG* fig. 4a vs KSG*-V fig.4b. Long-run average frequencies for a sample of 100 individual-based runs are also reported for KSG* in fig. 4c and for KSG*-V in fig. 4d.

  (a) Individual-based simulation run Dynamics for PKSG1-V. (b) Individual-based simulation run Dynamics for PKSG2-V. (c) Long-run average frequencies for PKSG1-V. (d) Long-run average frequencies for PKSG2-V.

Figure 6 :

 6 Figure 6: Population dynamics for voluntary interactions in PKSG1-V fig.6a vs PKSG2-V fig. 6b. Long-run average frequencies for a sample of 100 individual-based runs are also reported for PKSG1-V in fig. 6c and for PKSG2-V in fig. 6d.

  (a) Individual-based simulation run Dynamics for PKSG*. (b) Individual-based simulation run Dynamics for PKSG*-V. (c) Long-run average frequencies for PKSG*. (d) Long-run average frequencies for PKSG*-V.

Figure 7 :Figure 10 :

 710 Figure 7: Population dynamics for interactions in PKSG* fig. 7a vs PKSG*-V fig.7b. Long-run average frequencies for a sample of 100 individual-based runs are also reported for PKSG* in fig. 7c and for PKSG*-V in fig. 7d

Figure 11 :

 11 Figure 11: 11a Long-run avg. frequencies and effects of 11b own language cost, 11c common language cost, and 11d translation cost, θ, β, α ∈ [10 -3 , 10 3 ] in PKSG*-V when every run starts from an equal distribution of strategies. Sample size 100 individual runs. Lines represents means and shaded areas represend ±1 SD from the mean.

  Figure 12: Effect of social learning s ∈ [10 -4 , 10 3 ] on 12a T R 2 and 12b CL 2 in PKSG*-V when every run starts from an equal distribution of strategies. Sample size 100 individual runs. Lines represents means and shaded areas represend ±1 SD from the mean.

  (a) Effect of agent compatibility θ on O. (b) Effect of agent compatibility θ on T R2.

Figure 13 :

 13 Figure 13: Effect of agent compatibility θ ∈ [0, 1] in PKSG*-V when every run starts from an equal distribution of strategies. Sample size 100 individual runs. Lines represents means and shaded areas represend ±1 SD from the mean.

Table 5 :

 5 Means, standard deviations, t-test, and One-way ANOVA results for KSG1 in Tab. 5a vs KSG2 in Tab. 5b. Significant results where p < 0.01 are represented by *.

	(a) One-way ANOVA result (statistic=1884732.98, pvalue=0.0).
	T-tests and standard deviations in-table.
		CL	T R	H	O
	mn, sd .0; .0 .96; .004	.03; .0	.0; .005
	CL	N/A	-2044.75* -658.75* -0.97
	T R	N/A	N/A	1949.60* 1394.90*
	H	N/A	N/A	N/A	74.88

Table 6 :

 6 Means, standard deviations, t-test, and One-way ANOVA results for KSG1-V in Tab. 6a vs KSG2-V in Tab. 6b. Significant results where p < 0.01 are represented by *.

	(a) One-way ANOVA result (statistic=4937.86, pvalue=0.0).
	T-tests and standard deviations in-table.
		CL	T R	H	O	L
	mn, sd .0; .0 .95; .09 .04; .0	.0; .09 .0; .0
	CL	N/A	-98.86* -112.39* -1.04	-0.09
	T R	N/A	N/A	94.76*	70.42* 98.86*
	H	N/A	N/A	N/A	3.14*	112.39*
	O	N/A	N/A	N/A	N/A	1.04

Table 8 :

 8 Means, standard deviations, t-test, and One-way ANOVA results for PKSG1 in Tab. 8a vs PKSG2 in Tab. 8b. Significant results where p < 0.01 are represented by *.

Table 9 :

 9 Means, standard deviations, t-test, and One-way ANOVA results for PKSG1-V in Tab. 9a vs PKSG2-V in Tab. 9b. Significant results where p < 0.01 are represented by *.

	(a) One-way ANOVA result (statistic=2873.48, pvalue=0.0).
	T-tests and standard deviations in-table.
		CL	T R	H		O	L
	mn, sd .0; .0 .94; .12 .04; .009 .017; .12 .0; .0
	CL	N/A	-76.38* -41.78*	-1.4	-1.48
	T R	N/A	N/A	73.02*	53.46*	76.38*
	H	N/A	N/A	N/A		1.8	41.75*
	O	N/A	N/A	N/A		N/A	1.39
	(b) One-way ANOVA result (statistic=28446.99, pvalue=0.0).
	T-tests and standard deviations in-table.
		CL 2	T R 2		H 2		O	L
	mn, sd .039; .005 .96; .04	.0; .03 .0; .01	.0;.0
	CL 2	N/A	-205.61* 10.5*	51.3*	70.72*
	T R 2	N/A	N/A		171.4* 214.53* 213.07*
	H 2	N/A	N/A		N/A		0.23	0.99
	O	N/A	N/A		N/A		N/A	0.98

Table 10 :

 10 Means, standard deviations, t-test, and One-way ANOVA results for PKSG* in Tab. 10a vs PKSG*-V in Tab. 10a. Significant results where p < 0.01 are represented by *. (a) One-way ANOVA result (statistic=781654.67, pvalue=0.0). T-tests and standard deviations in-table.

		CL	CL 2	T R	T R 2	O
	mn, sd .0; .0 .038; .0	.0; .007 .96; .007	.0;.007
	CL	N/A	-719.52* 0.25	-1297.13* -0.97
	CL 2	N/A	N/A	718.95* -1242*	48.54*
	T R	N/A	N/A	N/A	-1297*	-0.97
	T R 2	N/A	N/A	N/A	N/A	893.06*
	(b) One-way ANOVA result (statistic=343995.38, pvalue=0.0).
	T-tests and standard deviations in-table.
		CL	CL 2	T R	T R 2	O	L
	mn, sd .0;.0 .038;.0	.0; .004 .96;.01	.0; .0	.0; .0
	CL	N/A -522.85* -0.97	-755.8*	-1	0.35
	CL 2	N/A N/A	91.9*	-724.16* 40.78*	523.76*
	T R	N/A N/A	N/A	-718.96* -0.51	0.98
	T R 2	N/A N/A	N/A	N/A	610.86* 755.8*
	O	N/A N/A	N/A	N/A	N/A	1

(b) One-way ANOVA result (statistic=862700.36, pvalue=0.0). Ttests and standard deviations in-table.

(b) One-way ANOVA result (statistic=22407.57, pvalue=0.0). Ttests and standard deviations in-table.

(a) One-way ANOVA result (statistic=181270.76, pvalue=0.0). Ttests and standard deviations in-table.

(b) One-way ANOVA result (statistic=118870.68, pvalue=0.0). Ttests and standard deviations in-table.

ACKNOWLEDGMENTS

We thank the 3IA Côte d'Azur ANR-19-P3IA-0002 and the HyperAgents project ANR-19-CE23-0030 for their support.

B Supplementary Material -Sensitivity Analysis

Next, we perform a sensitivity analysis for PKSG*-V, as it is the closest the most representative KSG given a fully open heterogrenous and self-organising multi-agent system that we have modeled where Common Language and Translation compete as knowledge sharing strategies. In summary, we show here the following significant effects:

• The particular case of when population size N drives the formation of silos if population distribution resets to O after every independent run. We also show here the effects of s and µ for a typical simulation run.

• For a controlled sensitivity analysis we also set up a simulation that starts and resets to keep an equal distribution of strategies in the population of N agents after every run. Hence every strategy will start with a distribution of N/6 agents. We show the avg. of the long-run avg. frequencies, and the effects of social learning s, compatibility, and strategy costs.

B.1


Nicole El Karoui
email: elkaroui@cmapx.polytechnique.fr

Monique Jeanblanc
email: monique.jeanblanc@univ-evry.fr

Ying Jiao
email: jiao@math.jussieu.fr

What happens after a default: the conditional density approach *

We present a general model for default time, making precise the role of the intensity process, and showing that this process allows for a knowledge of the conditional distribution of the default only "before the default". This lack of information is crucial while working in a multi-default setting. In a single default case, the knowledge of the intensity process does not allow to compute the price of defaultable claims, except in the case where immersion property is satisfied. We propose in this paper the density approach for default time. The density process will give a full characterization of the links between the default time and the reference filtration, in particular "after the default time". We also investigate the description of martingales in the full filtration in terms of martingales in the reference filtration, and the impact of Girsanov transformation on the density and intensity processes, and also on the immersion property.

Introduction

Modelling default time for a single credit event has been largely studied in the literature, the main approaches being the structural, the reduced-form and the intensity ones. In this context, most works are concentrated (for pricing purpose) on the computation of conditional expectation of payoffs, given that the default has not occurred, in the case where immersion property is satisfied. In this paper, we are also interested in what happens after a default occurs: we find it important to investigate the impact of a default event on the rest of the market and what goes on afterwards.

Furthermore, in a multi-default setting, it will be important to compute the prices of a portfolio derivative on the disjoint sets before the first default, after the first and before the second and so on. Our work will allow us to use a recurrence procedure to provide these computations, which will be presented in a companion paper [START_REF] El Karoui | Successive default times[END_REF].

We start with the knowledge of the conditional distribution of the default time τ , with respect to a reference filtration F = (F t ) t≥0 and we assume that this conditional distribution admits a density (see the first section below for a precise definition). We firstly reformulate the classical computation result of conditional expectations with respect to the observation σalgebra G t := F t ∨ σ(τ ∧ t) before the default time τ , i.e., on the set {t < τ }. The main purpose is then to deduce what happens after the default occurs, i.e., on the set {τ ≤ t}. We shall emphasize that the density approach is suitable in this after-default study and explain why the intensity approach is inadequate for this case. We present computation results of G = (G t ) t≥0 conditional expectations on the set {τ ≤ t} by using the conditional density of τ and point out that the whole term structure of the density is needed. By establishing an explicit link between (part of) density and intensity, which correspond respectively to the additive and multiplicative decomposition related to the survival process (Azéma supermartingale of τ ), we make clear that the intensity can be deduced from the density, but that the reverse does not hold, except when certain strong assumption, as the H-hypothesis, holds.

Note that, even if the "density" point of view is inspired by the enlargement of filtration theory, we shall not use classical results on the progressive enlargement of filtration. In fact, we take the opposite point of view: we are interested in G-martingales and their characterization in terms of F-(local) martingales. Moreover, these characterization results allow us to give a proof of a decomposition of F-(local) martingales in terms of G-semimartingales.

We study how the parameters of the default (i.e., the survival process, the intensity, the density) are modified by a change of probability in a general setting (we do not assume that we are working in a Brownian filtration, except for some examples), and we characterize changes of probability that do not affect the intensity process. We pay attention to the specific case where the dynamics of underlying default-free processes are changed only after the default.

The paper is organized as follows. We first introduce in Section 2 the different types of information we are dealing with and the key hypothesis of density. In Section 3, we establish results on computation of conditional expectations, on the "before default" and "after default" sets. The H-hypothesis is then discussed. The dynamic properties of the density process are presented in Section 4 where we make precise the links between this density process and the intensity process. In the last section, we present the characterization of G-martingales in terms of F-local martingales. We give a Girsanov type property and discuss the stability of immersion property and invariance of intensity.

The Different Sources of Information

In this section, we specify the link between the two filtrations F and G, and make some hypotheses on the default time. Our aim is to measure the consequence of a default event in terms of pricing various contingent claims.

We start as usual with a filtered probability space (Ω, A, F, P). Before the default time τ , i.e., on the set {t < τ }, the σ-algebra F t represents the information accessible to the investors at time t. When the default occurs, the investors will add this new information (i.e., the knowledge of τ ) to the σ-algebra F t .

More precisely, a strictly positive and finite random variable τ (the default time) is given on the probability space (Ω, A, P). This space is supposed to be endowed with a filtration F = (F t ) t≥0 which satisfies the usual conditions, that is, the filtration F is right-continuous and F 0 contains all P-null sets of A.

One of our goals is to show how the information contained in the reference filtration F can be used to obtain information on the distribution of τ . We assume that, for any t, the conditional distribution of τ with respect to F t is smooth, i.e., that the F t -conditional distribution of τ admits a density with respect to some positive σ-finite measure η on R + . As an immediate consequence, the unconditional distribution of τ is absolutely continuous w.r.t. η. Another consequence is that τ can not be an F-stopping time.

In other terms, we introduce the following hypothesis1 , that we call density hypothesis. This hypothesis will be in force in the paper. Hypothesis 2.1 (Density hypothesis.) We assume that η is a non-negative non-atomic measure on R + such that, for any time t ≥ 0, there exists an

F t ⊗ B(R + )-measurable function (ω, θ) → α t (ω, θ) which satisfies P(τ ∈ dθ|F t ) =: α t (θ)η(dθ), P -a.s. (1)
The family α t (.) is called the conditional density of τ with respect to η given F t (in short the density of τ if no ambiguity). Then, the distribution of τ is given by P(τ > θ) = ∞ θ α 0 (u)η(du). Note that, from the definition and the hypothesis that τ is finite, for any t, ∞ 0 α t (θ)η(dθ) = 1 (a.s.). By definition of the conditional expectation, for any (bounded) Borel function f ,

E[f (τ )|F t ] = ∞ 0 f (u)α t (u)η(du)(a.s.
). The conditional distribution of τ is also characterized by the survival probability function

S t (θ) := P(τ > θ|F t ) = ∞ θ α t (u)η(du) (2)
The family of random variables

S t := S t (t) = P(τ > t|F t ) = ∞ t α t (u)η(du)
plays a key role in what follows. Observe that one has

{τ > t} ⊂ {S t > 0} =: A t (3)
(where the inclusion is up to a negligible set) since P(A c t ∩ {τ > t}) = 0. Note also that S t (θ) = E(S θ |F t ) for any θ ≥ t.

More generally, if an

F t ⊗ B(R + )-measurable function (ω, θ) → Y t (ω, θ) is given, the F t - conditional expectation of the r.v. Y t (τ ) := Y t (ω, τ (ω)), assumed to be integrable, is given by E[Y t (τ )|F t ] = ∞ 0 Y t (u)α t (u)η(du) . (4)

Notation:

In what follows, we shall simply say that Y t (θ) is an F t ⊗ B(R + )-random variable and even that Y t (τ ) is an F t ⊗ σ(τ )-random variable as a short cut for Y t (θ) is an F t ⊗ B(R + )measurable function.

Corollary 2.2

The default time τ avoids F-stopping times, i.e., P(τ = ξ) = 0 for every Fstopping time ξ.

Proof: Let ξ be an F-stopping time bounded by a constant T . Then, the random variable

H ξ (t) = 1 1 {ξ=t} is F T ⊗ B(R +
)-measurable, and, η being non-atomic

E[H ξ (τ )|F t ] = E[ E[H ξ (τ )|F T ] |F t ] = E[ ∞ 0 H ξ (u)α T (u)η(du)|F t ] = 0 .
Hence, E[H ξ (τ )] = P(ξ = τ ) = 0.

Remark 2.3 By using density, we adopt an additive point of view to represent the conditional probability of τ : the conditional survival function S t (θ) = P(τ > θ | F t ) is written on the form S t (θ) = ∞ θ α t (u)η(du). In the default framework, the "intensity" point of view is often preferred, and one uses the multiplicative representation S t (θ) = exp(-θ 0 λ t (u)η(du)). The family of F t -measurable random variables λ t (u) = -∂ u ln S t (u) is called the "forward hazard rate". We shall discuss and compare these two points of view further on.

Computation of conditional expectations in a default setting

The specific information related to the default time is the knowledge of this time when it occurs.

It is defined in mathematical terms as follows: let D = (D t ) t≥0 be the smallest right-continuous filtration such that τ is a D-stopping time; in other words, D t is given by D t = D 0 t+ where D 0 t = σ(τ ∧ t). This filtration D represents the default information, that will be "added" to the reference filtration; the filtration G := F ∨ D is the smallest filtration containing F and making τ a stopping time. Moreover, any G t -measurable r.v. H G t may be represented as

H G t = H F t 1 1 {τ >t} + H t (τ )1 1 {τ ≤t} where H F t is an F t -measurable random variable and H t (τ ) is F t ⊗ σ(τ )-measurable. In particular, H G t 1 1 {τ >t} = H F t 1 1 {τ >t} a.s. , (5)
where the random variable H F t is the F t -conditional expectation of H G t given the event {τ > t}, i.e.,

H F t = E[H G t 1 1 {τ >t} |F t ] P(τ > t|F t ) = E[H G t 1 1 {τ >t} |F t ]
S t a.s. on A t ; H F t = 0 on the complementary set .

Conditional expectations

The definition of G allows us to compute conditional expectations with respect to G t in terms of conditional expectations with respect to F t . This will be done in two steps, depending whether or not the default has occurred: as we explained above, before the default, the only information contained in G t is F t , after the default, the information contained in G t is, roughly speaking,

F t ∨ σ(τ ).
The G t -conditional expectation of an integrable σ(τ )-measurable r.v. (of the form f (τ )) is

given by Recall the notation S t = P(τ > t|F t ). On the set A t , the "before the default" conditional distribution α bd t admits a density α bd t (u) with respect to η, given by

α G t (f ) := E[f (τ )|G t ] = α bd t (f ) 1 1 {τ >t} + f (τ )
α bd t (u) = 1 S t 1 [t,∞) (u)α t (u)η(du) a.s. .
The same calculation as in (4) can be performed in this new framework and extended to the computation of G t -conditional expectations for a bounded F T ⊗ σ(τ )-r.v..

Theorem 3.1 Let Y T (τ ) be a bounded F T ⊗ σ(τ )-random variable. Then, for t ≤ T E[Y T (τ )|G t ] = Y bd t 1 1 {t<τ } + Y ad t (T, τ )1 1 {τ ≤t} dP -a.s.
where

Y bd t = E ∞ t Y T (u)α T (u)η(du)|F t ] S t dP -a.s. on A t , Y ad t (T, θ) = E Y T (θ)α T (θ) F t α t (θ) 1 1 {αt(θ)>0} dP -a.s.. (7)
Proof: The computation on the set {t < τ } (the pre-default case) is obtained following ( 5), [START_REF] Elliott | On models of default risk[END_REF] and using [START_REF] Coculescu | Default times, non arbitrage conditions and change of probability measures[END_REF]. For the after-default case, we note that, by definition of G, any G t -measurable r.v. can be written on the set {τ ≤ t} as H t (τ )1 1 {τ ≤t} . Assuming that H t (τ ) is positive or bounded, and using the density α t (θ), we obtain

E[H t (τ )1 1 {τ ≤t} Y T (τ )] = ∞ 0 dθ E[H t (θ)1 1 {θ≤t} Y T (θ)α T (θ)] = ∞ 0 dθ E H t (θ)1 1 {θ≤t} E[Y T (θ)α T (θ)|F t ] = ∞ 0 dθ E H t (θ)1 1 {θ≤t} Y ad t (T, θ)α t (θ) = E H t (τ )1 1 {τ ≤t} Y ad t (T, τ ) ,
which implies immediately (7).

Immersion property or H-hypothesis

In the form of the density α t (θ) = P(τ ∈ dθ|F t )/η(dθ), the parameter θ plays the role of the default time. It is hence natural to consider the particular case where

α t (θ) = α θ (θ), ∀ θ ≤ t , (8)
i.e., the case when the information contained in the reference filtration after the default time does not give new information on the conditional distribution of the default. In that case

S t = P(τ > t|F t ) = 1 - t 0 α t (u)η(du) = 1 - t 0 α u (u)η(du)
and, in particular S is decreasing2 . Furthermore,

S t = 1 - t 0 α T (u)η(du) = P(τ > t|F T ) = S T (t) a.s.
for any T ≥ t and it follows that P(τ > t|F t ) = P(τ > t|F ∞ ). This last equality is known to be equivalent to the immersion property ( [START_REF] Brémaud | Changes of filtrations and of probability measures[END_REF]), also known as the H-hypothesis, stated as: for any fixed t and any bounded

G t -measurable r.v. Y G t , E[Y G t |F ∞ ] = E[Y G t |F t ] a.s.. (9)
Conversely, if immersion property holds, then (8) holds. In that case, the conditional survival functions S t (θ) are constant in time on [θ, ∞), i.e., S t (θ) = S θ (θ) for t > θ. Then the previous result ( 7) takes a simpler form:

Y ad t (T, θ) = E[Y T (θ)|F t ], a.s. for θ ≤ t ≤ T , on the set {α θ (θ) > 0}.
Under immersion property, the knowledge of S implies that of the conditional distribution of τ for all positive t and θ: indeed, one has

S t (θ) = E[S θ |F t ] (note that, for θ < t, this equality reduces to S t (θ) = S θ (θ) = S θ ). Remarks 3.2 1)
The decreasing property of S (equivalent to the fact that τ is a pseudo-stopping time (see [START_REF] Nikeghbali | A definition and some properties of pseudo-stopping times[END_REF])) does not imply the H-hypothesis, but only that F-bounded martingales stopped at τ are G-martingales (see also [START_REF] Elliott | On models of default risk[END_REF]). We shall revisit this property in Remarks 4.2 and 4.9 and Corollary 5.4.

2) The most important example where immersion holds is the widely studied Cox-process model introduced by Lando [START_REF] Lando | On Cox processes and credit risky securities[END_REF].

Dynamic point of view and density process

Our aim is here to give a dynamic study of the previous results. We shall call (S t , t ≥ 0) the survival process, which is an F-supermartingale. We have obtained equalities for fixed t, we would like to study the conditional expectations as processes. One of the goals is to recover the value of the intensity of the random time, and the decompositions of S. Another one is to study the link between G and F martingales: this is of main interest for pricing.

In this section, we present the dynamic version of the previous results in terms of F or G martingales or supermartingales. To be more precise, we need some "universal" regularity on the paths of the density process. We shall treat some technical problems in Subsection 4.1 which can be skipped for the first reading.

Regular Version of Martingales

One of the major difficulties is to prove the existence of a universal càdlàg martingale version of this family of densities. Fortunately, results of Jacod [START_REF] Jacod | Grossissement initial, Hypothèse (H') et théorème de Girsanov[END_REF] or Stricker and Yor [START_REF] Stricker | Calcul stochastique dépendant d'un paramètre[END_REF] help us to solve this technical problem.

Jacod ([9], Lemme 1.8) establishes the existence of a universal càdlàg version of the density process in the following sense: there exists a non negative function α t (ω, θ) càdlàg in t, optional w.r.t. the filtration F on Ω = Ω × R + , generated by F t ⊗ B(R + ), such that

• for any θ, α . (θ) is an F-martingale; moreover, denoting ζ θ = inf{t : α t-(θ) = 0}, then α . (θ) > 0, and α -(θ) > 0 on [0, ζ θ ), and α . (θ) = 0 on [ζ θ , ∞).
• For any bounded family (Y t (ω, θ), t ≥ 0) measurable w.r.t. P(F) ⊗ B(R + ), (where P(F)

is the F-predictable σ-field), the F-predictable projection of the process Y t (ω, τ ) is the process Y (p) t = α t-(θ)Y t (θ)η(dθ).
In particular, for any t,

P(ζ τ < t) = E[ ∞ 0 α t-(θ)1 1 {ζ θ <t} η(dθ)] = 0. So, ζ τ is infinite a.s.
We are also concerned with the càdlàg version of the martingale (S t (u), t ≥ 0) for any u ∈ R + .

By the previous result, we have a universal version of their predictable projections,

S t-(u) = S (p) t (u) = ∞ u α t-(θ)η(dθ).
It remains to define S t (u) = lim q∈Q + , q↓t S (p) q (u) to obtain a universal càdlàg version of the martingales S . (u).

Remark that to show directly that ∞ u α t (θ)η(dθ) is a càdlàg process, we need stronger assumption on the process α t (θ) which allows us to apply the Lebesgue theorem w.r.t. η(dθ).

We say that the process

(Y t (ω, θ), t ≥ 0) is F-optional if it is O(F) ⊗ B(R + )-measurable,
where O(F) is the optional σ-field of F. In particular, the process (Y t (ω, t), t ≥ 0) is optional.

Density and intensity processes

We are now interested in the relationship between the density and the intensity process of τ .

As we shall see, this is closely related to the (additive and multiplicative) decompositions of the supermartingale S.

F-decompositions of the survival process S

In this section, we characterize the martingale and the predictable increasing part of the additive and multiplicative Doob-Meyer decomposition of the supermartingale S in terms of the density. Proposition 4.1

1) The Doob-Meyer decomposition of the survival process S is given by

S t = 1 + M F t - t 0 α u (u)η(du)
where M F is the càdlàg square-integrable F-martingale defined by

M F t = - t 0 α t (u) -α u (u) η(du) = E[ ∞ 0 α u (u)η(du)|F t ] -1, a.s.
2) Let ζ F := inf{t : S t-= 0} and define λ F t := αt(t) S t-for any t < ζ F and λ F t := λ F t∧ζ F for any t ≥ ζ F . The multiplicative decomposition of S is given by

S t = L F t e - R t 0 λ F s η(ds) (10)
where

L F is the F-local martingale solution of dL F t = e R t 0 λ F s η(ds) dM F t , L F 0 = 1.
Proof: 1) First notice that ( t 0 α u (u)η(du), t ≥ 0) is an F-adapted continuous increasing process (the measure η does not have any atom). By the martingale property of (α t (θ), t ≥ 0), for any fixed t,

S t = P(τ > t|F t ) = ∞ t α t (u)η(du) = E[ ∞ t α u (u)η(du)|F t ], a.s.. Therefore, the non-negative process S t + t 0 α u (u)η(du) = E[ ∞ 0 α u (u)η(du)|F t ] is a square- integrable martingale since E ∞ 0 α u (u)η(du) 2 = 2E ∞ 0 ∞ u α s (s)η(ds)α u (u)η(du) = 2E ∞ 0 S u α u (u)η(du) ≤ 2.
We shall choose its càdlàg version if needed. Using the fact that

∞ 0 α t (u)η(du) = 1, we obtain ∀t, E[ ∞ 0 α u (u)η(du)|F t ] = 1 - t 0 (α t (θ) -α θ (θ))η(dθ), a.s.
and the result follows.

2) Setting L F t = S t e R t 0 λ F s η(ds) , integration by parts formula and 1) yield to

dL F t = e R t 0 λ F s η(ds) dS t + e R t 0 λ F s η(ds) λ F t S t η(dt) = e R t 0 λ F s η(ds) dM F t ,
which implies the result.

Remarks 4.2

1) Note that, from (3), P(ζ F ≥ τ ) = 1.
2) The survival process S is a decreasing process if and only if the martingale M F is constant (M F ≡ 0) or equivalently if and only if the martingale L F is constant (L F ≡ 1). In that case, by Proposition 4.1, S is the continuous decreasing process S t = e - R t 0 λ F s η(ds) . Moreover, for any pair (t, θ), t ≤ θ, the conditional distribution is given by S

t (θ) = E[e - R θ 0 λ F s η(ds) |F t ].
3) The condition M F ≡ 0 can be written as t 0 (α t (u) -α u (u))η(du) = 0 and is satisfied if, for t ≥ u, α t (u) -α u (u) = 0 (immersion property), but the converse is obviously not true (Remark 3.2).

Relationship with the G-intensity

The intensity approach has been largely studied in the credit literature. We study now in more details the relationship between the density and the intensity, and notably between the F-density process of τ and its intensity process with respect to G. We first recall some definitions.

Definition 4.3 Let τ be a G-stopping time. The G-compensator of τ is the G-predictable increasing process Λ G such that the process (N G t = 1 1 {τ ≤t} -Λ G t , t ≥ 0) is a G-martingale. If Λ G is
absolutely continuous with respect to the measure η, the G-adapted process λ G such that

Λ G t = t 0 λ G s η(ds)
is called the (G, η)-intensity process or the G-intensity if there is no ambiguity. The G-compensator is stopped at τ , i.e., Λ G t = Λ G t∧τ . Hence, λ G t = 0 when t > τ .

The following results give the G-intensity of τ in terms of F-density, and conversely the F-density α t (θ) in terms of the G-intensity, but only for θ ≥ t.

Proposition 4.4

1) The random time τ admits a (G, η)-intensity given by

λ G t = 1 1 {τ >t} λ F t = 1 1 {τ >t} α t (t) S t , η(dt) a.s.. ( 11
)
The processes

(N G t := 1 1 {τ ≤t} - t 0 λ G s η(ds), t ≥ 0), and (L G t := 1 1 {τ >t} e R t 0 λ G s η(ds) , t ≥ 0) are G-local martingales.
2) For any t < ζ F and θ ≥ t, we have:

α t (θ) = E[λ G θ |F t ].
Then, the F-optional projections of the local martingales N G and L G are the F-local martingales -M F and L F .

Proof: 1) The G-local martingale property of N G is equivalent to the G-local martingale property of L G t = 1 1 {τ >t} e R t 0 λ G s η(ds) = 1 1 {τ >t} e R t 0 λ F s η(ds) , since dL G t = -L G t-d1 {τ ≤t} + 1 {τ >t} e R t 0 λ F s η(ds) λ F t η(dt) = -L G t-dN G t
Since the process t 0 λ F s η(ds) is continuous, we can proceed by localization introducing the Gstopping times τ n = τ 1 1 {τ ≤Tn} + ∞1 1 {τ >Tn} where T n = inf{t :

t 0 λ F s η(ds) > n}.
Then, the martingale property of the stopped process

L G t∧τ n = L G,n t follows from the F-martingale property of L F t∧T n = L F,n t , since for any s ≤ t, E[L G,n t |G s ] = E[1 1 {τ >t∧T n } e R t∧T n 0 λ F u η(du) |G s ] = 1 1 {τ >s∧T n } E[1 1 {τ >t∧T n } e R t∧T n 0 λ F u η(du) |F s ] S s = 1 1 {τ >s} E[S t∧T n e R t∧T n 0 λ F u η(du) |F s ] S s = 1 1 {τ >s∧T n } L F,n s S s
where the last equality follows from the F-martingale property of L F,n .

Then, the form of the intensities follows from the definition.

2) By the martingale property of density, for any

θ ≥ t, α t (θ) = E[α θ (θ)|F t ].
Using the definition of S, and the value of λ G given in 1), we obtain

α t (θ) = E α θ (θ) 1 1 {τ >θ} S θ |F t = E[λ G θ |F t ], ∀t < ζ F , a.s.
Hence, the value of the density can be partially deduced from the intensity.

The F-projection of the local martingale

L G t = 1 1 {τ >t} e R t 0 λ G s η(ds) is the local martingale S t e R t 0 λ F s η(ds) = L F t by definition of the survival process S. Similarly, since α t (θ) = E[λ G θ |F t ], the F-projection of the martingale N G t = 1 1 {τ ≤t} - t 0 λ G s η(ds) is 1 -S t - t 0 α s (s)η(ds) = -M F t .
Remarks 4.5

1) Since the intensity process is continuous, τ is a totally inaccessible G-stopping time.

2) The density hypothesis, and the fact that η is non-atomic allow us to choose α s (s)/S s as an intensity, instead of α s (s)/S s-as it is usually done (see [START_REF] Elliott | On models of default risk[END_REF] in the case where the numerator α s (s) represents the derivative of the compensator of S).

3) Proposition 4.1 shows that density and intensity approaches correspond respectively to the additive and the multiplicative decomposition point of view of the survival process S.

We now use the density-intensity relationship to characterize the pure jump G-martingales having only one jump at τ .

Corollary 4.6

1) For any locally bounded G-optional process H G , the process

N H,G t := H G τ 1 1 {τ ≤t} - t∧τ 0 α s (s) S s H G s η(ds) = t 0 H G s dN G s , t ≥ 0 ( 12
)
is a G-local martingale.

2) Conversely, any pure jump G-martingale M G which has only one locally bounded jump at τ can be written on the form [START_REF] Lando | On Cox processes and credit risky securities[END_REF], with

H G τ = M G τ -M G τ -. 3) Any nonnegative pure jump G-martingale U G such that U G 0 = 1,

with only one jump at time τ has the following representation

U G t = u τ 1 {τ ≤t} + 1 {t<τ } e - R t∧τ 0 (us-1)λ F s η(ds)
where u is a positive F-optional process associated with the relative jump such that

u τ = U G τ /U G τ -.
Proof: 1) The G-martingale property of N G implies that N H,G defined in ( 12) is a G-martingale for any bounded predictable process H G . From a reinforcement of ( 5), if

H G is a G-predictable process (typically H G t 0 1 1 ]t 0 ,∞]
), there exists an F-predictable process H F such that H G τ = H F τ , a.s.. Then the process H G may be replaced by its representative H F in the previous relations.

Let Y G

s be a bounded G s -random variable, expressed in terms of F-random variables as

Y G s = Y F s 1 {s<τ } + Y F s (τ )1 {τ ≤s} where Y F s ∈ F s and Y F s (θ) ∈ F s ⊗ B([0, s]), (typically Y F s (θ) = Y F s × g(θ)1 [0,s]
). Then, the G-martingale property still holds for the process N H,G where H G is the ) . The optional σ-field being generated by such processes, the assertion holds for any G-optional process.

G-optional process H G s = Y G s 1 [s,∞
2) For the converse, observe that the locally bounded jump H G τ of the martingale M G at time τ is the value at time τ of some locally bounded F-optional process H F . Then the difference M G -N H,G is a finite variation local martingale without jump, that is a constant process.

3) It is easy to calculate the differential of the finite variation process U G as

dU G t = -U G t (u t -1)λ G t η(dt) + U G t-((u t -1)(dN G t + λ G t η(dt)) = U G t-(u t -1)dN G t .
Then U G is the exponential martingale of the purely jump martingale (u t -1)dN G t .

An example of HJM type

We now give some examples, where we point out similarities with Heath-Jarrow-Morton models.

Here, our aim is not to present a general framework, therefore, we reduce our attention to the case where the reference filtration F is generated by a multidimensional standard Brownian motion W .

The following two propositions, which model the dynamics of the conditional probability S(θ), correspond respectively to the additive and multiplicative points of view. From the predictable representation theorem in the Brownian filtration, applied to the family of bounded martingales (S t (θ), t ≥ 0), θ ≥ 0, there exists a family of F-predictable processes (Z t (θ), t ≥ 0) such that

dS t (θ) = Z t (θ)dW t , a.s. ( 13
)
Proposition 4.7 Let dS t (θ) = Z t (θ)dW t be the martingale representation of (S t (θ), t ≥ 0)

and assume that the processes (Z t (θ); t ≥ 0) are differentiable in the following sense: there exists a family of processes (z t (θ), t ≥ 0), bounded by an integrable process, such that Z t (θ) = θ 0 z t (u)η(du). Then, 1) The density martingales have the following dynamics dα t (θ) = -z t (θ)dW t .

2) The survival process S evolves as dS t = -α t (t)η(dt) + Z t (t)dW t .

3) With more regularity assumptions, if (∂ θ α t (θ)) θ=t is simply denoted by ∂ θ α t (t), then the process α t (t) is driven by :

dα t (t) = ∂ θ α t (t)η(dt) -z t (t)dW t
Proof: Observe that Z(0) = 0 since S(0) = 1, hence the existence of z is related with some smoothness conditions. Then

S t (θ) = S 0 (θ) + t 0 Z u (θ)dW u = S 0 (θ) + θ 0 η(dv) t 0 z u (v)dW u
and 1) follows. Furthermore, by using Proposition 4.1 and integration by parts,

M F t = t 0 (α t (u) -α u (u))η(du) = t 0 η(du) t u z s (u)dW s = t 0 dW s s 0 z s (u)η(du)
which implies 2).

3) Let us use the short notation introduced above. We follow the same way as for the decomposition of S, by studying the process

α t (t) - t 0 (∂ θ α s )(s)η(dθ) = α t (0) + t 0 (∂ θ α t )(s)η(ds) - t 0 (∂ θ α s )(s)η(ds)
Using martingale representation of α t (θ) and integration by parts, (assuming that smoothness hypothesis allows these operations) the integral in the RHS is a stochastic integral,

t 0 (∂ θ α t )(s) -(∂ θ α s )(s) η(ds) = - t 0 η(ds)∂ θ ( t s z u (θ)dW u ) = - t 0 η(ds) t s ∂ θ z u (s)dW u = - t 0 dW u u 0 η(ds)∂ θ z u (s) = - t 0 dW u (z u (u) -z u (0))
The stochastic integral t 0 dW u z u (0) is the stochastic part of the martingale α t (0), and so the property 3) holds true.

We now consider (S t (θ), t ≥ 0) in the classical HJM models (see [START_REF] Schönbucher | Credit derivatives pricing models, models, pricing and implementation[END_REF]) where its dynamics is given in multiplicative form. By definition, the forward hazard rate λ t (θ) of τ is given by λ t (θ) = -∂ θ ln S t (θ) and the density can then be calculated as α t (θ) = λ t (θ)S t (θ). As noted in Remark 2.3, λ(θ) plays the same role as the spot forward rate in the interest rate models.

Classically, HJM framework is studied for time smaller than maturity, i.e. t ≤ T . Here we consider all positive pairs (t, θ). Proposition 4.8 For any t, θ ≥ 0, let Ψ t (θ) = Zt(θ) St(θ) with the notation of Proposition 4.7. We assume that ψ t (θ) defined by Ψ t (θ) = θ 0 ψ t (u)η(du) is bounded by some integrable process. Then

1) S t (θ) = S 0 (θ) exp t 0 Ψ s (θ)dW s -1 2 t 0 |Ψ s (θ)| 2 ds ;
2) The forward hazard rate λ(θ) has the dynamics:

λ t (θ) = λ 0 (θ) - t 0 ψ s (θ)dW s + t 0 ψ s (θ)Ψ s (θ) * ds; 3) S t = exp - t 0 λ F s η(ds) + t 0 Ψ s (s)dW s -1 2 t 0 |Ψ s (s)| 2 ds ;
Proof: By choice of notation, the process S t (θ) is the solution of the equation

dS t (θ) S t (θ) = Ψ t (θ)dW t , ∀ t, θ ≥ 0. ( 14
)
Hence 1), from which we deduce immediately 2) by differentiation w.r.t. θ.

3) This representation is the multiplicative version of the additive decomposition of S. There is not technical difficulties because S is continuous.

Remarks 4.9 If Ψ s (s) = 0, then S t = exp(-t 0 λ F s η(ds)), which is decreasing. For the (H)hypothesis to hold, it needs Ψ s (θ) = 0 for any s ≥ θ.

As a conditional survival probability, S t (θ) is decreasing on θ, which is equivalent to that λ t (θ) is positive. When θ > t, this property is implied by the weaker condition λ t (t) ≥ 0. That is similar as for the zero coupon bond prices. But when θ < t, additional assumption is necessary.

We do not characterize this condition.

Remark 4.10 The above results are not restricted to the Brownian filtration and can be easily extended to more general filtrations under similar representation dS t (θ) = Z t (θ)dM t where M is a martingale which can include jumps. In this case, Proposition 4.7 can be generalized in a similar form; for Proposition 4.8, more attention should be payed to Doléans-Dade exponential martingales with jumps.

Example: We now give a particular example which provides a large class of forward rate processes. The non-negativity of λ is satisfied, by 2) of Proposition 4.8, if

• for any θ, the process ψ(θ)Ψ(θ) is non negative, or if ψ(θ) is non negative;

• for any θ, the local martingale ζ t (θ) = λ 0 (θ)-t 0 ψ s (θ)dW s is a Doléans-Dade exponential of some martingale, i.e., is solution of

ζ t (θ) = λ 0 (θ) + t 0 ζ s (θ)b s (θ)dW s , that is, if - t 0 ψ s (θ)dW s = t 0 b s (θ)ζ s (θ)dW s .
Here the initial condition is a positive constant λ 0 (θ). Hence, we set

ψ t (θ) = -b t (θ)ζ t (θ) = -b t (θ)λ 0 (θ) exp t 0 b s (θ)dW s - 1 2 t 0 b 2 s (θ)ds
where λ 0 is a positive intensity function and b(θ) is a non-positive F-adapted process. Then, the family

α t (θ) = λ t (θ) exp - θ 0 λ t (v) dv ,
where

λ t (θ) = λ 0 (θ) - t 0 ψ s (θ) dW s + t 0 ψ s (θ)Ψ s (θ) ds
satisfies the required assumptions.

Characterization of G-martingales in terms of F-martingales

In the theory of pricing and hedging, martingale properties play a very important role. In this section, we study the martingale characterization when taking into account information of the default occurrence. The classical question in the enlargement of filtration theory is to give decomposition of F-martingales in terms of G-semimartingales. For the credit problems, we are concerned with the problem in a converse sense, that is, with the links between G-martingales and F-(local) martingales. In the literature, G-martingales which are stopped at τ have been investigated, particularly in the credit context. For our analysis of after-default events, we are furthermore interested in the martingales which start at the default time τ and in martingales having one jump at τ , as the ones introduced in Corollary 4.6. We shall give characterization results for these types of G-martingales in the following, by using a coherent formulation in the density framework.

G-martingale characterization

Any G-martingale may be split into two martingales, the first one stopped at time τ and the second one starting at time τ , that is

Y G t = Y bd,G t + Y ad,G t where Y bd,G t := Y G t∧τ and Y ad,G t := (Y G t -Y G τ )1 1 {τ ≤t} .
We now study the two types of martingales respectively.

The density hypothesis allows us to provide easily a characterization3 of G-martingales stopped at time τ .

process Z such that Y G t = Y t 1 1 {τ >t} + Z τ 1 1 {τ ≤t} a.s.

and that

(U t := Y t S t + t 0 Z s α s (s)η(ds), t ≥ 0) is an F-martingale on [0, ζ F ). (15)
Equivalently, using the multiplicative decomposition of S as

S t = L F t e - R t 0 λ F s η(ds) on [0, ζ F ), the above condition (15) is equivalent to (L F t [Y t + t 0 (Z s -Y s )λ F s η(ds)], t ≥ 0) is an F-local martingale on [0, ζ F ). ( 16
) Proof: The conditional expectation of Y G t given F t is the F-martingale defined on [0, ζ F ) as Y F t = E[Y G t |F t ] = Y t S t + t 0 Z s α t (
s)η(ds) by using the F t -density of τ . Notice that Y F t differs from U t by ( t 0 Z s (α t (s) -α s (s))η(ds), t ≥ 0), which is an F-local martingale (this can be easily checked using that Z is locally bounded and (α t (s), t ≥ 0) is F-martingale). So U is also an F-local martingale. Moreover, since E[|Y G t |] < ∞, for any F-stopping time ϑ, the quantity Y ϑ 1 1 {τ >ϑ} is integrable, hence Y ϑ S ϑ is also integrable, and

E[ ζ F 0 |Z s |α s (s)η(ds)] = E[|Y G τ |] < ∞,
which establishes that U is a martingale.

Conversely, if U is an F-local martingale, it is easy to verify by Theorem 3.

1 that E[Y G T -Y G t |G t ] = 0,
d(Y t L F t ) = e Λ F t d(Y t S t ) + e Λ F t Y t S t λ F t η(dt) = e Λ F t dU t + (Y t -Z t )λ F t L F t η(dt).
The local martingale property of the process U is then equivalent to that of (Y t L F t -t 0 (Y s -Z s )λ F s L F s η(ds), t ≥ 0), and then to the condition (16). Indeed, integration by parts formula proves that (L F t t 0 (1 -Z s )λ F s η(ds), t ≥ 0) is a local martingale if and only if the continuous bounded variation process (

t 0 L F s (1 -Z s )λ F s η(ds), t ≥ 0) is a local-martingale, that is if L F s (1 -Z s )λ F s = 0, which implies that Z s = 1 on [0, ζ F ).
The before-default G-martingale Y bd,G can always be separated into two parts: a martingale which is stopped at τ and is continuous at τ ; and a martingale which has a jump at τ .

Lemma 5.3 Let Y bd,G be a G-martingale stopped at τ of the form Y bd,G t = Y t 1 1 {τ >t} + Z τ 1 1 {τ ≤t} .
Then there exist two G-martingales Y c,bd and Y d,bd such that Y bd,G = Y c,bd +Y d,bd which satisfy the following conditions:

1) (Y d,bd t = (Z τ -Y τ )1 1 {τ ≤t} - t∧τ 0 (Z s -Y s )λ F s η(ds), t ≥ 0) is a G-martingale with a single jump at τ ; 2) (Y c,bd t = Y τ ∧t , t ≥ 0) is continuous at τ , where Y t = Y t + t 0 (Z s -Y s )λ F s η(ds).
Proof: From Corollary 4.6, Y d,bd is a martingale. The result follows.

Corollary 5.4 With the above notation, a martingale Y G which is stopped and continuous at τ is characterized by: (L F t Y t , t ≥ 0) is an F-local martingale. Furthermore, if L F is a martingale, then this condition is equivalent to that Y is an F-local martingale w.r.t. the probability measure P L = L F T P. In particular, under the immersion assumption, the G-martingales stopped at time τ and continuous at τ are F-martingales stopped at τ . Combining the previous results, we give the characterization of general G-martingale.

(Y t S t + t 0 Y s (s)α s (s)η(ds), t ≥ 0) or equivalently (L F t [Y t + t 0 (Y s (s) - Y s )λ F s η(ds)], t ≥ 0) is an F-local martingale; 2) for any θ ≥ 0, (Y t (θ)α t (θ), t ≥ θ) is an F-martingale on [θ, ζ θ ). Proof: Notice that Y ad,G t = (Y t (τ ) -Y τ (τ ))1 1 {τ ≤t} .
Then the theorem follows directly by applying Propositions 5.1 and 5.6 on Y bd,G and Y ad,G respectively. Remark 5.8 We observe again the fact that to characterize what goes on before the default, it suffices to know the survival process S or the intensity λ F . However, for the after-default studies, we need the whole conditional distribution of τ , i.e., α t (θ) where θ ≤ t.

Decomposition of F-(local) martingale

An important result in the enlargement of filtration theory is the decomposition of F-(local) martingales as G-semimartingales. Using the above results, we provide an alternative proof for a result established in [START_REF] Jeanblanc | Progressive enlargement of filtration with initial times[END_REF], simplified by using the fact that any F-martingale is continuous at time τ . Our method is interesting, since it gives the intuition of the decomposition without using any result on enlargement of filtrations. Proposition 5.9 Any F-martingale Y F is a G-semimartingale can be written as

Y F t = M Y,G t + A Y,G t where M Y,G is a G-martingale and (A Y,G t := A t 1 1 {τ >t} + A t (τ )1 1 {τ ≤t} , t ≥ 0) is an optional
process with finite variation. Here

A t = t 0 d[Y F , S] s S s and A t (θ) = t θ d[Y F , α(θ)] s α s (θ) . ( 17
)
Proof: On the one hand, assuming that Y F is a G-semimartingale, it can be decomposed as the sum of a G-(local)martingale and a G-optional process A Y,G with finite variation which can be written as A t 1 1 {τ >t} + A t (τ )

Y F t -A Y,G t = (Y F t -A t )1 1 {τ >t} + (Y F t -A t (τ ))1 1 {τ ≤t}
for any t ∈ R + and Q F be the restriction of Q to F, which has Radon-Nikodým density Q F , given by the projection of Q G on F, that is Q F t = q t S t + t 0 q t (u)α t (u)η(du). Then (Ω, Q, G, F, τ ) satisfies the density hypothesis with the (F, Q)-density of τ given in closed form after the default, that is for θ ≤ t by

α Q t (θ) = α t (θ) q t (θ) Q F t , η(dθ)-a.s.;
and, only via a conditional expectation before the default, that is for t ≤ θ, by

α Q t (θ) = 1 Q F t E P [α θ (θ)q θ (θ)|F t ].
Furthermore:

1) the Q-conditional survival process is defined on [0, ζ F ) by S Q t = S t q t Q F t
, and is null after

ζ F ; 2) the (F, Q)-local martingale L F,Q is (L F,Q t = L F t q t Q F t exp t 0 (λ F,Q s -λ F s )η(ds), t ∈ [0, ζ F )); 3) the (F, Q)-intensity process is λ F,Q t = λ F t q t (t) q t , η(dt)-a.s.
Proof: The expression of the density process after the default (α Q t (θ), θ ≤ t) is an immediate consequence of definition. Before the default, the density may be only obtained via a conditional expectation form given by

α Q t (θ) = E Q α θ (θ) q θ (θ) Q F θ |F t = 1 Q F t E P Q F θ α θ (θ) q θ (θ) Q F θ |F t = 1 Q F t E P [α θ (θ)q θ (θ)|F t ].
For any t ∈ [0, ζ F ), the Q-conditional survival probability can be calculated by

S Q t = Q(τ > t|F t ) = E P [1 1 {τ >t} Q G t |F t ] Q F t = q t S t Q F t and finally, we use λ F,Q t = α Q t (t)/S Q t and L F,Q t = S Q t e R t 0 λ F,Q s η(ds) to complete the proof.
It is known, from [START_REF] Grorud | Asymmetrical information and incomplete markets[END_REF], that under density hypothesis, there exists at least a change of probability, such that immersion property holds under this change of probability. Theorem 5.11 provides a full characterization of such changes of probability.

Corollary 5.12 We keep the notation of Theorem 5.11. The change of probability measure generated by the two processes

q t = (L F t ) -1 , q t (θ) = α θ (θ) α t (θ) (19)
Proof: The first part is a direct consequence of the previous results. It remains to note that

Q F t = St S Q t
and, for t ≥ θ, α Q t (θ) = α θ (θ) qt(θ) Q F t = α θ (θ) qt(θ) St S Q t ; hence the martingale properties follow from the ones of Q F t and q t (θ) .

This result admits a converse. For the sake of simplicity we assume the condition S * t > 0, ∀t ∈ R + . This assumption can be removed, using the terminal time ζ It remains to note that the condition α * s (s) = S * s λ F s is equivalent to the fact that the intensity of τ under Q is λ F .

Conclusion

Our study relies on the impact of information related to the default time on the market.

Starting from a default-free model, where some assets are traded with the knowledge of a reference filtration F, we consider the case where the participants of the market take into account the possibility of a default in view of trading default-sensitive asset. If we are only concerned by what happens up to the default time, the natural assumption is to assume immersion property with stochastic intensity process adapted to the default-free market evolution.

The final step is to anticipate that the default should have a large impact on the market, as now after the crisis. In particular, with the non constant "after default" density, we express how the default-free market is modified after the default. In addition, hedging strategies of default-free contingent claims are not the same in the both universes.

In a following paper [START_REF] El Karoui | Successive default times[END_REF], we shall apply this methodology to several default times, making this tool powerful for correlation of defaults. In another paper, we shall provide explicit examples of density processes, and give some general construction of these processes.

Proposition 5 . 1 A

 51 G-adapted càdlàg process Y G is a closed G-martingale stopped at time τ if and only if there exist an F-adapted càdlàg process Y defined on [0, ζ F ) and an F-optional

Remark 5 . 2 A

 52 G-martingale stopped at time τ and equal to 1 on [0, τ ) is constant on [0, τ ].

Remark 5 . 5 1 . 5 . 6

 55156 Under immersion, L F t = 1. So a process Y G stopped at τ and continuous at time τ is a G-martingale if and only if Y is an F-local martingale.We now concentrate on the G-martingales starting at τ , which, as we can see below, are easier to characterize. The following proposition is a direct consequence of Theorem 3.Proposition Any càdlàg integrable process Y G is a G-martingale starting at τ with Y τ = 0 if and only if there exists an O(F) ⊗ B(R + )-optional process (Y t (.), t ≥ 0) such that Y t (t) = 0 and Y G t = Y t (τ )1 1 {τ ≤t} and that, for any θ > 0, (Y t (θ)α t (θ), t ≥ θ ≥ 0) are F-martingales on [θ, ζ θ ), where ζ θ is defined as in Section 4.1.

Theorem 5 . 7 A

 57 càdlàg process Y G is a G-martingale if and only if there exist an F-adapted càdlàg process Y and an O(F) ⊗ B(R + )-optional process Y t (.) such that Y G t = Y t 1 1 {τ >t} + Y t (τ )1 1 {τ ≤t} and 1) the process

  1 1 {τ ≤t} where α bd t is the value of the G t -conditional distribution before the default, given by

	α bd t (f ) :=	E[f (τ )1 1 {τ >t} |F t ] P(τ > t|F t )	a.s. on A t ; α bd t (f ) := 0 on the complementary set.

  a.s..

	The second formulation is based on the multiplicative representation S t = L F t e -Λ F t where Λ F t =
	t 0 λ F s η(ds) is a continuous increasing process. Since e Λ F t Y t S t = Y t L F t and α t (t) = λ F t S t , we have

  1 1 {τ ≤t} where A and A(θ) are still unknown. Note that, since Y F has no jump at τ (indeed, τ avoids F-stopping times -see Corollary 2.2), we can choose M Y,G such that M Y,G and hence A Y,G have no jump at τ . Applying the martingale characterization result obtained in Theorem 5.7 to the G-local martingale

  * ,F . Proposition 5.15 Let (Ω, P, F, G, τ ) be a model satisfying the H-hypothesis, with the decreasing survival process S t = exp(-* t (θ), t ≥ θ) be a given family, where, for all θ > 0, α * (θ) is a non-negative process and define S * t = 1 -Assume that S * ∞ = 0 and S * t > 0, ∀t ∈ R + and that s λ F s = S * s α s (s)/S s . For every θ, the processes (q t (θ), t ≥ θ) are martingales since (α * t (θ)S t /S * t , t ≥ θ) are martingales. From Theorem 5.11, the F-projection of the Radon-Nikodým density Q G is

						t 0 λ F s η(ds)).
	Let (α   	∀θ, α * θ (θ) = S * θ λ F θ = α θ (θ) the processes St S * t , t ≥ 0 and α * S * θ S θ t (θ) St S *	(22)
	Let								Q G t := 1 1 {τ >t} +	α * t (τ ) α τ (τ )	S t t S *	1 1 {τ ≤t} ,	(23)
	Proof: We set									q t (θ) =	α * t (θ) α θ (θ)	S t S *
		Q F t = S t +	0	t	α * t (s) α t (s)	S t S * t	α t (s)η(ds) = S t 1 +	0	t	α * t (s)	1 t S *	η(ds) ,
	and the survival probability is
		S t Q F t	= 1 +	1 S * t		0	t	α * t (s)η(ds)	-1 = S * t S * t +	0	t	α * t (s)η(ds)	-1 = S * t .

t 0 α * t (θ)η(dθ). t , t ≥ θ are (F, P)-martingales.

and Q be the probability measure with Radon-Nikodỳm density the (G, P)-martingale Q G . Then,

Q is equal to P on G τ and λ Q,F = λ F , α Q t (θ) = α * t (θ) , ∀t ≥ θ and S Q = S *

(24)

t .

Note that q t (t) = 1 since α * s (s) = S *

This hypothesis has been discussed by Jacod[START_REF] Jacod | Grossissement initial, Hypothèse (H') et théorème de Girsanov[END_REF] in the initial enlargement of filtration framework. The same assumption also appears in the dynamic Bayesian framework[START_REF] Gasbarra | Enlargement of filtration and additional information in pricing models: Bayesian approach[END_REF]. We do not assume that η is finite, allowing for the specific case of the Lebesgue measure.

and continuous, this last property will be useful later.

The following proposition was established in[START_REF] Bielecki | Credit risk[END_REF] Lemma

4.1.3] in a hazard process setting.

* This work has benefited from financial support by La Fondation du

leads to the fact that the two processes

are F-(local) martingales. Since

based on the intuition given by the Girsanov theorem, natural candidate for the finite variation processes A is dA t = d[Y F , L F ] t /L F t where [ , ] denotes the co-variation process. Similarly, dA t (θ) = d[Y F , α(θ)] t /α t (θ). Then, using the fact that Y F , L F , α(θ) are F-local martingales, we

and

Then, since S is the product of the martingale L F and an continuous increasing process e Λ F , we have

, S] t /S t and obtain the first equality in (17).

On the other hand, define the optional process A Y,G by using (17). It is not difficult to verify by Theorem 5.7 that

Remark 5.10 Note that our decomposition differs from the usual one, since our process A is optional (and not predictable) and that we are using the co-variation process, instead of the predictable co-variation process. As a consequence our decomposition is not unique.

Girsanov theorem

Change of probability measure is a key tool in derivative pricing as in martingale theory. In credit risk framework, we are also able to calculate parameters of the conditional distribution of the default time w.r.t. a new probability measure. The links between change of probability measure and the initial enlargement have been established, in particular, in [START_REF] Jacod | Grossissement initial, Hypothèse (H') et théorème de Girsanov[END_REF] and [START_REF] Ankirchner | Enlargement of filtrations and continuous Girsanov-type embeddings[END_REF]. In statistics, it is motivated by the Bayesian approach [START_REF] Gasbarra | Enlargement of filtration and additional information in pricing models: Bayesian approach[END_REF].

We present a Girsanov type result, where the Radon-Nikodým density is given in an additive form instead of in a multiplicative one as in the classical literature. This makes the density of τ having simple form under the new probability measure.

provides a model where the immersion property holds true under Q, and where the intensity processes does not change, i.e., remains λ F .

More generally, the only changes of probability measure for which the immersion property holds with the same intensity process are generated by a process q such that (q t L F t , t ≥ 0) is a uniformly integrable martingale.

Proof: : Any change of probability measure with immersion property and the same intensity processes is characterized by the martingale property of the product Q F = q. L F . Moreover, given q, the immersion property determines in a unique way the process (q t (θ); t ≥ θ) via the boundary condition q θ (θ) = q θ and the equalities

The martingale Q F = q. L F has to satisfy the compatibility condition

where the last equality comes from the identities [START_REF] Jeanblanc | Progressive enlargement of filtration with initial times[END_REF] and [START_REF] Kusuoka | A remark on default risk models[END_REF]. The term in the bracket in (20) is of finite variation and is hence equal to 1. Then the Q F -compatibility condition is always satisfied. So the only constraint on the process q is the martingale property of q. L F .

It is well known, from Kusuoka [START_REF] Kusuoka | A remark on default risk models[END_REF], that immersion property is not stable by a change of probability. In the following, we shall in a first step characterize, under density hypothesis, changes of probability which preserve this immersion property, that is, H-hypothesis is satisfied under both P and Q. (See also [START_REF] Coculescu | Default times, non arbitrage conditions and change of probability measures[END_REF] for a different study of changes of probabilities preserving immersion property.) In a second step, we shall study changes of probability which preserve the information before the default, and give the impact of a change of probability after the default.

Corollary 5. [START_REF] Nikeghbali | A definition and some properties of pseudo-stopping times[END_REF] We keep the notation of Theorem 5.11, and assume immersion property under P.

1) Let the Radon-Nikodým density (Q G t , t ≥ 0) be a pure jump martingale with only one jump at time τ . Then, the (F, P)-martingale

q t , η(dt)-a.s., and the immersion property still holds.

2) Conversely, the only changes of probability measure compatible with immersion property have

Radon-Nikodým densities that are the product of a pure jump positive martingale with only one jump at time τ , and a positive F-martingale.

Proof: : The previous Girsanov theorem 5.11 gives immediately the intensity characterization.

From Lemma 5.3, the pure jump martingale (Q G t , t ≥ 0) is a finite variation process and (Q G t = q t , t < τ ) is a continuous process with bounded variation. Since immersion property holds, S is a continuous decreasing process (see footnote 2), and Q F t = q t S t + t 0 q u (u)α t (u)η(du) = q t S t + t 0 q u (u)α u (u)η(du) is a continuous martingale with finite variation. Since Q F 0 = 1, then at any time, Q F t = 1, a.s. By the other results established in Girsanov's theorem 5.11, this key point implies that the new density is constant after the default, so that the immersion property still holds.

2) Thanks to the first part of this corollary, we can restrict our attention to the case when in the both universe the intensity processes are the same. Then the Radon-Nikodým density is continuous at time τ and the two processes (q t , t ≥ 0) and (q t (θ), t ≥ θ) are F-(local) martingales.

Assume now that the immersion property holds also under the new probability measure Q. Both martingales L F,P and L F,Q are constant, and Q F t = q t . Moreover the Q-density process being constant after the default (θ < t), q t (θ)/q t = q θ (θ)/q θ = 1, a.s.. The processes Q G , Q F and q are undistinguishable.

As shown in this paper, the knowledge of the intensity does not allow to give full information on the law of the default, except if immersion property holds. Starting with a model under which immersion property holds, taking q t (t) = q t in Theorem 5.11 will lead us to a model where the default time admits the same intensity whereas immersion property does not hold, and then the impact of the default changes the dynamics of the default-free assets. We present a specific case where, under the two probability measures, the dynamics of these assets are the same before the default but are changed after the default, a phenomenon that is observed in the actual crisis.

We impose that the new probability Q coincide with P on the σ-algebra G τ . In particular, if m is an (F, P)-martingale, the process (m t∧τ , t ≥ 0) will be an ((G t∧τ , t ≥ 0),Q)-martingale (but not necessarily an (G, Q)-martingale). From Theorems 5.7 and 5.11 and Corollary 5.13, one gets the obvious proposition. Proposition 5.14 Let (Ω, P, F, G, τ ) be a model satisfying the immersion property.

Let (q t (θ), t ≥ θ) be a family of positive (F, P)-martingales such that q θ (θ) = 1 and let Q be the probability measure with Radon-Nikodým density equal to the (G, P)-martingale

Then, Q and P coincide on G τ and the P and Q intensities of τ are the same.

Furthermore, if S Q is the Q-survival process, the processes (S t /S Q t , t ≥ 0) and the family (α Q t (θ) S t /S Q t , t ≥ θ) are (F, P)-martingales.