JEANBLANC Monique

We continue to study the credit risk model of a financial market introduced in [19] in which the dynamics of intensity rates of two default times are described by linear combinations of three independent geometric Brownian motions. The dynamics of two default-free risky asset prices are modeled by two geometric Brownian motions which are dependent of the ones describing the default intensity rates. We obtain closed form expressions for the no-arbitrage prices of some first-and second-to-default European style contingent claims given the reference filtration initially and progressively enlarged by the two successive default times. The accessible default-free reference filtration is generated by the standard Brownian motions driving the model.

We consider the initial and progressive enlargements of a filtration generated by a marked point process (called the reference filtration) with a strictly positive random time. We assume Jacod's equivalence hypothesis, that is, the existence of a strictly positive conditional density for the random time with respect to the reference filtration. Then, starting with the predictable integral representation of a martingale in the initially enlarged reference filtration, we derive explicit expressions for the coefficients which appear in the predictable integral representations for the optional projections of the martingale on the progressively enlarged filtration and on the reference filtration. We also provide similar results for the optional projection of a martingale in the progressively enlarged filtration on the reference filtration.

The paper studies thin times which are random times whose graph is contained in a countable union of the graphs of stopping times with respect to a reference filtration F. We show that a generic random time can be decomposed into thin and thick parts, where the second is a random time avoiding all F-stopping times. Then, for a given random time τ , we introduce F τ , the smallest right-continuous filtration containing F and making τ a stopping time, and we show that, for a thin time τ , each F-martingale is an F τ-semimartingale, i.e., the hypothesis (H) for (F, F τ) holds. We present applications to honest times, which can be seen as last passage times, showing classes of filtrations which can only support thin honest times, or can accommodate thick honest times as well.

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We present some results on enlargement of filtration in discrete time. Many results known in continuous time extend immediately in a discrete time setting. Here, we provide direct proofs which are much more simpler. We study also arbitrages conditions in a financial setting and we present some specific cases, as immersion and pseudo-stopping times for which we obtain new results.

We study a credit risk model of a financial market in which the dynamics of intensity rates of two default times are described by linear combinations of three independent geometric Brownian motions. The dynamics of two default-free risky asset prices are modeled by two geometric Brownian motions which are dependent of the ones describing the default intensity rates. We obtain closed form expressions for the rational prices of both risk-free and risky credit default swaps given the reference filtration initially and progressively enlarged by the two default times. The accessible default-free reference filtration is generated by the standard Brownian motions driving the model.

It is known since Kellerer (1972) that for any peacock process there exist mar-tingales with the same marginal laws. Nevertheless, there is no general method for finding such martingales that yields diffusions. Indeed, Kellerer's proof is not constructive: finding the dynamics of processes associated to a given peacock is not trivial in general. In this paper we are interested in the uniform peacock that is, the peacock with uniform law at all times on a generic time-varying support [a(t), b(t)]. We derive explicitly the corresponding Stochastic Differential Equations (SDEs) and prove that, under certain conditions on the boundaries a(t) and b(t), they admit a unique strong solution yielding the relevant diffusion process. We discuss the relationship between our result and the previous derivation of diffusion processes associated to square-root and linear time-boundaries, emphasizing the cases where our approach adds strong uniqueness, and study the local time and activity of the solution processes. We then study the peacock with uniform law at all times on a constant support [−1, 1] and derive the SDE of an associated mean-reverting diffusion process with uniform margins that is not a martingale. For the related SDE we prove existence of a solution in [0, T ]. Finally, we provide a numerical case study showing that these processes have the desired uniform behaviour. These results may be used to model random probabilities, random recovery rates or random correlations.

The first goal of this article is to identify, for different defaultable claims, the fundamental processes which uniquely determine the pre-default price and therefore require to be modelled. The main message to the reader is that although the use of the default intensity or hazard process is ubiquitous, it may not uniquely characterise the price of some defaultable claims. The second goal is to better consolidate the reduced form approach with the structural approach, by extending the reduced form approach to allow for default times which can occur at stopping times and do not satisfy the immersion property.

We study a credit risk model of a financial market in which the dynamics of intensity rates of two default times are described by linear combinations of three independent geometric Brownian motions. The dynamics of two default-free risky asset prices are modeled by two geometric Brownian motions which are dependent of the ones describing the default intensity rates. We obtain closed form expressions for the rational prices of both risk-free and risky credit default swaps given the reference filtration initially and progressively enlarged by the two default times. The accessible default-free reference filtration is generated by the standard Brownian motions driving the model.

In this paper we propose a new methodology for solving an uncertain stochastic Marko-vian control problem in discrete time. We call the proposed methodology the adaptive robust control. We demonstrate that the uncertain control problem under consideration can be solved in terms of associated adaptive robust Bellman equation. The success of our approach is to the great extend owed to the recursive methodology for construction of relevant confidence regions. We illustrate our methodology by considering an optimal portfolio allocation problem, and we compare results obtained using the adaptive robust control method with some other existing methods.

In this paper, we consider two kinds of enlargements of a Brownian filtration F: the initial enlargement with a random time τ , denoted by F (τ) , and the progressive enlargement with τ , denoted by G. We assume Jacod's equivalence hypothesis, that is, the existence of a positive F-conditional density for τ. Then, starting with the predictable representation of an F (τ)-martingale Y (τ) in terms of a standard F (τ)-Brownian motion, we consider its projection on G, denoted by Y G , and on F, denoted by y. We show how to obtain the coefficients which appear in the predictable representation property for Y G (and y) in terms of Y (τ) and its predictable representation. In the last part, we give examples of conditional densities.

Let X be a point process and let F denote the filtration generated by X. In this paper we study martingale representation theorems in the filtration G obtained as an initial and progressive enlargement of the filtration F. In particular, the progressive enlargement is done by means of a whole point process H. We work here in full generality, without requiring any further assumption on the process H and we recover the special case in which X is enlarged progressively by a random time τ.

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We consider a complete probability space (Ω, F, P), which is endowed with two fitrations, G and F, assumed to satisfy the usual conditions and such that F ⊂ G. On this probability space we consider a real valued special G-semimartingale X. The results can be generalized to the case of R^n valued special semimartingales, in a straightforward manner. We fix a truncation function with respect to which the semimartingale characteristics are computed. The purpose of this work is to study the following two problems: A. If X is F-adapted, compute the F-semimartingale characteristics of X in terms of the G-semimartingale characteristics of X. B. If X is not F-adapted, given that the F-optional projection of X is a special semimartingale, compute the F-semimartingale characteristics of F-optional projection of X in terms of the G-canonical decomposition and G-semimartingale characteristics of X.

Under short sales prohibitions, no free lunch with vanishing risk (NFLVR-S) is known to be equivalent to the existence of an equivalent supermartingale measure for the price processes (Pulido [23]). For two given price processes, we translate the property (NFLVR-S) in terms of so called structure conditions and we introduce the concept of fundamental supermartingale measure. When a certain condition necessary to the construction of the fundamental supermartingale measure is not fulfilled, we provide the corresponding arbitrage portfolios. The motivation of our study lies in understanding the particular case of converging prices, i.e., that coincide at a bounded random time.

The paper studies thin times which are random times whose graph is contained in a countable union of the graphs of stopping times with respect to a reference filtration F. We show that a generic random time can be decomposed into thin and thick parts, where the second is a random time avoiding all F-stopping times. Then, for a given random time τ , we introduce F τ , the smallest right-continuous filtration containing F and making τ a stopping time, and we show that, for a thin time τ , each F-martingale is an F τ-semimartingale, i.e., the hypothesis (H) for (F, F τ) holds. We present applications to honest times, which can be seen as last passage times, showing classes of filtrations which can only support thin honest times, or can accommodate thick honest times as well.

We begin this thesis report by evaluating the expected exposure, which represents one of the major components of XVA adjustments. Under the assumption of independence between exposure and financing and credit costs, we derive in Chapter 3 a new representation of expected exposure as the solution of an ordinary differential equation with respect to the time of default observation. For the one-dimensional case, we rely on arguments similar to those for Dupire's local volatility. And for the multidimensional case, we refer to the Co-aire formula. This representation allows us to explain the impact of volatility on the expected exposure: this time value involves the volatility of the underlyings as well as the first-order sensitivity of the price, evaluated on a finite set of points. Despite numerical limitations, this method is an accurate and fast approach for valuing unit XVA in dimension 1 and 2.The following chapters are dedicated to the risk aspects of correlations between exposure envelopes and XVA costs. We present a model of the general correlation risk through a multivariate stochastic diffusion, including both the underlying assets of the derivatives and the default intensities. In this framework, we present a new approach to valuation by asymptotic developments, such that the price of an XVA adjustment corresponds to the price of the zero-correlation adjustment, plus an explicit sum of corrective terms. Chapter 4 is devoted to the technical derivation and study of the numerical error in the context of the valuation of default contingent derivatives. The quality of the numerical approximations depends solely on the regularity of the credit intensity diffusion process, and is independent of the regularity of the payoff function. The valuation formulas for CVA and FVA are presented in Chapter 5. A generalization of the asymptotic developments for the bilateral default framework is addressed in Chapter 6.We conclude this dissertation by addressing a case of the specific correlation risk related to rating migration contracts. Beyond the valuation formulas, our contribution consists in presenting a robust approach for the construction and calibration of a rating transition model consistent with market implied default probabilities.

For a finite state Markov process X and a finite collection {Γ k , k ∈ K} of subsets of its state space, let τ k be the first time the process visits the set Γ k. In general, X may enter some of the Γ k at the same time and therefore the vector τ := (τ k , k ∈ K) may put nonzero mass over lower dimensional regions of R |K| + . these regions are of the form R s = {t ∈ R |K| + : t i = t j , i, j ∈ s(1)} ∩ |s| l=2 {t : t m < t i = t j , i, j ∈ s(l), m ∈ s(l − 1)} where s is any ordered partition of the set K and s(j) denotes the j th subset of K in the partition s. When |s| < |K|, the density of the law of τ over these regions is said to be " singular " because it is with respect to the |s|-dimensional Lebesgue measure over the region R s. We derive explicit/recursive and simple to compute formulas for these singular densities and their corresponding tail probabilities over all R s as s ranges over ordered partitions of K. We give a numerical example and indicate the relevance of our results to credit risk modeling. * The research of T.R.

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This paper quantifies the interplay between the no-arbitrage notion of no-unbounded-profit-with-bounded-risk (NUPBR hereafter) and additional progressiveinformation generated by a randomtime. This study complements the one of Aksamit et al. in which the authors have studied similar topics for the case of stopping at the randomtime instead, while herein we deal with the part after the occurrence of the randomtime. Given that all the literature, up to our knowledge, proves that NUPBR is always violated after honest times that avoid stopping times in a continuous filtration, herein we propose a new class of honest times for which NUPBR can be preserved for some models. For these honest times, we obtain two principal results. The first result characterizes the pairs of initial market and honest time for which the resulting model preserves NUPBR, while the second result characterizes honest times that do not affect NUPBR of any quasi-left-continuous model (i.e., model in which the assets’ price process has no predictable jump times). Furthermore, we construct explicitly local martingale deflators for a large class of models.

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We consider a multivariate default system where random environmental information is available. We study the dynamics of the system in a general setting and adopt the point of view of change of probability measures. We also make a link with the density approach in the credit risk modelling. In the particular case where no environmental information is concerned, we pay a special attention to the phenomenon of system weakened by failures as in the classical reliability system.

In this paper we propose a new methodology for solving an uncertain stochastic Marko-vian control problem in discrete time. We call the proposed methodology the adaptive robust control. We demonstrate that the uncertain control problem under consideration can be solved in terms of associated adaptive robust Bellman equation. The success of our approach is to the great extend owed to the recursive methodology for construction of relevant confidence regions. We illustrate our methodology by considering an optimal portfolio allocation problem, and we compare results obtained using the adaptive robust control method with some other existing methods.

This thesis addresses various issues related to collateral management in the context of centralized trading through clearing houses. First, we present the notions of capital cost and funding cost for a bank, by placing them in an elementary Black-Scholes framework where the payoff of a standard call takes the place of a counterparty default exposure. We assume that the bank only imperfectly hedges this call and faces a funding cost higher than the risk-free rate, hence the FVA and KVA pricing corrections with respect to the Black-Scholes price. We then focus on the costs that a bank faces when trading in a CCP. To this end, we transpose the XVA framework of bilateral trading to centralized trading. The total cost for a member to trade through a CCP is thus decomposed into a CVA corresponding to the cost for the member to replenish its contribution to the guarantee fund in case of losses due to defaults by other members, an MVA corresponding to the cost of financing its initial margin and a KVA corresponding to the cost of capital put at risk by the member in the form of its contribution to the guarantee fund. We then question the regulatory assumptions previously used, looking at alternatives in which members would use a third party for their initial margin, who would post the margin in the member's place in exchange for a fee. We also consider a method of calculating the guarantee fund and its allocation that takes into account the risk of the chamber in the sense of the fluctuations of its P&L over the following year, as it results from the combination of the market risk and the default risk of the members. Finally, we propose the application of multivariate risk measure methodologies for the calculation of members' margins and/or guarantee funds. We introduce a notion of systemic risk measures in the sense that they are sensitive not only to the marginal risks of the components of a financial system (e.g., but not necessarily the positions of the members of a CCP), but also to their dependence.

We consider a multivariate default system where random environmental information is available. We study the dynamics of the system in a general setting and adopt the point of view of change of probability measures. We also make a link with the density approach in the credit risk modelling. In the particular case where no environmental information is concerned, we pay a special attention to the phenomenon of system weakened by failures as in the classical reliability system.

It is known since Kellerer (1972) that for any peacock process there exist mar-tingales with the same marginal laws. Nevertheless, there is no general method for finding such martingales that yields diffusions. Indeed, Kellerer's proof is not constructive: finding the dynamics of processes associated to a given peacock is not trivial in general. In this paper we are interested in the uniform peacock that is, the peacock with uniform law at all times on a generic time-varying support [a(t), b(t)]. We derive explicitly the corresponding Stochastic Differential Equations (SDEs) and prove that, under certain conditions on the boundaries a(t) and b(t), they admit a unique strong solution yielding the relevant diffusion process. We discuss the relationship between our result and the previous derivation of diffusion processes associated to square-root and linear time-boundaries, emphasizing the cases where our approach adds strong uniqueness, and study the local time and activity of the solution processes. We then study the peacock with uniform law at all times on a constant support [−1, 1] and derive the SDE of an associated mean-reverting diffusion process with uniform margins that is not a martingale. For the related SDE we prove existence of a solution in [0, T ]. Finally, we provide a numerical case study showing that these processes have the desired uniform behaviour. These results may be used to model random probabilities, random recovery rates or random correlations.

Under short sales prohibitions, no free lunch with vanishing risk (NFLVR-S) is known to be equivalent to the existence of an equivalent supermartingale measure for the price processes (Pulido [22]). For two given price processes, we translate the property (NFLVR-S) in terms of so called structure conditions and we introduce the concept of fundamental supermartingale measure. When a certain condition necessary to the construction of the fundamental martingale measure is not fulfilled, we provide the corresponding arbitrage portfolios. The motivation of our study lies in understanding the particular case of converging prices, i.e., that are going to cross at a bounded random time.

For a finite state Markov process X and a finite collection {Γ k , k ∈ K} of subsets of its state space, let τ k be the first time the process visits the set Γ k. In general, X may enter some of the Γ k at the same time and therefore the vector τ := (τ k , k ∈ K) may put nonzero mass over lower dimensional regions of R |K| + . these regions are of the form R s = {t ∈ R |K| + : t i = t j , i, j ∈ s(1)} ∩ |s| l=2 {t : t m < t i = t j , i, j ∈ s(l), m ∈ s(l − 1)} where s is any ordered partition of the set K and s(j) denotes the j th subset of K in the partition s. When |s| < |K|, the density of the law of τ over these regions is said to be " singular " because it is with respect to the |s|-dimensional Lebesgue measure over the region R s. We derive explicit/recursive and simple to compute formulas for these singular densities and their corresponding tail probabilities over all R s as s ranges over ordered partitions of K. We give a numerical example and indicate the relevance of our results to credit risk modeling. * The research of T.R.

In this paper we introduce the concept of \textit{conic martingales}. This class refers to stochastic processes having the martingale property, but that evolve within given (possibly time-dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. Specific attention is paid to martingales in $[0,1]$. One of these martingales proves to be analytically tractable. It is shown that up to shifting and rescaling constants, it is the only martingale (with the trivial constant, Brownian motion and Geometric Brownian motion) having a separable coefficient $\sigma(t,y)=g(t)h(y)$ and that can be obtained via a time-homogeneous mapping of \textit{Gaussian diffusions}. The approach is exemplified to the modeling of stochastic conditional survival probabilities in the univariate (both conditional and unconditional to survival) and bivariate cases.

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In this work, for a reference filtration F, we develop a method for computing the semimartingale decomposition of F-martingales in a specific type of enlargement of F. As an application, we study the progressive enlargement of F with a sequence of non-ordered default times and we show how to deduce results concerning the first-to-default, k-th-to-default, k-out-of-n-to-default or the all-to-default events. In particular, using this method, we compute explicitly the semimartingale decomposition of F-martingales under the absolute continuity condition of Jacod.

In this paper we introduce the concept of \textit{conic martingales}. This class refers to stochastic processes having the martingale property, but that evolve within given (possibly time-dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. Specific attention is paid to martingales in $[0,1]$. One of these martingales proves to be analytically tractable. It is shown that up to shifting and rescaling constants, it is the only martingale (with the trivial constant, Brownian motion and Geometric Brownian motion) having a separable coefficient $\sigma(t,y)=g(t)h(y)$ and that can be obtained via a time-homogeneous mapping of \textit{Gaussian diffusions}. The approach is exemplified to the modeling of stochastic conditional survival probabilities in the univariate (both conditional and unconditional to survival) and bivariate cases.

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This paper studies the impact, on no-arbitrage conditions, of stopping the price process at an arbitrary random time. As price processes, we consider the class of quasi-left-continuous semimartingales, i.e., semimartingales that do not jump at predictable stopping times. We focus on the condition of no unbounded profit with bounded risk (called NUPBR), also known in the literature as no arbitrage of the first kind. The first principal result describes all the pairs of quasi-left-continuous market models and random times for which the resulting stopped model fulfils NUPBR. Furthermore, for a subclass of quasi-left-continuous local martingales, we construct explicitly martingale deflators, i.e., strictly positive local martingales whose product with the price process stopped at a random time is a local martingale. The second principal result characterises the random times that preserve NUPBR under stopping for any quasi-left-continuous model. The analysis carried out in the paper is based on new stochastic developments in the theory of progressive enlargements of filtrations.

In this paper, we are interested by advanced backward stochastic differential equations (ABSDE), in a probability space equipped with a Brownian motion and a single jump process. The solution of the ABSDE is a triple (Y, Z, U) where Y is a semimartingale, Z is the diffusion coefficient and U the size of the jump. We allow the generator to depend on the future paths of the solution.

This paper completes the studies undertaken in [3, 4] and [8], where the authors quantify the impact of a random time on the No-Unbounded-Risk-with-Bounded-Profit concept (called NUPBR hereafter) for quasi-left-continuous models and discrete-time market models respectively. Herein, we focus on the NUPBR for semimartingales models that live on thin predictable sets only and when the extra information about the random time is added progressively over time. For this setting, we explain how far the NUPBR property is affected when one stops the model by an arbitrary random time or when one incorporates fully an honest time into the model. Furthermore, we show how to construct explicitly local martingale deflator under the bigger filtration from those of the smaller filtration. As consequence, by combining the current results on the thin case and those of [3, 4], we elaborate universal results for general semimartingale models.

This paper quantifies the interplay between the non-arbitrage notion of No-Unbounded-Profit-with-Bounded-Risk (NUPBR hereafter) and additional information generated by a random time. This study complements the one of Aksamit/Choulli/Deng/Jeanblanc [1] in which the authors studied similar topics for the case of stopping with the random time instead, while herein we are concerned with the part after the occurrence of the random time. Given that all the literature —up to our knowledge— proves that the NUPBR notion is always violated after honest times that avoid stopping times in a continuous filtration, herein we propose a new class of honest times for which the NUPBR notion can be preserved for some models. For this family of honest times, we elaborate two principal results. The first main result characterizes the pairs of initial market and honest time for which the resulting model preserves the NUPBR property, while the second main result characterizes the honest times that preserve the NUPBR property for any quasi-left continuous model. Furthermore , we construct explicitly " the-after-τ " local martingale deflators for a large class of initial models (i.e.,models in the small filtration) that are already risk-neutralized.

The paper gathers together ideas related to thin random time, i.e., random time whose graph is contained in a thin set. The concept naturally completes the studies of random times and progressive enlargement of filtrations. We develop classification and (*)-decomposition of random times, which is analogous to the decomposition of a stopping time into totally inaccessible and accessible parts, and we show applications to the hypothesis (H ′), honest times and informational drift via entropy.

We introduce a way to design Stochastic Differential Equations of diffusion type admitting a unique strong solution distributed as a uniform law with conic time-boundaries. We connect this general result to some special cases that where previously found in the peacock processes literature, and with the square root of time boundary case in particular. We introduce a special case with linear time boundary. We further introduce general mean-reverting diffusion processes having a constant uniform law at all times. This may be used to model random probabilities, random recovery rates or random correlations. We study local time and activity of such processes and verify via an Euler scheme simulation that they have the desired uniform behaviour.

We present some results on enlargement of filtration in discrete time. Many results known in continuous time extend immediately in a discrete time setting. Here, we provide direct proofs which are much more simpler. We study also arbitrages conditions in a financial setting and we present some specific cases, as immersion and pseudo-stopping times for which we obtain new results.

This thesis consists of four independent parts. The main thread of this one is the filtration magnification. In the first part, we present classical results of filtration magnification in discrete time. We study some examples in the context of initial filtration magnification. In the framework of progressive magnification we give conditions to obtain the immersion property of martingales. We also give various characterizations of pseudo stopping times and we state properties for honest times.In the second part, we are interested in the pricing of variable annuity products in the context of life insurance. For this we consider two models, in both of which we consider the market to be incomplete and adopt the indifference pricing approach. In the first model we assume that the insured makes random withdrawals and we compute the indifference premium by standard methods in stochastic control. We solve stochastic backward differential equations (SDEs) with a jump. We provide a verification theorem and we give the optimal strategies associated with our control problems. From these, we derive a computational method to obtain the indifference premium. In the second model we propose the same approach as in the first model but we assume that the insured makes withdrawals that correspond to the worst case for the insurer. In the third part, we study the relationship between the EDSR solutions in two different filterings. We then study the relationship between these two solutions. We apply these results to obtain the indifference price in the two filtrations, i.e. the price at which an agent would have the same level of expected utility using additional information.In the fourth part, we consider advanced stochastic backward differential equations (EDSRAs) with one jump. We study the existence and uniqueness of a solution to these EDSRAs. For this purpose we use the decomposition of jump processes related to the progressive coarsening of filtration to bring us back to the study of Brownian EDSRAs before and after the jump time.

In this paper, we study the classical problem of maximization of the sum of the utility of the terminal wealth and the utility of the consumption, in a case where a sudden jump in the risk-free interest rate creates incompleteness. The value function of the dual problem is proved to be solution of a BSDE and the duality between the primal and the dual value functions is exploited to study the BSDE associated to the primal problem.

We consider the problem of maximizing the expected amount of time an exponential martingale spends above a constant threshold up to a finite time horizon. We assume that at any time the volatility of the martingale can be chosen to take any value between σ 1 and σ 2 , where 0 < σ 1 < σ 2. The optimal control consists in choosing the minimal volatility σ 1 when the process is above the threshold, and the maximal volatility if it is below. We give a rigorous proof using classical verification and provide integral formulas for the maximal expected occupation time above the threshold.

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The central objective of the thesis is to study various measures of model risk, expressed in monetary terms, that can be consistently applied to a heterogeneous collection of financial products. The first two chapters deal with this problem, first from a theoretical point of view, and then by conducting an empirical study focused on the natural gas market. The third chapter focuses on a theoretical study of the so-called basis risk. In the first chapter, we focused on the valuation of complex financial products, taking into account the model risk and the availability in the market of basic derivatives, also called vanilla. In particular, we have pursued the optimal transport approach (known in the literature) for the computation of price bounds and model-risk robust over- (under-) hedging strategies. In particular, we revive a construction of martingale probabilities under which the price of an exotic option reaches the so-called price bounds, focusing on the case of positive martingales. We also highlight significant symmetry properties in the study of this problem. In the second chapter, we approach the model risk problem from an empirical point of view, by studying the optimal management of a unit of natural gas and quantifying the effect of this risk on its optimal value. In this study, the valuation of the storage unit is based on the spot price, while its hedging is done with forward contracts. As mentioned before, the third chapter focuses on the basis risk, which arises when one wants to hedge a contingent asset based on an unprocessed asset (e.g. temperature) using a portfolio of processed assets in the market. A hedging criterion in this context is that of variance minimization, which is closely related to the so-called Föllmer-Schweizer decomposition. This decomposition can be deduced from the solution of a certain stochastic backward differential equation (SDE) directed by a possibly jumping martingale. When this martingale is a standard Brownian motion, the SRDEs are strongly associated with semilinear parabolic PDEs. In the general case we formulate a deterministic problem that extends the mentioned PDEs. We apply this approach to the important special case of the Föllmer-Schweizer decomposition, for which we give explicit expressions for the payoff decomposition of an option when the underlyings are exponential of additive processes.

This volume is dedicated to the memory of Marc Yor, who passed away in 2014. The invited contributions by his collaborators and former students bear testament to the value and diversity of his work and of his research focus, which covered broad areas of probability theory. The volume also provides personal recollections about him, and an article on his essential role concerning the Doeblin documents. With contributions by P. Salminen, J-Y. Yen & M. Yor. J. Warren. T. Funaki. J. Pitman& W. Tang. J-F.

We study problems related to the predictable representation property for a progressive enlargement G of a reference filtration F through observation of a finite random time τ. We focus on cases where the avoidance property and/or the continuity property for F-martingales do not hold and the reference filtration is generated by a Poisson process. Our goal is to find out whether the predictable representation property (PRP), which is known to hold in the Poisson filtration, remains valid for a progressively enlarged filtration G with respect to a judicious choice of G-martingales.

Given a reference filtration F, we consider the cases where an enlarged filtration G is constructed from F in two different ways: progressively with a random time or initially with a random variable. In both situations, under suitable conditions, we present a G-optional semimartingale decomposition for F-local martingales. Our study is then applied to answer the question of how an arbitrage-free semimartingale model is affected when stopped at the random time in the case of progressive enlargement or when the random variable used for initial enlargement satisfies Jacod's hypothesis. More precisely, we focus on the No-Unbounded-Profit-with-Bounded-Risk (NUPBR) condition. We provide alternative proofs of some results from [5], with a methodology based on our optional semimartingale decomposition, which reduces significantly the length of the proof.

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We apply the default density framework developed in El Karoui et al. \cite{ejj1} to modelling of multiple defaults, which can be adapted to both top-down and bottom-up models. We present general pricing results and establish links with the classical intensity approach. Explicit models are also proposed by using the methods of change of probability measure or dynamic copula.

In this thesis we consider the utility maximization and indifference price formation problem for exponential semimartingale models depending on a random factor ξ. The challenge is to solve the indifference price problem using space magnification and filtration. We reduce the maximization problem in the magnified filtration to the conditional problem, knowing {ξ = v}, which we solve using a dual approach. For HARA-utility we introduce information such as relative entropies and Hellinger-type integrals, as well as the corresponding information processes, enfin to express, via these processes, the maximal utility. In particular, we study exponential Lévy models, where the information processes are deterministic which considerably simplifies the calculations of indifference prices. Infin, we apply the results to the geometric Brownian motion model and the diffusion-jump model that includes Brownian motion and Poisson processes. In the logarithmic, power, and exponential utility cases, we provide the explicit formulas for the information, and then, using numerical methods, we solve the equations to obtain the indifference prices in the case of selling a European option.

For a finite state Markov process X and a finite collection {Γ k , k ∈ K} of subsets of its state space, let τ k be the first time the process visits the set Γ k. In general, X may enter some of the Γ k at the same time and therefore the vector τ := (τ k , k ∈ K) may put nonzero mass over lower dimensional regions of R |K| + . these regions are of the form R s = {t : t i = t j , i, j ∈ s(1)}∩ |s| l=2 {t : t m < t i = t j , i, j ∈ s(l), m ∈ s(l −1)} where s is any ordered partition of the set K and s(j) denotes the j th subset of K in the partition s. When |s| < |K|, the density of the law of τ over these regions is said to be " singular " because it is with respect to the |s|-dimensional Lebesgue measure over the region R s. We derive explicit/recursive and simple to compute formulas for these singular densities and their corresponding tail probabilities over all R s as s ranges over ordered partitions of K. We give a numerical example and indicate the relevance of our results to credit risk modeling.

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In this paper we study a utility maximization problem with random horizon and reduce it to the analysis of a specific BSDE, which we call BSDE with singular coefficients, when the support of the default time is assumed to be bounded. We prove existence and uniqueness of the solution for the equation under interest. Our results are illustrated by numerical simulations.

In the context of a general continuous financial market model, we study whether the additional information associated with an honest time gives rise to arbitrage profits. By relying on the theory of progressive enlargement of filtrations, we explicitly show that no kind of arbitrage profit can ever be realised strictly before an honest time, while classical arbitrage opportunities can be realised exactly at an honest time as well as after an honest time. Moreover, stronger arbitrages of the first kind can only be obtained by trading as soon as an honest time occurs. We carefully study the behavior of local martingale deflators and consider no-arbitrage-type conditions weaker than NFLVR.Comment: 25 pages, revised versio.

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We study the stability of several no-arbitrage conditions with respect to absolutely continuous, but not necessarily equivalent, changes of measure. We first consider models based on continuous semimartingales and show that no-arbitrage conditions weaker than NA and NFLVR are always stable. Then, in the context of general semimartingale models, we show that an absolutely continuous change of measure does never introduce arbitrages of the first kind as long as the change of measure density process can reach zero only continuously.

In this Note we study a class of BSDEs which admits a particular singularity in their driver. More precisely, we assume that the driver is not integrable and degenerates when approaching to the terminal time of the equation.

This thesis deals with problems associated with the theory of filtration magnification. It is divided into two parts: the first part is devoted to random times. We study the properties of the different classes of random times from the point of view of the filtration magnification. The second part concerns the study of the stability of the arbitration condition on the filtration magnification.

This paper quantifies the impact of stopping at a random time on non-arbitrage, for a class of semimartingale models. We focus on No-Unbounded-Profit-with-Bounded-Risk (called NUPBR hereafter) concept, also known in the literature as the arbitrage of the first kind. The first principal result lies in describing the pairs of market model and random times for which the resulting stopped model fulfills the NUPBR condition. The second principal result characterises the random time models that preserve the NUPBR property after stopping for any quasi-left-continuous market model. The analysis that drives these results is based on new stochastic developments in martingale theory with progressive enlargement of filtration. Furthermore, we construct explicit martingale densities (deflators) for a subclass of local martingales when stopped at a random time.

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We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time \tau. Under minimal assumptions on the random time and on the driving Brownian motions, we study the behavior of the model in all the filtrations which naturally arise in this setting, establishing martingale representation results and characterizing the validity of the NA1 and NFLVR no-arbitrage conditions.

This thesis deals with several topics in financial mathematics: credit risk, portfolio optimization and interest rate modeling. Chapter 1 consists of three studies in the area of credit risk. The most innovative is the first one in which we build a model such that the immersion property is not verified under any equivalent martingale measure. Chapter 2 studies the problem of maximizing the sum of a terminal wealth utility and a consumption utility. Chapter 3 studies the valuation of interest rate derivatives in a multi-curve framework, which takes into account the difference between a risk-free rate curve and Libor rate curves of different tenors.

The financial markets have expanded considerably over the last three decades and have seen the emergence of a variety of derivative products. The most widely used of these derivatives are American options.

This thesis deals with density models for default times and their application to credit and counterparty risk. The first part is a theoretical contribution to the study of projections on different filtrations of the Radon-Nikodym density, in the form of Doleans-Dade exponential, occurring during measurement changes. The main result is the characterization of the measurement changes that preserve the immersion, obtained by applying our projection formulas. The second part aims at an informational dynamization of the static Gaussian copula model applied to a credit portfolio, which can be seen as a density model allowing to deal with CDO hedging by CDS or counterparty risk on credit derivatives. The main contributions are the introduction of the dynamic perspective, which gives a theoretical justification to the Gaussian copula bump-sensitivities used by practitioners, and the application to CVA calculations on a CDS.

In this Note we study a class of BSDEs which admits a particular singularity in their driver. More precisely, we assume that the driver is not integrable and degenerates when approaching to the terminal time of the equation.

In this paper, we study the classical problem of maximization of the sum of the utility of the terminal wealth and the utility of the consumption, in a case where a sudden jump in the risk-free interest rate creates incompleteness. The value function of the dual problem is proved to be solution of a BSDE and the duality between the primal and the dual value functions is exploited to study the BSDE associated to the primal problem.

This work is concerned with the theory of initial and progressive enlargements of a reference filtration F with a random time tau. We provide, under an equivalence assumption, slightly stronger than the absolute continuity assumption of Jacod, alternative proofs to results concerning canonical decomposition of an F-martingale in the enlarged filtrations. Also, we address martingales' characterization in the enlarged filtrations in terms of martingales in the reference filtration, as well as predictable representation theorems in the enlarged filtrations.

In this thesis work the information process regarding a default instant τ in a credit risk model is described by a Brownian bridge on the stochastic interval [0, τ]. Such a bridge process is characterized as more suitable in modeling than the classical model considering the indicator I[0, τ]. After studying the associated Bayes formulas, this approach to modeling default time information is linked with other financial market information. This is done using the theory of filtration magnification, where the filtration generated by the information process is expanded by the reference filtration describing other information not directly related with the default. Special attention is devoted to the classification of the defect time with respect to the minimum filtration but also to the expanded filtration. Sufficient conditions, under which τ is totally unreachable, are discussed, but also an example is given in which τ avoids downtime, is totally unreachable with respect to minimum filtration and predictable with respect to extended filtration. Finally, financial contracts such as, for example, private bonds and credit default swaps, are studied in the context described above.

This thesis deals with the modeling of credit derivatives and consists of two parts: The first part concerns the density model, recently proposed by El Karoui et al. where we make the assumption that the conditional law of default time knowing reference filtration is equivalent to its (unconditional) law. Under this assumption, we give different (and simpler) proofs to the already existing results in the theory of initial and progressive coarsening of filtrations. In addition, we present new results such as the predictable representation theorem for progressively coarsened filtration in the multidimensional case. We then propose several methods for constructing density models in both the one-dimensional and multi-dimensional cases. Finally, we show that the density model is an efficient approach for dynamic hedging of multi-name credit derivatives. In the second part, in order to study the counterparty risk in a CDS contract, we propose a Markov model in which simultaneous defaults are possible. The wrong-way risk is thus represented by the fact that, at the time of the counterparty default, there is a strictly positive probability that the reference entity will also default. We begin by considering a Markov chain with four states corresponding to two names. In this simple case, we obtain semi-explicit formulas for most important quantities, such as price, CVA, EPE, or hedge ratios. We then generalize this framework to account for spread risk by introducing stochastic factors. We treat a Markovian copula model with stochastic intensities. We also address the issue of dynamic CVA hedging with a written CDS on the counterparty. For the implementation of the model, we specify the intensities by affine processes, which given the dynamic copula property of the model, makes the calibration of this model efficient. Numerical results are presented to show the relevance of the CVA behavior in the model with the stylized market facts.

This thesis deals with the optimization of asset portfolios subject to default risk. The current crisis has allowed us to understand that it is important to take into account the risk of default to be able to give the real value of its portfolio. Indeed, due to the different exchanges of the financial market actors, the financial system has become a network of several connections which it is essential to identify in order to evaluate the risk of investing in a financial asset. In this thesis, we define a financial system with a finite number of connections and we propose a model of the dynamics of an asset in such a system by taking into account the connections between the different assets. The measurement of the correlation will be done through the jump intensity of the processes. Using Stochastic Differential Backward Equations (SDGE), we will derive the price of a contingent asset and take into account the model risk in order to better evaluate the optimal consumption and wealth if one invests in such a market.

This thesis consists of five independent parts dedicated to the modeling and study of the problems related to the risk of default, in partial information. The first part constitutes the Introduction. The second part is dedicated to the calculation of the survival probability of a firm, conditional on the information available to the investor, in a structural model with partial information. We use a hybrid numerical technique based on the Monte Carlo method and optimal quantization. In the third part we treat, with the Dynamic Programming approach, a discrete time problem of maximizing the utility of terminal wealth, in a market where securities subject to default risk are traded. The risk of contagion between defaults is modeled, as well as the possible uncertainty of the model. In the fourth part, we address the problem of uncertainty related to the investment time horizon. In a complete market subject to the risk of default, we solve, either with the martingale method or with Dynamic Programming, three problems of maximizing the utility of consumption: when the time horizon is fixed, finite but uncertain and infinite. Finally, in the fifth part we deal with a purely theoretical problem. In the context of the coarsening of filtrations, our goal is to re-demonstrate, in a specific framework, the already known results on the characterization of martingales, the decomposition of martingales with respect to the reference filtration as semimartingales in progressively and initially coarsened filtrations and the Predictable Representation Theorem.

This thesis deals - in three essays - with portfolio choice problems in a situation of partial information, a theme that we present in a short introduction. The essays developed each address a particularity of this problem. The first essay (co-written with M. Jeanblanc and V. Lacoste) deals with the question of choosing the optimal strategy for a terminal utility maximization problem when the evolution of prices is modeled by an Itô-Lévy process whose trend and intensity of jumps are not observed. The approach consists in rewriting the initial problem as a reduced problem in the filtration generated by the prices. This requires the derivation of the nonlinear filtering equations, which we develop for a Lévy process. The problem is then solved using dynamic programming by the Bellman and HJB equations. The second essay addresses in a Gaussian framework the question of the cost of uncertainty, which we define as the difference between the optimal strategies (or maximum wealth) of a perfectly informed agent and a partially informed agent. The properties of this cost of information are studied in the context of the three standard forms of utility functions and numerical examples are presented. Finally, the third essay addresses the issue of portfolio choice when market price information is only available at discrete and random dates. This amounts to assuming that price dynamics follow a marked process. In this framework, we develop the filtering equations and rewrite the initial problem in its reduced form in discrete price filtration. The optimal strategies are then computed using the Malliavin calculus for random measures and an extension of the Clark-Ocone-Haussman formula is presented for this purpose.

This thesis deals with the calculation and allocation of regulatory and economic capital for credit risk in retail banking. For the calculation of regulatory capital, we propose an approach that relies on a model of the loss on the portfolio at the horizon of one year. Thus, the model for portfolio aggregation is developed according to two methods: Basel II and a more general model called Basel II "extended". Capital requirements are calculated using indicators constructed with quantiles of the loss distribution. We emphasize the benefits of our correlation structure, the core of the Basel II extended approach, to take into account diversification effects. We calculate an economic capital from a valuation of the debt portfolios. The economic capital is defined as the average value of the portfolio at the horizon, minus the Expected Shortfall of the value distribution.

This thesis is composed of two parts: in the first part, we study a complete market whose risky asset is a discontinuous process. The second part is devoted to a default risk model. We emphasize the difference between default and non-default market information. Chapters four and five deal respectively with the case where the information is coarse and the case where the default time is a stopping time for the filtration generated by the information available in the default-free market. In the next chapter, we study the conservation property of martingales (assumption (H)). In this framework, we establish a predictable representation theorem and make the link between hypothesis (H) and the absence of arbitrage. The next two chapters generalize these results to the presence of several default moments and to the case where hypothesis (H) is not verified. The ninth chapter studies the incompleteness generated by the default. In particular, we characterize the set of equivalent measure martingales. We determine the range of prices for some contingent assets. Using the predictability theorems, we show that the market can be completed by a zero-coupon with default and we explain the hedging of assets. The last chapter is first devoted to the problem of optimizing the expected wealth utility in the presence of a default. We show that the use of a utility function allows the agent to set a unique martingale equivalent measure. Then, we solve a maximization problem with a random horizon.

This thesis contains two parts: the first part deals with numerical methods and the second part studies their applications in finance. The first chapters are devoted to a description of Monte-Carlo, quasi Monte-Carlo and hybrid methods. We give an estimate of the variation of a function and techniques to reduce it. We also give an estimate of the extended discrepancy of one-dimensional sequences, in particular those whose terms are a sum of components of a multidimensional sequence with low discrepancy. Then, the last chapters focus on the valuation and hedging of options with one or more risky assets, such as a European call in a full market model with jumps, an Asian call in an incomplete market model with jumps or a basket call in a multidimensional Black-Scholes model. We obtain many numerical results and prove that some functions from finance are not finite variable.

This thesis studies markets whose risk assets are discontinuous processes. The interest in such modelizations is justified by numerous statistical studies, which show discontinuities in the observed price trajectories. The first two chapters are devoted to valuation problems in an incomplete market with a risk-free asset and a risk asset. We assume that the risk asset is a jump spread. In this incomplete market, we determine the range of viable prices (i.e., not creating arbitrage opportunities) of European contingent assets, and we study some properties of these prices. We then analyze the case of Asian and American finite-maturity options, as well as American perpetual put options. The third chapter examines the completion of such markets on the one hand, and on the other hand presents some examples of complete markets driven by a single jumping asset. The following chapters are devoted to the hedging problem. We consider an agent whose strategy is specified by a utility function and we study the maximization of the expected utility of his terminal wealth. When the market is incomplete and driven by a mixed diffusion, we establish the existence of an optimal strategy that we characterize. We prove that the use of utility functions brings a significant reduction in the range of viable prices. Finally, we consider the hedging failure in markets driven by mixed diffusion.

This thesis has five chapters. In the first chapter, we return to the theory of expectations. After a reminder of the basic notions, we give two new formulations of the expectations hypotheses. The first one is obtained in the framework of the quadratic Gaussian model. The second one is obtained in the case of a diffusion model with jumps. In the second chapter, two problems are exposed: the long rate problem and the problem of affine factorial models in the presence of jumps. First, we show that the long rate is a random process. Then, we obtain new constraints on the rate curve in the framework of affine factorial models. The third chapter is inspired by a work done by el karoui and cherif (1992) on multi-currency arbitrage. After a reminder of some known results, we give the new arbitrage relations in the framework of a discontinuous model. Then, we study some quanto options. The first part of the fourth chapter is a complement to the third. In the second part, we study some quanto-barrier options and level barriers. In the last part, examples on the terminal wealth maximization problem are presented in detail. The last chapter studies the exchange rate: the problems of intervention in a target zone regime and the problem of the single currency are discussed.