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Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Stochastic equations in infinite dimensions

This PhD dissertation presents two independent research topics dealing with contemporary issues in mathematical finance, the second one being divided into into two distinct problems. Throughout the first part of the dissertation, we study the notion of second order backward stochastic differential equations (2BSDE in the following), first introduced by Cheredito, Soner, Touzi and Victoir, then reformulated by Soner, Touzi and Zhang. We start by proving an extension of their existence and uniqueness results to the case of a continuous generator with linear growth. Then, we pursue our study with another extension to the case of a quadratic generator. The theoretical results obtained in that chapter allow us to solve a problem of utility maximization for an investor in an incomplete market, the source of incompleteness being on one hand the restrictions on the class of admissible trading strategies, and on the other hand the fact that the volatility of the market is uncertain. We prove the existence of optimal strategies, we characterize the value function of the problem thanks to a 2BSDE and solve explicetely several examples which give further insight into the main modifications introduced by the uncertain volatility framework. We conclude the first part of the dissertation by introducing the notion of 2BSDEs reflected on an obstacle. We prove existence and uniqueness of the solutions of those equations and propose an application to the pricing problem of American options under volatility uncertainty. The first chapter of the second part of the dissertation deals with a problem of option pricing in an illiquidity model. We provide asymptotic expansions of those prices in the infinite liquidity limit and highlight a transition phase effect depending on the regularity of the payoff considered. We also give numerical results. Finally, the last chapter of this thesis is devoted to a Principal/Agent problem with moral hazard. A bank (the agent) has a certain number of defaultable loans and is ready to exchange their interests with the promess of payments. The bank can influence the default probabilities by choosing whether it monitors the loans or not, this monitoring being costly for the bank. Those choices are only known by the bank itself. Investors (the principal) want to design contracts which maximize their utility while implicitely giving incentives to the bank to monitor all the loans at all times. We solve explicitely this optimal control problem, we describe the associated optimal contract and its economic implications and provide some numerical simulations.

https://pastel.archives-ouvertes.fr/pastel-00651589/document

A journey through second order BSDEs and other contemporary issues in mathematical finance.

In this paper, we extend the optimal securitisation model of Pages [50] and Possamai and Pages [51] between an investor and a bank to a setting allowing both moral hazard and adverse selection. Following the recent approach to these problems of Cvitanic, Wan and Yang [14], we characterise explicitly and rigorously the so-called credible set of the continuation and temptation values of the bank, and obtain the value function of the investor as well as the optimal contracts through a recursive system of first-order variational inequalities with gradient constraints. We provide a detailed discussion of the properties of the optimal menu of contracts.

Bank monitoring incentives under moral hazard and adverse selection

https://www.math.hu-berlin.de/~fiebig/paper_Imkeller/IMPR1.pdf

A note on the Malliavin-Sobolev spaces

In this paper we provide new conditions for the Malliavin differentiability of solutions of Lipschitz or quadratic BSDEs. Our results rely on the interpretation of the Malliavin derivative as a G{\^a}teaux derivative in the directions of the Cameron-Martin space. Incidentally, we provide a new formulation for the characterization of the Malliavin-Sobolev type spaces $D^{1,p}$ .

On the Malliavin differentiability of BSDEs

The aim of this paper is twofold. First, we extend the results of [33] concerning the existence and uniqueness of second-order reflected 2BSDEs to the case of two obstacles. Under some regularity assumptions on one of the barriers, similar to the ones in [10], and when the two barriers are completely separated, we provide a complete wellposedness theory for doubly reflected second-order BSDEs. We also show that these objects are related to non-standard optimal stopping games, thus generalizing the connection between DRBSDEs and Dynkin games first proved by Cvitanic and Karatzas [11]. More precisely, we show under a technical assumption that the second order DRBSDEs provide solutions of what we call uncertain Dynkin games and that they also allow us to obtain super and subhedging prices for American game options (also called Israeli options) in financial markets with volatility uncertainty

Dylan Possamaï

Papers

A journey through second order BSDEs and other contemporary issues in mathematical finance.

Bank monitoring incentives under moral hazard and adverse selection

A note on the Malliavin-Sobolev spaces

On the Malliavin differentiability of BSDEs

Second-order BSDEs with general reflection and game options under uncertainty