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

Abstract : 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.
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Submitted on : Tuesday, December 13, 2011 - 8:13:57 PM
Last modification on : Wednesday, March 27, 2019 - 4:08:30 PM
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Dylan Possamaï. A journey through second order BSDEs and other contemporary issues in mathematical finance.. Probability [math.PR]. Ecole Polytechnique X, 2011. English. ⟨pastel-00651589⟩

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