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Some contributions to backward stochastic differential equations and applications

Abstract : This thesis is dedicated to the study of backward stochastic differential equations (BSDEs) and their applications. In chapter 1, we study the problem of maximizing the utility from terminal wealth where the stock price may jump and there are investment constraints on the agent 's strategies. We focus on the BSDE whose solution represents the maximal utility, which allows transferring results on quadratic BSDEs, in particular the stability results, to the problem of utility maximisation. In chapter 2, we consider the problem of pricing American options from theoretical and numerical sides based upon an alternative representation of the value of the option in the form of a viscosity solution of a parabolic equation with a nonlinear reaction term. We extend the viscosity solution characterization proved in [Benth, Karlsen and Reikvam 2003] for call/put American option prices to the case of a general payoff function in a multi-dimensional setting. We address two new numerical schemes inspired by the branching processes. Our numerical experiments show that approximating the discontinuous driver of the associated reaction/diffusion PDE by local polynomials is not efficient, while a simple randomization procedure provides very good results. In chapter 3, we prove existence and uniqueness results for a general class of coupled mean-field forward-backward SDEs with jumps under weak monotonicity conditions and without the non-degeneracy assumption on the forward equation and we give an application in the field of storage in smart grids in the case where the production of electricity is unpredictable.
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Submitted on : Friday, November 20, 2020 - 4:08:07 PM
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Arij Manai. Some contributions to backward stochastic differential equations and applications. General Mathematics [math.GM]. Le Mans Université; Université de Tunis El-Manar. Faculté des Sciences de Tunis (Tunisie), 2019. English. ⟨NNT : 2019LEMA1022⟩. ⟨tel-03016927⟩



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