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EDSs réfléchies en moyenne avec sauts et EDSs rétrogrades de type McKean-Vlasov : étude théorique et numérique

Abstract : This thesis is devoted to the theoretical and numerical study of two main subjects in the context of stochastic differential equations (SDEs): mean reflected SDEs with jumps and McKean-Vlasov backward SDEs.The first part of my thesis establishes the propagation of chaos for the mean reflected SDEs with jumps. First, we study the existence and uniqueness of a solution. Then, we develop a numerical scheme based on the particle system. Finally, we obtain the rate of convergence of this scheme.The second part of my thesis studies the McKean-Vlasov backward SDEs. In this case, we prove the existence and uniqueness of a solution for such equations. Then, thanks to the Wiener chaos expansion, we provide a numerical approximation. Moreover, the convergence rate of this approximation is also determined.The third part of my thesis proposes another type of simulation for the McKean-Vlasov backward SDEs. Due to the approximation of Brownian motion by a scaled random walk, we develop a numerical scheme and we get its convergence rate.In addition, a few numerical examples in these three parts are given to illustrate the efficiency of our schemes and their convergence rates stated by the theoretical results.
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Abir Ghannoum. EDSs réfléchies en moyenne avec sauts et EDSs rétrogrades de type McKean-Vlasov : étude théorique et numérique. Mathématiques générales [math.GM]. Université Grenoble Alpes; Université libanaise, 2019. Français. ⟨NNT : 2019GREAM068⟩. ⟨tel-02951412⟩

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