Contributions à l'étude des équations différentielles stochastiques rétrogrades fléchies et applications aux équations et dérivées partielles PDF Download

Author: Mingyu Xu
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Languages : en
Pages : 218

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Book Description
In the first chapter, we consider the reflected backward stochastic differential equation (BSDEsin short) with one or two right continuous and left limited (RCLL in short) barriers. Using the Picarditeration method, we obtained the existence and uniqueness of the solution of the reflected BSDEwith two RCLL barriers. Then we use the penalization method to the case of one RCLL barrier.Considering the solutions (Y n,Zn,Kn) of penalized equations as solutions of reflected BSDEs,we prove that the limit (Y,Z,K) is the solution of equation, by properties of Snell envelope andmonotonic limit theorem (Peng S., 1999). In the case of equation with two RCLL barriers, by theanalogue method, we prove the limit (Y,Z,K) of penalized equation is the solution of problem,by the representation of solutions via Dynkin game. Here we need a generalized monotonic limittheorem, which permit us to pass the limit for penalized equations.In a second work, we have generalized this type of result to the case where barriers are just inL2, by the method of penalization and the theory of g-supersolution.In the second chapter, we consider the reflected BSDEs with one continuous barrier, associatedto (_, f,L), when _ 2 L2(FT ), f(t, !, y, z) is continuous, satisfies monotonic and general increasingconditions on y, and Lipschitz condition on z, and when the barrier (Lt)0_t_T is a progressivelymeasurable continuous process, which verifies certain integrability condition.We have also notable prove the existence and uniqueness of solution in L2, for this reflectedequation with determinist terminal time. The proof of existence is effected by four steps. The firststep consists to prove the result under the boundness condition of _, f(t, 0) et L+. The second step(the most delicate) consists to relax the boundness condition of L+ ; the following two step permitus to obtain the general result, relaxing the boundness condition on _ and f(t, 0). The comparisontheorems play important roles, which help us to pass the limit in the equations. Then we study thecase when the terminal time is a stopping time. The existence and uniqueness of the solution arealso proved.In the third chapter, we have studied the reflected BSDEs with one barrier, whose generator fsatisfies the monotonic and general increasing condition on y, and quadratic and linear condition onz, when the barrier L is uniformly bounded. We prove the existence of a solution by approximation,under these conditions. We also find a necessary and sufficient condition for the case f(t, !, y, z) =|z|2, and construct its solution explicitly. For the case f(t, !, y, z) = |z|p, p 2 (1, 2), we prove asufficient condition.In the forth chapter, we treat the reflected BSDE with two barrier, when f satisfies the mono-tonic, continuous and general increasing conditions on y, and Lipschitz condition on z, like in thesecond chapter. For the barriers, we suppose that L and U are continuous, L