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Backward Stochastic Differential Equations Associated to a Symmetric Markov Process

Vlad Bally 1 Etienne Pardoux L. Stoica
1 MATHFI - Financial mathematics
Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech, UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12
Abstract : We consider a second order semi-elliptic differential operator L with measurable coefficients, in divergence form, and the semilinear parabolic PDE \begin{eqnarray*} (_t+L)u(t,x)+f(t,x,u,u) &=&0,0tT u(T,x) &=&(x) \end{eqna- rray*} and employ the symmetric Markov process of infinitesimal operator L in order to give a probabilistic interpretation for the solution u, i.e. we solve the corresponding BSDE. We obtain also a representation theorem for martingales which represents a generalization of the representation theorem given by Fukushima for additive functional martingales. This permits us to solve general (non-Markov) BSDE's with semi-linear terms. The nonlinear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to u. Finally we prove a comparison theorem and use it in order to solve a stochastic control problem.
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https://hal.inria.fr/inria-00072163
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Submitted on : Tuesday, May 23, 2006 - 7:56:55 PM
Last modification on : Wednesday, September 4, 2019 - 1:52:07 PM
Long-term archiving on: : Sunday, April 4, 2010 - 10:56:24 PM

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Vlad Bally, Etienne Pardoux, L. Stoica. Backward Stochastic Differential Equations Associated to a Symmetric Markov Process. [Research Report] RR-4425, INRIA. 2002. ⟨inria-00072163⟩

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