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* Corresponding author 1 TOSCA INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502

Abstract : In the first part of this article, we present the main tools and definitions of Markov processes- theory: transition semigroups, Feller processes, infinitesimal generator, Kolmogorov-s backward and forward equations and Feller diffusion. We also give several classical examples including stochastic differential equations SDEs and backward SDEs BSDEs. The second part of this article is devoted to the links between Markov processes and parabolic partial differential equations PDEs. In particular, we give Feynman-Kac formula for linear PDEs, we present Feynman-Kac formula for BSDEs, and we give some examples of the correspondance between stochastic control problems and Hamilton-Jacobi-Bellman HJB equations and between optimal stopping problems and variational inequalities. Several examples of financial applications are given to illustrate each of these results, including European options, Asian options and American put options.

Keywords : Markov processes parabolic partial differential equations semigroup Feller processes strong Markov property infinitesimal generator Kolmogorov-s backward and forward equations Fokker-Planck equation stochastic differential equations backward stochastic differential equations heat equation Feynman-Kac formula Hamilton-Jacobi-Bellman equation variational inequality

Author: Mireille Bossy - Nicolas Champagnat -



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