Markov processes and parabolic partial differential equations

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 correspondence between stochastic control problems and Hamilton-Jacobi-Bellman (HJB) equations and between optimal stopping problems and variational inequalities. Several examples of finan∗TOSCA project-team INRIA Sophia Antipolis – Méditerranée Research Center 2004 route des Lucioles, BP-93 06902 Sophia Antipolis Cedex, France [email protected] and [email protected] 1 in ria -0 01 93 88 3, v er si on 2 25 J an 2 00 8 cial applications are given to illustrate each of these results, including European options, Asian options and American put options.