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Markov processes and parabolic partial differential equations

Mireille Bossy 1, * Nicolas Champagnat 1, *
* 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.
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Mireille Bossy, Nicolas Champagnat. Markov processes and parabolic partial differential equations. Cont Rama. Encyclopedia of Quantitative Finance, John Wiley & Sons Ltd. Chichester, UK, pp.1142-1159, 2010. ⟨inria-00193883v2⟩

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