Markov processes and parabolic partial differential equations
Résumé
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.
Mots clés
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
Domaines
Probabilités [math.PR]Origine | Fichiers produits par l'(les) auteur(s) |
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