Markov Processes

Abstract

Abstract We present the main concepts of the theory of Markov processes: 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 stochastic differential equations (BSDEs) and describe the links between Markov processes and parabolic partial differential equations (PDEs). In particular, we state the Feynman–Kac formula for linear PDEs and 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 financial applications are given to illustrate each of these results, including European options, Asian options, and American put options.