Backward and Forward equations for Diffusion processes

This section is devoted to the discussion of two fundamental (partial) differential equations, that arise in the context of Markov diffusion processes. After giving a brief introduction of continuous-time continuous state Markov processes, we introduce the forward and backward equation, and provide a heuristic derivation of these equations for diffusion processes. We also discuss some examples and features of these two equations. In this section we discuss two partial differential equations (PDE) that arise in the theory of continuous-time continuous-state Markov processes, which was introduced by Kolmogorov in 1931. Here, we focus only on Markov diffusion processes (see Section 2.1.6.1) and describe the forward and backward equation for such processes. The forward equation is also known as Fokker-Planck equation (and was already known in the physics literature before Kolmogorov formulated these). We begin by a brief introduction to continuous-time continuous-state Markov processes which are continuous analogs of Discrete Time Markov Chains (DTMC) and Continuous Time Markov Chains (CTMC) discussed earlier in Section 2.1.1 and 2.1.2 followed by some basic properties of Markov processes. Then we state the two equations and provide sketches of the proofs. Finally, we conclude the section with some specific examples and features of these equations. Preliminaries. Diffusion processes have been discussed in Section 2.1.6.1. For simplicity of the exposition, we consider the following time-homogeneous version of the diffusion process for this section: A (time-homogeneous) ltô diffusion is a stochastic process {X(t)} satisfying a stochastic differential equation of the form dX(t) = b(X(t))dt+ σ(X(t))dW (t), t > 0; X(0) = x, (1) where {W (t)} is a (standard) Brownian motion and b, σ are functions that satisfy : |σ(x)− σ(y)| < D|x− y|; x, y ∈ IR.