Introduction to Stochastic Analysis: Integrals and Differential Equations

Chapter 10

Solutions of SDEs as Markov Diffusion Processes

 

 

10.1. Introduction

One consequence of the independence of increments of Brownian motion B is their orthogonality to the past or, in other words, that Brownian motion is a martingale. Another consequence is the so-called Markov property. Intuitively, the Markov property of a random process is described as follows: its probabilistic behavior after any moment s (in the future) depends on the value of the process at the time moment s (in the present) and does not depend on the values of the process until the moment s(in the past). There are many more or less rigorous ways to define the Markov property. It is relatively simple to do for random processes having densities. A process X is called a (real) Markov process if its conditional density satisfies the equality

[10.1]

 

for all 0 t₁ < t₂ < … < t_k < s < t, x₁, x₂, …, x_k, x, y ∈ Then the function p = p(s,x,t,y), 0 < s < t,x,y ∈ is called a transition density of the Markov process X. Most often, the Markov property is applied when written in terms of conditional expectations ...