Introduction to Stochastic Calculus with Applications, Second Edition
Containing many solved examples and exercises, this book gives a simple but rigorous treatment of stochastic calculus and its applications, including a range of advanced topics.
TABLE OF CONTENTS Introduction to Stochastic Calculus with Applications, Second Edition
Chapter 6: Diffusion Processes
In this chapter various properties of solutions of stochastic differential equations are studied. The approach taken here relies on martingales obtained by means of It 's formula. Relationships between stochastic differential equations (SDEs) and partial differential equations (PDEs) are given, but no prior knowledge of PDEs is required. Solutions to SDEs are referred to as diffusions.
6.1 Martingales and Dynkin's Formula
It 's formula provides a source for construction of martingales. Let X ( t) solve the stochastic differential equation (SDE)
and L t be the generator of X ( t), that is, the second order differential operator associated with SDE (6.1),
It 's formula (4.65) takes a compact form
Theorem 6.1
For any twice continuously differentiable in x, and once in t function f(x, t)
Since, under appropriate conditions, the It integral is a martingale (see Theorem 4.7), by isolating the It integral martingales are obtained.
To illustrate this simple idea, let f have a bounded (by K) derivative, and use It 's formula for f( B( t)). Then
The It integral ? t 0 f ? ( B( s)) dB( s) is a martingale on [0, T], because condition (4.10) holds, ? T 0( f ? ( B( s))) 2 ds < K 2 T < ?. Thus 
is a martingale. The next result is more general.
Theorem 6.2
Let X (t) be a solution to SDE (6.1)...
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