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 4: Brownian Motion Calculus
In this chapter stochastic integrals with respect to Brownian motion are introduced and their properties are given. They are also called It integrals, and the corresponding calculus It calculus.
4.1 Definition of It Integral
Our goal is to define the stochastic integral ? 0 T X(t)dB(t), also denoted ? XdB or X B. This integral should have the property that if X(t) = 1 then ? 0 T dB(t) = B(T) ? B(0). Similarly, if X(t) is a constant c, then the integral should be c(B(T) ? B(0)). In this way we can integrate constant processes with respect to B. The integral over (0, T] should be the sum of integrals over two subintervals (0,a] and (a, T]. Thus if X(t) takes two values c 1 on (0, a], and c 2on (a, T], then the integral of X with respect to B is easily defined. In this way the integral is defined for simple processes, that is, processes which are constant on finitely many intervals. By the limiting procedure the integral is then defined for more general processes.
It Integral of Simple Processes
Consider first integrals of a non-random simple process X(t), which is a function of t and does not depend on B(t). By definition a simple non-random process X(t) is a process for which there exist times 0 = t 0
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