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 14: Applications in Engineering and Physics
In this chapter methods of Stochastic Calculus are applied to the Filtering problem in Engineering and Random Oscillators in Physics. The Filtering problem consists of finding the best estimator of a signal when observations are contaminated by noise. For a number of classical equations of motions in Physics we find stationary densities when the motion is subjected to random excitations.
14.1 Filtering
The filtering problem is the problem of estimation of a signal contaminated by noise. Let Y(t) be the observation process, and 
denote the information available by observing the process up to time t, that is 
= ? ( Y( s), s ? t). The observation Y(t) at time t is the result of a deterministic transformation of the signal process X(s), s ? t, typically a linear transformation, to which a random noise is added. The filtering problem is to find the "best" estimate ? x (t) of the signal X(t) on the basis of all the observations Y(s), s ? t, or 
. The "best" is understood in the sense of the smallest estimation error 
, when Z(t) varies over all 
-measurable processes. Denote for an adapted process h(t)
Then by Theorem 2.26 (see also Exercise 14.2) the filtering problem is in finding ? t (X).
Two main results of stochastic calculus are used to solve the filtering problem, Levy's characterization of Brownian motion and the predictable representation property of Brownian...
Copyright Imperial College Press 2005 under license agreement with Books24x7
