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
8.3: Doob-Meyer Decomposition
8.3 Doob-Meyer Decomposition
Recall that a process is a submartingale if for all s < t, E( X( t) 
) ? X( s) almost surely
Theorem 8.4
If X is a submartingale or a local submartingale, then there exists a local martingale M(t) and a unique increasing predictable process A(t), locally integrable, such that
If X( t) is a submartingale of Dirichlet class ( D) (see Definition 7.25), then the process A is integrable, that is, sup tE A( t) < ?, and M( t) is a uniformly integrable martingale.
Example 8.6
-
Let X( t) = B 2( t) on a finite interval t ? T. X( t) is a submartingale. Decomposition (8.3) holds with M( t) = B 2( t) ? t and A( t) = t. Since the interval is finite, M is uniformly integrable and A is integrable.
-
Let X( t) = B 2( t) on the infinite interval t ? 0. Then (8.3) holds with M( t) = B 2( t) ? t and A( t) = t. Since the interval is infinite M is a martingale, and A is locally integrable; for example take the localizing sequence ? n = n.
-
Let X( t)...
Copyright Imperial College Press 2005 under license agreement with Books24x7
