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
8.9: Compensators and Sharp Bracket Process
8.9 Compensators and Sharp Bracket Process
A process N is called increasing if all of its realizations N(t) are non-decreasing functions of t. A process N is of finite variation if all of its realizations N( t) are functions of finite variation, V N ( t) < ? for all t, where V N ( t) is the variation of N on [0, t].
Definition 8.19
An increasing process N, t ? 0, is called integrable if sup t ?0 E N( t) < ?.
A finite variation process N is of integrable variation if its variation process is integrable, sup t?0EV N(t) < ?.
A finite variation process N is of locally integrable variation if there is a sequence of stopping times ? n such that ? n ? ? so that N(t ? ? n) is of integrable variation, that is, sup t ? 0EV N(t ? ? n) < ?.
Example 8.14
A Poisson process N( t) with parameter ? is of finite but not integrable variation, since for any t, V N( t) = N( t) < ?, but sup t ? 0E V N( t) = ?. It is of locally integrable variation, since sup t ? 0 E V N
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