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 8: Calculus for Semimartingales
In this chapter rules of calculus are given for the most general processes for which stochastic calculus is developed, called semimartingales. A semimartingale is process consisting of a sum of a local martingale and a finite variation process. Integration with respect to semimartingales involves integration with respect to local martingales, and these integrals generalize the It integral where integration is done with respect to a Brownian motion. Important concepts, such as compensators and the sharp bracket processes are introduced, and It 's formula in its general form is given.
8.1 Semimartingales
In stochastic calculus only regular processes are considered. These are either continuous processes, or right-continuous with left limits, or left-continuous with right limits. The regularity of the process implies that it can have at most countably many discontinuities, and all of them are jumps (Chapter 1). The definition of a semimartingale presumes a given filtration and processes which we consider are adapted to it. Following the classical approach, see for example, Metivier (1982), Liptser and Shiryayev (1989) p.85, a semimartingale, is a local martingale plus a process of finite variation. More precisely,
Definition 8.1
A regular right-continuous with left limits (c dl g) adapted process is a semimartingale if it can be represented as a sum of two processes: a local martingale M(t) and a process of finite variation A(t), with M(0) = A(0) = 0, and
Example 8.1
(Semimartingales)
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S( t) = B 2( t), where B( t) is a Brownian motion is a semimartingale.
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