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
8.6: It 's Formula for Continuous Semimartingales
8.6 It 's Formula for Continuous Semimartingales
If X( t) is a continuous semimartingale and f is a twice continuously differentiate function, then Y( t) = f( X( t)) is a semimartingale and admits the following representation
In differential form this is written as
It follows, in particular, that f( X( t)) is also a semimartingale, and its decomposition into the martingale part and the finite variation part can be obtained from It 's formula by splitting the stochastic integral with respect to X( t) into the integral with respect to a local martingale M( t) and a finite variation process A( t).
We have given a justification of It 's formula and examples of its use in Chapters 4 and 6.
Remark 8.1
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The differentiability properties of f may be relaxed. If, for example, X is of finite variation, then f needs to be only once continuously differentiable. f can be defined only on an open set, rather than a whole line, but then X must take its values almost surely in this set. For example, if X is a positive semimartingale, then It 's formula can be used with f = ln.
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It 's formula holds for convex functions (Protter (1992) p. 163), and more generally, for functions which are the difference of two convex functions. This is the Meyer-It (It -Tanaka) formula, see for example, Protter (1992) p.167,...
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