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
8.7: Local Times
8.7 Local Times
Let X( t) be a continuous semimartingale. Consider X( t) ? a, a ? 
The function x ? a is not differentiable at a, but at any other point its derivative is given by sign( x ? a), where sign( x) = 1 for x > 0 and sign( x) = ?1 for x ? 0. It is possible to extend It 's formula for this case and prove (see Rogers and Williams (1990), p.95 102, Protter (1992) p.165 167)
Theorem 8.9: (Tanaka's Formula)
Let X(t) be a continuous semimartingale. Then for any a ? 
there exists a continuous non-decreasing adapted process L a(t), called the local time at a of X, such that
As a function in a, L a(t) is right-continuous with left limits. For any fixed a as a function in t L a(t) increases only when X(t) = a, that is, L a(t) = ? t 0 I(X(s) = a)dL a(s). Moreover, if X(t) is a continuous local martingale, then L a(t) is jointly continuous in a and t.
Remark 8.2
Heuristically Tanaka's formula can be justified by a formal application of It 's formula to the function sign ( x). The derivative of sign ( x) is zero everywhere but at zero, where it is not defined. However, it is possible to define the derivative as a generalized function or a...
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