Stochastic Integration with Respect to Local Time

Abstract

Let {B_t: t ≥ 0} be a standard Brownian motion with B₀ = O, defined on a probability space (ΩF,P). Define the occupation time below x by

$$H_t^X\; = \;\int\limits_0^t {{I_{{{\{ {B_S} \le X\} }^{ds}}}}}$$

and let L ^x_t be the local time at x:

$$L_t^X\; = \;{d \over {dx}}\;H_t^X.$$