## The Annals of Probability

### Local Time and Stochastic Area Integrals

#### Abstract

If $(B_t)_{t\geq 0}$ is Brownian motion on $\mathbb{R},$ if $A(t, x) \equiv \int^t_0I_{\{B_s\leq x\}} ds$ and if $\tau(\cdot, x)$ is the right-continuous inverse to $A(\cdot, x)$, then the process $\tilde{B}(t, x) \equiv B(\tau(t, x))$ is a reflecting Brownian motion in $(-\infty, x\rbrack$. If $\mathscr{E}_x$ denotes the $\sigma$-field generated by $\tilde{B}(\cdot, x)$, then $(\mathscr{E}_x)_{x\in\mathbb{R}}$ forms a filtration. It has been proved recently that all $(\mathscr{E}_x)$-martingales are continuous, in common with the martingales on the Brownian filtration. Here we shall prove that, as with the Brownian filtration, all $(\mathscr{E}_x)$-martingales can be written as stochastic area integrals with respect to local time. This requires a theory of such integrals to be developed; the first version of this was given by Walsh some years ago, but we consider the account presented here to be definitive. We apply this theory to an investigation of stochastic line integrals of local time along curves which need not be adapted processes and illustrate these constructs by identifying the compensator of the supermartingale $(L(\tau(t, x), x))_{x\geq a}$ previously studied by McGill.

#### Article information

Source
Ann. Probab., Volume 19, Number 2 (1991), 457-482.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176990435

Digital Object Identifier
doi:10.1214/aop/1176990435

Mathematical Reviews number (MathSciNet)
MR1106270

Zentralblatt MATH identifier
0729.60073

JSTOR