## The Annals of Probability

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

### Local Time and Stochastic Area Integrals

L. C. G. Rogers and J. B. Walsh

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60J55: Local time and additive functionals

Secondary: 60J65: Brownian motion [See also 58J65] 60H05: Stochastic integrals 60G05: Foundations of stochastic processes 60G07: General theory of processes

**Keywords**

Identifiable stochastic area integral Brownian motion Brownian local time stochastic line integral intrinsic local time excursion filtration

#### Citation

Rogers, L. C. G.; Walsh, J. B. Local Time and Stochastic Area Integrals. Ann. Probab. 19 (1991), no. 2, 457--482. doi:10.1214/aop/1176990435. https://projecteuclid.org/euclid.aop/1176990435