The Annals of Probability

Local Time and Stochastic Area Integrals

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

Full-text: Open access

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


Export citation