The Annals of Probability

Local Time and Stochastic Area Integrals

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

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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

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

First available in Project Euclid: 19 April 2007

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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

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


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.

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