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April, 1991 Local Time and Stochastic Area Integrals
L. C. G. Rogers, J. B. Walsh
Ann. Probab. 19(2): 457-482 (April, 1991). DOI: 10.1214/aop/1176990435


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.


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L. C. G. Rogers. J. B. Walsh. "Local Time and Stochastic Area Integrals." Ann. Probab. 19 (2) 457 - 482, April, 1991.


Published: April, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0729.60073
MathSciNet: MR1106270
Digital Object Identifier: 10.1214/aop/1176990435

Primary: 60J55
Secondary: 60G05 , 60G07 , 60H05 , 60J65

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

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 2 • April, 1991
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