LOCAL TIME AND STOCHASTIC AREA INTEGRALS

• 成果类型: Article
• 署名作者: ROGERS, LCG; WALSH, JB
• 署名单位: University of British Columbia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176990435
• 发表日期: 1991
• 页码: 457-482
• 关键词: brownian excursion filtration; martingales

摘要

If (B(t))t greater-than-or-equal-to 0 is Brownian motion on R, if A(t,x) = integral-0t(I){B(s) less-than-or-equal-to x} ds and if tau(.,x) is the right-continuous inverse to A(.,x), then the process BBAR(t,x) = B(tau(t,x)) is a reflecting Brownian motion in (-infinity, x]. If E(x) denotes the sigma-field generated by BBAR(.,x), then (E(x))x-element-of-R forms a filtration. It has been proved recently that all (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 (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 greater-than-or-equal-to a previously studied by McGill.