Some limit theorems connected with Brownian local time

[摘要] LetB=(Bt)t≥0be a standard Brownian motion and let(Ltx;t≥0,x∈ℝ)be a continuous version of its local time process. We show that the following limitlim⁡ε↓0(1/2ε)∫0t{F(s,Bs−ε)−F(s,Bs+ε)}dsis well defined for a large class of functionsF(t,x), and moreover we connect it with the integration with respect to local timeLtx. We give an illustrative example of the nonlinearity of the integration with respect to local time in the random case.

[学科分类] 应用数学