This article is cited in 6 papers

On local times for functions and stochastic processes. I

F. S. Nasyrov

Ufa State Aviation Technical University

Abstract: Let $X(t)$, $0\le t\le 1$, be a real-valued measurable function having a local time $\alpha (t,u)$, $0\le t\le 1$, $u\in\mathbf{R}$. If the latter is continuous in $t$ for a.e. $u$, then the distribution. $F(t,x)=\int_\mathbf{R}\mathbb{I}(\alpha(t,u)>x)\,du$ and the monotone rearrangement $\alpha^*(t,u)=\inf\{x:F(t,x)<u\}$ of the local time $\alpha(t,u)$ are the local times for $\xi(s)=\alpha(s,X(s))$ and $\xi^*(s)=F(s,X(s))$, $0\le s\le 1$, respectively.

Keywords: local time, distribution and monotone rearrangement of a function, orthogonal decomposition, Brownian motion.

Received: 06.12.1991

 English version: Theory of Probability and its Applications, 1995, 40:4, 702–713

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