The Annals of Probability

Extreme Sojourns of Diffusion Processes

Simeon M. Berman

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Abstract

Let $(X_t)$ be a diffusion process on the interval $(r_1, r_2)$ where $r_2$ is inaccessible. For $r_1 < z < r_2$, let $T_z$ be the first passage time to $z$, and define $L_u = \int_{\{t: 0 \leq t \leq T_z, X_t > u\}} J(X_t) dt$, where $J$ is a particular function determined by the generator of the diffusion. An explicit asymptotic expression is obtained for the probability $P(L_u > y|X_0 = x)$, for $u \rightarrow r_2$, fixed $y > 0$, and $r_1 < x < r_2$. From this the corresponding asymptotic form of the distribution of the sojourn time, $mes(t: 0 \leq t \leq T_z, X_t > u)$ is determined when $r_2 = \infty$. Related theorems are given for the distribution of $L_u - L_v$, for $u, v \rightarrow \infty, u \leq v$ and $u/v \rightarrow 1$. Finally, the results are extended to the long-term sojourn integral $L^\ast_u = \int_{\{t: 0 \leq t \leq S(u), X_t > u\}} J(X_t) dt$, where $S(u) \rightarrow \infty$ for $u \rightarrow r_2$.

Article information

Source
Ann. Probab., Volume 16, Number 1 (1988), 361-374.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991908

Digital Object Identifier
doi:10.1214/aop/1176991908

Mathematical Reviews number (MathSciNet)
MR920278

Zentralblatt MATH identifier
0637.60089

JSTOR
links.jstor.org

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60G17: Sample path properties

Keywords
Diffusion process sojourn time first passage time local time scale function

Citation

Berman, Simeon M. Extreme Sojourns of Diffusion Processes. Ann. Probab. 16 (1988), no. 1, 361--374. doi:10.1214/aop/1176991908. https://projecteuclid.org/euclid.aop/1176991908


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