Extreme Sojourns of Diffusion Processes

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 $u \rightarrow r_2$.