A formula for wanderings of a regular Markov process

M. I. Taksar

Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 4, pp. 839-845

Résumé

Let $(z_t,\mathbf P)$ be a regular Markov process on the time interval $T=(-\infty,\infty)$ with sample space $\Omega$ and state space $Z_t$ at time $t$. Let $\Gamma$ be a measurable subset of $\displaystyle Z=\bigcup_t Z_t$. Assume that $M(\omega)=\{t\colon z_t(\omega)\in\Gamma\}$ is closed a. s. $\mathbf P$. The complement of $M(\omega)$ is a countable union of open intervals $(\gamma,\delta)$. The collections of paths $w_{\delta}^{\gamma}=z_t(\omega)$ ($t\in(\gamma,\delta)$; the birth time of $w_{\delta}^{\gamma}$ is $\gamma$ and the death time is $\delta$) are called wanderings of $z_t$ in $Z\setminus\Gamma$. Let $W$ be the set of all paths in $Z\setminus\Gamma$ defined on all open time intervals. Let $f(t,\omega,w)$ be a function on $T\times\Omega\times W$. Denote by $S_f$ the sum $\sum f(\gamma,\omega,w_{\delta}^{\gamma})$ taken over all the wanderings $w_{\delta}^{\gamma}$. We calculate the expectation of $S_f$ (formula (2)) under some special assumptions of measurability of $f(t,\omega,w)$.