On the number of boundary out-crossings of a~region by a~vector stochastic process

Yu. K. Belyaev

Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 333-337

Résumé

It is shown that under some restrictions (see the conditions $C_\Phi$, $C_\xi$, $C_{\xi\xi}$) the moments of the number of crossing of a set $\Gamma$ with a smooth boundary $S_\Phi=\{\mathbf x\colon\Phi(\mathbf x)=0\}$. $\mathbf x\in R^m$, by a continually differentiable vector stochastic process $\xi_i$ can be found explicitly. For example, the intensity $\mu^+(\Gamma,t)$ of the number of out-crossings of $\Gamma$ from the region $\Phi(x)0$ at time $t$ is expressed by a surface integral of the first kind: $$ \mu^+(\Gamma,t)=\int_{x\in\Gamma}\mathbf M\{(\mathbf n_\Phi(\mathbf x)'\xi_t)^+\mid\xi_t'=\mathbf x\}p_t(\mathbf x)\,ds(\mathbf x). $$ At the end of the paper examples are given, which illustrate advantages of the obtained formulas.