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Teor. Veroyatnost. i Primenen., 1968, Volume 13, Issue 2, Pages 333–337 (Mi tvp850)  

Short Communications

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

Yu. K. Belyaev

Moscow

Abstract: 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.

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English version:
Theory of Probability and its Applications, 1968, 13:2, 320–324

Bibliographic databases:

Received: 24.08.1967

Citation: Yu. K. Belyaev, “On the number of boundary out-crossings of a region by a vector stochastic process”, Teor. Veroyatnost. i Primenen., 13:2 (1968), 333–337; Theory Probab. Appl., 13:2 (1968), 320–324

Citation in format AMSBIB
\Bibitem{Bel68}
\by Yu.~K.~Belyaev
\paper On the number of boundary out-crossings of a~region by a~vector stochastic process
\jour Teor. Veroyatnost. i Primenen.
\yr 1968
\vol 13
\issue 2
\pages 333--337
\mathnet{http://mi.mathnet.ru/tvp850}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=229286}
\zmath{https://zbmath.org/?q=an:0181.20503}
\transl
\jour Theory Probab. Appl.
\yr 1968
\vol 13
\issue 2
\pages 320--324
\crossref{https://doi.org/10.1137/1113036}


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