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Teor. Veroyatnost. i Primenen., 1964, Volume 9, Issue 2, Pages 352–357 (Mi tvp381)  

Short Communications

On the Probability of a Markov Point Falling on a Plane Region with Small Diameter

M. S. Nikol'skiĭ

Moscow

Abstract: The following problem arises in the field of optimum control (see [1]). A point in a plane with a probability density $p(\sigma,x,\tau ,y)$, that satisfies Kolmogorov's equation
$$ \frac{\partial p}{\partial\sigma}+\sum_{i,j=1}^2 a^{ij}(\sigma,x)\frac{\partial^2p}{\partial x^i\partial x^j}+\sum_{i=1}^2 b^i(\sigma,x)\frac{\partial p}{\partial x^i}=0. $$
A second point $z$ moves in the same plane in accordance with the equation $z=z(t)$. A closed curve $S_t=z(t)+\varepsilon S$ moves together with $z$. It is similar to a stationary curve $S$ with a small similarity coefficient $\varepsilon$. It is required to calculate the probability $\varphi(\sigma,x,\tau)$ that a random point will intersect curve $S_A$ during the time interval $\sigma\leqq t\leqq\tau$ if at time $\sigma$ the point $z$ is at $z(\sigma)$ and the random point is at $x$. It is shown in the paper that with some restrictions imposed on the coefficients in Kolmogorov's equation for $|x-z(\sigma)|>r_0$, where $r_0$ is any non-zero constant, the following is true:
$$ \varphi(\sigma,x,\tau)=\frac{2\pi}{|{\ln\varepsilon}|}\int_\sigma^\tau p(\sigma,x,s,z(s)) ds+o(\frac1{|{\ln\varepsilon}|}). $$


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English version:
Theory of Probability and its Applications, 1964, 9:2, 320–325

Bibliographic databases:

Received: 13.11.1963

Citation: M. S. Nikol'skiǐ, “On the Probability of a Markov Point Falling on a Plane Region with Small Diameter”, Teor. Veroyatnost. i Primenen., 9:2 (1964), 352–357; Theory Probab. Appl., 9:2 (1964), 320–325

Citation in format AMSBIB
\Bibitem{Nik64}
\by M.~S.~Nikol'ski{\v\i}
\paper On the Probability of a~Markov Point Falling on a~Plane Region with Small Diameter
\jour Teor. Veroyatnost. i Primenen.
\yr 1964
\vol 9
\issue 2
\pages 352--357
\mathnet{http://mi.mathnet.ru/tvp381}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=163359}
\zmath{https://zbmath.org/?q=an:0141.15404}
\transl
\jour Theory Probab. Appl.
\yr 1964
\vol 9
\issue 2
\pages 320--325
\crossref{https://doi.org/10.1137/1109045}


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