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

M. S. Nikol'skiǐ

Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 2, pp. 352-357

Résumé

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: $$ \varphi(\sigma,x,\tau)=\frac{2\pi}{|{\ln\varepsilon}|}\int_\sigma^\tau p(\sigma,x,s,z(s))\,ds+o\left(\frac1{|{\ln\varepsilon}|}\right). $$