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Short Communications

Estimating the Probability Density for Random Processes in Systems with Nonlinear Reformers of the Piecing-linear Type

È. M. Khazen

Moscow

Abstract: A system of stochastic Ito differential equations is dealt with in this paper:
$$dy_i=F(y_1,\dots,y_n ,t)\,dt+\sum\limits_{j=1}^n{a_{ij}\,d\zeta_j (t),}$$
$i=1,2,\dots n$, where ${\zeta_j (t)}$ are independent Wiener processes; or,
$$\frac{dy_i }{dt}=F_i(y_1,\dots,y_n ,t)+\sum\limits_{j=1}^n{a_{ij}\zeta_j(t),}$$
where ${\zeta_j (t)}$ are Gaussian “white noise” processes. The functions $F_i(y_1,\dots,y_n )$ are piecewise-linear, and $a_{ij}$ are piecewise-constant.
The problem of estimating the probability density for Markov random processes $(y_1(t),\dots,y_n (t))$ is reduced to the solution of a system of Volterra linear integral equations of second kind.

Received: 20.10.1960

 English version: Theory of Probability and its Applications, 1961, 6:2, 214–220