Estimating the Probability Density for Random Processes in Systems

È. M. Khazen

Teoriâ veroâtnostej i ee primeneniâ, Tome 6 (1961) no. 2, pp. 234-242

Résumé

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.