Estimating the Density of the Probability Distributions for Random Processes in Systems with Non-Linearities of Piecewise-Linear Type

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