A simulation of integral and derivative of the solution of a stochastici integral equation

A stochastic integral equation corresponding to a probability space ${\displaystyle (\Omega ,{\Sigma }_{\omega },P_{\omega })}$ is considered. This equation plays the role of a dynamical system in many problems of stochastic control with the control variable ${\displaystyle u(·):{ℝ}^1\to {ℝ}^m}$. One constructs stochastic processes ${\displaystyle {\eta }^{\left(1\right)}\left(t\right)}$, ${\displaystyle {\eta }^{\left(2\right)}\left(t\right)}$ connected with a Markov chain and with the space ${\displaystyle (\Omega ,{\Sigma }_{\omega },P_{\omega })}$. The expected values of ${\displaystyle {\eta }^{\left(i\right)}\left(t\right)}$ (i = 1,2) are respectively the expected value of an integral representation of a solution x(t) of the equation and that of its derivative ${\displaystyle x{\prime }_u\left(t\right)}$.