Convergence of stochastic process with Markov switching (Q402873)

The small parameter method is used to establish sufficient conditions for the convergence of the solution to a stochastic differential equation with Markov switching to a diffusion process. The stochastic differential equation is defined as follows: NEWLINE\[NEWLINE d u^\varepsilon=C(u^\varepsilon(t), x(t/\varepsilon))dt+ \sigma(u^\varepsilon(t))dw(t). NEWLINE\]NEWLINE Here \(u^\varepsilon(t)\) is a random evolution, \(x(t) \in X\) is a switching Markov process, \(w(t)\) is a Wiener process and \(\varepsilon\) is a small parameter.NEWLINENEWLINEThe average regression function is defined as NEWLINE\[NEWLINE C(u)=\int_X\pi(dx)C(u,x), NEWLINE\]NEWLINE where \(\pi(\cdot)\) is a stationary distribution of \(x(t)\). Then, under mild conditions, \(u^\varepsilon(t)\) converges weakly to the random process \(\zeta(t)\) as \(\varepsilon \to 0\). It is a solution to the stochastic differential equation NEWLINE\[NEWLINE d \zeta(t)=C(\zeta(t))dt+ \sigma(\zeta(t))dw(t). NEWLINE\]