On functional limit theorems for solutions of stochastic equations (Q1592448)

A stochastic differential equation \[ x(t) = x + \int ^{t}_{0} b(x(s),y(s)) ds + \int ^{t}_{0} r(x(s),y(s)) d\beta (s) \] in \(\mathbb R^{p}\) and an \(\mathbb R^{d}\)-valued integral functional \[ F(t) = \int ^{t}_{0} g(x(s),y(s)) ds + \int ^{t}_{0} h(x(s),y(s)) dw \] are considered. Here, \(\beta \) and \(w\) are independent Wiener processes and \(y\) is a jump process with a finite state space. Set \(F^\varepsilon (t) = \varepsilon F(\varepsilon ^{-2}t)\), let \(\mu ^\varepsilon \) be the measure induced on \(C([0,T]; \mathbb R^{d})\) by the process \(F^\varepsilon (t) - \varepsilon ^{-1}\lambda t\). Under suitable assumptions on the coefficients \(b\), \(r\), \(g\) and \(h\) it is shown that the measures \(\mu ^\varepsilon \) converge weakly as \(\varepsilon \searrow 0\) to the law of \(\sqrt {D}\hat w\), \(\hat w\) being a standard \(d\)-dimensional Wiener process. Further, a large deviations principle is established for the process \(\varphi (\varepsilon) (F^\varepsilon (t) - \varepsilon ^{-1}\lambda t)\) with a nonnegative function \(\varphi \) such that \(\varphi (\varepsilon) \to 0\) and \(\varepsilon /\varphi (\varepsilon)\to 0\) as \(\varepsilon \) tends to zero.