Weak convergence of integral functionals constructed from solutions of Itô’s stochastic differential equations with non-regular dependence on a parameter

The weak convergence of the functionals $\int _0^tg_T(\xi _T (s)) dW_T(s)$, $t\ge 0$, is studied as $T\to \infty$, where $\xi _T(t)$ is a strong solution of the stochastic differential equation $d\xi _T (t)=a_T(\xi _T(t)) dt+dW_T(t)$ and $T>0$ is a parameter. Here $a_T (x)$, $x\in \mathbb {R}$, are some real-valued measurable functions such that $\left |a_T(x)\right |\leq C_T$ for all $x$, $W_T(t)$ are standard Wiener processes, and $g_T (x)$ are real-valued measurable locally bounded non-random functions. The explicit form of the limit processes is found in the case where both $g_T (x)$ and $a_T (x)$ depend on the parameter in a non-regular way.