Asymptotic behavior of functionals of the solutions to inhomogeneous It\^{o} stochastic differential equations with nonregular dependence on parameter

Abstract

The asymptotic behavior, as $T\to\infty$, of some functionals of the form $I_T(t)=F_T(\xi_T(t))+\int_0^tg_T(\xi_T(s))\,dW_T(s)$, $t\ge0$ is studied. Here $\xi_T(t)$ is the solution to the time-inhomogeneous It\^{o} stochastic differential equation \[d\xi_T(t)=a_T\bigl(t,\xi_T(t)\bigr)\,dt+dW_T(t),\quad t\ge0, \xi_T(0)=x_0,\] $T>0$ is a parameter, $a_T(t,x),x\in\mathbb{R}$ are measurable functions, $|a_T(t,x)|\leq C_T$ for all $x\in\mathbb{R}$ and $t\ge0$, $W_T(t)$ are standard Wiener processes, $F_T(x),x\in\mathbb{R}$ are continuous functions, $g_T(x),x\in\mathbb{R}$ are measurable locally bounded functions, and everything is real-valued. The explicit form of the limiting processes for $I_T(t)$ is established under nonregular dependence of $a_T(t,x)$ and $g_T(x)$ on the parameter $T$.