Asymptotic behavior of homogeneous additive functionals of the solutions of Itô stochastic differential equations with nonregular dependence on parameter

Abstract

We study the asymptotic behavior of mixed functionals of the form $I_T(t)=F_T(ξ_T(t))+\int_0^tg_T(ξ_T(s))\,dξ_T(s)$, $t\ge0$, as $T\to\infty$. Here $ξ_T(t)$ is a strong solution of the stochastic differential equation $dξ_T(t)=a_T(ξ_T(t))\,dt+dW_T(t)$, $T>0$ is a parameter, $a_T=a_T(x)$ are measurable functions such that $\left|a_T(x)\right|\leq C_T$ for all $x\in \mathbb {R}$, $W_T(t)$ are standard Wiener processes, $F_T=F_T(x)$, $x\in \mathbb {R}$, are continuous functions, $g_T=g_T(x)$, $x\in \mathbb {R}$, are locally bounded functions, and everything is real-valued. The explicit form of the limiting processes for $I_T(t)$ is established under very nonregular dependence of $g_T$ and $a_T$ on the parameter $T$.