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

TL;DR

This paper investigates the asymptotic behavior of additive functionals of solutions to stochastic differential equations with nonregular parameter dependence, establishing explicit limits under challenging conditions.

Contribution

It provides explicit forms of the limiting processes for additive functionals of SDE solutions with highly nonregular parameter dependence.

Findings

Explicit limiting processes are derived for the additive functionals.

The results apply under very nonregular dependence of coefficients on the parameter.

The study extends understanding of asymptotic behavior in complex stochastic systems.

Abstract

We study the asymptotic behavior of mixed functionals of the form $I_T(t)=F_T(\xi_T(t))+\int_0^tg_T(\xi_T(s))\,d\xi_T(s)$, $t\ge0$, as $T\to\infty$. Here $\xi_T(t)$ is a strong solution of the stochastic differential equation $d\xi_T(t)=a_T(\xi_T(t))\,dt+dW_T(t)$, $T>0$ 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 $T$.