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

TL;DR

This paper investigates the long-term asymptotic behavior of certain functionals of solutions to inhomogeneous Itô stochastic differential equations with nonregular parameter dependence, providing explicit limiting process forms.

Contribution

It introduces new methods to analyze the asymptotics of functionals of inhomogeneous SDE solutions with nonregular parameter dependence, deriving explicit limit processes.

Findings

Explicit forms of limiting processes are established.

Asymptotic behavior as T approaches infinity is characterized.

Results apply to SDEs with nonregular dependence on parameters.

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 $T$.