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 $I_T(t)=F_T({\xi }_T(t))+{\int }_0^t{g}_T({\xi }_T(s))d{\xi }_T(s)$, $t\ge 0$, as $T\to \infty$. Here ${\xi }_T(t)$ is a strong solution of the stochastic differential equation $d{\xi }_T(t)=a_T({\xi }_T(t))dt+d{W}_T(t)$, $T>0$ is a parameter, $a_T=a_T(x)$ are measurable functions such that $\left|a_T(x)\right|\le C_T$ 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 $T$.

Published at http://dx.doi.org/10.15559/16-VMSTA58 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)

Cited by in corpus (1)

• Asymptotic behavior of functionals of the solutions to inhomogeneous Itô stochastic differential equations with nonregular dependence on parameter