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

Abstract

The asymptotic behavior, as $T\to \infty$, of some functionals of the form $I_T(t)=F_T({\xi }_T(t))+{\int }_0^t{g}_T({\xi }_T(s))d{W}_T(s)$, $t\ge 0$ is studied. Here ${\xi }_T(t)$ is the solution to the time-inhomogeneous Itô stochastic differential equation \[dξ_T(t)=a_T\bigl(t,ξ_T(t)\bigr)\,dt+dW_T(t),\quad t\ge0, ξ_T(0)=x_0,\] $T>0$ is a parameter, $a_T(t,x),x\in \mathbb{R}$ are measurable functions, $|a_T(t,x)|\le C_T$ for all $x\in \mathbb{R}$ and $t\ge 0$, $W_T(t)$ 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 $T$.

Published at http://dx.doi.org/10.15559/17-VMSTA83 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/). arXiv admin note: text overlap with arXiv:1607.03661

References in corpus (1)

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