Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to Inhomogeneous Itô SDEs with Non-regular Dependence on a Parameter

Abstract

In this chapter, we consider the asymptotic behavior, as T → +∞, of some functionals of the form $I_T(t)=F_T(\xi _T(t))+\int \limits _{0}^{t} g_T(\xi _T(s))\,dW_T(s)$, t ≥ 0. Here ξ _T(t) is the solution to the time-inhomogeneous Itô stochastic differential equation

$$\displaystyle d\xi _T (t)=a_T(t,\,\xi _T(t))\,dt+dW_T(t),\;\; t\ge 0,\;\; \xi _T (0)=x_0, $$

where T > 0 is a parameter, $a_T (t,x), x\in \mathbb R$ are measurable functions, $\left |a_T(t,x)\right |\leq L_T$ for all $x\in \mathbb R$ and t ≥ 0, W _T are standard Wiener processes, $F_T(x),\, x\in \mathbb R$ are continuous functions, and $g_T(x),\, x\in \mathbb R$ are measurable locally bounded functions. Section 6.1 contains some preliminary remarks, notations, and basic definitions. The asymptotic behavior of the integral functionals of the Lebesgue integral type is investigated in Sect. 6.3. Section 6.4 contains some results about the weak convergence of the martingale type functionals and the mixed functionals. Section 6.5 includes several examples. Auxiliary results are collected in Sect. 6.6.