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

In this chapter, we consider homogeneous one-dimensional stochastic differential equations with non-regular dependence on a parameter. The asymptotic behavior of the mixed functionals of the form \(I_T(t)=F_T(\xi _T(t))+\int \limits _{0}^{t} g_T(\xi _T(s))\,d\xi _T(s)\), t ≥ 0 is studied as T → +∞. Here ξT is a strong solution to the stochastic differential equation dξT(t) = aT(ξT(t)) dt + dWT(t), T > 0 is a parameter, aT = aT(x) is a set of measurable functions, \(\left |a_T(x)\right |\leq L_T\) for all \(x\in \mathbb R\), WT = WT(t) are standard Wiener processes, \( F_T= F_T(x), x\in \mathbb R\) are continuous functions, and \( g_T= g_T(x), x\in {\mathbb R} \) are locally bounded real-valued functions. Section 5.1 contains some preliminary remarks. We prove a theorem about the weak compactness of the family of some processes in Sect. 5.2. Section 5.3 includes a theorem concerning the weak convergence of some stochastic processes to the solutions of Ito SDEs. In Sect. 5.4 we consider the asymptotic behavior of integral functionals of Lebesgue integral type. Section 5.5 is devoted to asymptotic behavior of the integral functionals of martingale type. The explicit form of the limiting processes for IT(t) is established in Sect. 5.6 under very non-regular dependence of gT and aT on the parameter T. This section summarizes the main results and their proofs. Section 5.7 contains several examples. Auxiliary results are collected in Sect. 5.8.