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

• Main Authors: Grigorij Kulinich, Svitlana Kushnirenko, Yuliia Mishura
• Format: Article
• Language: English
• Published: VTeX 2016-07-01
• Series: Modern Stochastics: Theory and Applications
• Subjects: Diffusion-type processes; asymptotic behavior of additive functionals; nonregular dependence on the parameter
• Online Access: https://vmsta.vtex.vmt/doi/10.15559/16-VMSTA58

We study the asymptotic behavior of mixed functionals of the form $I_{T}(t)=F_{T}(\xi _{T}(t))+{\int _{0}^{t}}g_{T}(\xi _{T}(s))\hspace{0.1667em}d\xi _{T}(s)$, $t\ge 0$, as $T\to \infty $. Here $\xi _{T}(t)$ is a strong solution of the stochastic differential equation $d\xi _{T}(t)=a_{T}(\xi _{T}(t))\hspace{0.1667em}dt+dW_{T}(t)$, $T>0$ 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.