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

Volume 4, Issue 3 (2017), pp. 199–217

Grigorij Kulinich

^{ }Svitlana Kushnirenko^{ }
Pub. online: 22 September 2017
Type: Research Article
Open Access

Received

29 May 2017

29 May 2017

Revised

6 August 2017

6 August 2017

Accepted

10 August 2017

10 August 2017

Published

22 September 2017

22 September 2017

#### Abstract

The asymptotic behavior, as $T\to \infty $, of some functionals of the form $I_{T}(t)=F_{T}(\xi _{T}(t))+{\int _{0}^{t}}g_{T}(\xi _{T}(s))\hspace{0.1667em}dW_{T}(s)$, $t\ge 0$ is studied. Here $\xi _{T}(t)$ is the solution to the time-inhomogeneous Itô stochastic differential equation

\[d\xi _{T}(t)=a_{T}\big(t,\xi _{T}(t)\big)\hspace{0.1667em}dt+dW_{T}(t),\hspace{1em}t\ge 0,\hspace{2.5pt}\xi _{T}(0)=x_{0},\]

$T>0$ 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*.#### References

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