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The asymptotic behavior, as $T\to\infty$, of some functionals of the form$I_T(t)=F_T(\xi_T(t))+\int_0^tg_T(\xi_T(s))\,dW_T(s)$, $t\ge0$ is studied. Here$\xi_T(t)$ is the solution to the time-inhomogeneous It\^{o} stochasticdifferential equation \[d\xi_T(t)=a_T\bigl(t,\xi_T(t)\bigr)\,dt+dW_T(t),\quadt\ge0, \xi_T(0)=x_0,\] $T>0$ is a parameter, $a_T(t,x),x\in\mathbb{R}$ aremeasurable functions, $|a_T(t,x)|\leq C_T$ for all $x\in\mathbb{R}$ and$t\ge0$, $W_T(t)$ are standard Wiener processes, $F_T(x),x\in\mathbb{R}$ arecontinuous functions, $g_T(x),x\in\mathbb{R}$ are measurable locally boundedfunctions, and everything is real-valued. The explicit form of the limitingprocesses for $I_T(t)$ is established under nonregular dependence of $a_T(t,x)$and $g_T(x)$ on the parameter $T$.

modern stochastics: theory and applications

Asymptotic behavior of functionals of the solutions to inhomogeneous  It\^{o} stochastic differential equations with nonregular dependence on  parameter