Asymptotic behavior of the martingale type integral functionals for unstable solutions to stochastic differential equations

We consider functionals of the type $\int _ {0} ^ {t} g (\xi (s)) dW (s)$, $t \ge 0$. Here $g$ is a real valued and locally square integrable function, $\xi$ is a unique strong solution of the Itô stochastic differential equation $d \xi (t) = a (\xi (t)) dt + dW (t)$, $a$ is a measurable real valued bounded function such that $| xa (x) | \le C$. The behavior of these functionals is studied as $t \to \infty$. The appropriate normalizing factor and the explicit form of the limit random variable are established.