Summary: We consider functionals of the type \(\int_{0}^{t}g(\xi(s))\,dW(s)\), \(t\geq0\). Here, \(g\) is a real valued and locally square integrable function, \(\xi\) is a unique strong solution of the Itō stochastic differential equation \(d\xi(t)=a(\xi(t))dt+dW(t)\) and \(a\) is a measurable real valued bounded function such that \(| xa(x)| \leq C\). 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.