On the limit behavior of the solution of a stochastic differential equation. (English. Russian original) Zbl 0629.60063

Diallo, Mamadou

Theory Probab. Math. Stat. 34, 51-56 (1987); translation from Teor. Veroyatn. Mat. Stat. 34, 47-53 (1986).

In this paper it is shown that if a and g are integrable and \(\int^{+\infty}_{-\infty}a(x)dx=0\) the solution of \(d\xi =[a(\xi)+g(W_ 1)]dt+dW\), \(\xi (0)=0\), with W and \(W_ 1\) independent Wiener processes, scaled as \(\xi_ T(t)=T^{-}\xi (tT)\), converge weakly as \(T\to \infty\) to the sum of a Wiener process \(\hat W\) and a jump process which is constant as long as another independent Wiener process \(\hat W_ 1\) does not cross the zero. This component is absent if and only if \(\int^{+\infty}_{-\infty}g(x)dx=0\).

Reviewer: M.Piccioni

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
34D05	Asymptotic properties of solutions to ordinary differential equations
93E15	Stochastic stability in control theory

Keywords:

asymptotics of stochastic differential equations; weak convergence