On the asymptotic behavior of solutions of a certain class of one-dimensional Itô stochastic differential equations. (English. Ukrainian original) Zbl 0923.60072

Kulinich, G. L.; Kas’kun, E. P.

Theory Probab. Math. Stat. 56, 97-105 (1998); translation from Teor. Jmovirn. Mat. Stat. 56, 96-104 (1997).

The authors consider the class of stochastic differential equations \[ d\xi (t)=a(\xi(t))dt+dw(t), \qquad t\geq 0, \] where \(a(x)\) is a measurable bounded function, \(w(t)\) is a Wiener process. Such stochastic differential equations are studied under two types of conditions: (a) \(\lim_{| x| \to \infty}({{1}\over{\ln | x| }}\int_0^{x}a(v)dv-c_0)=0\) or (b) \(\lim_{| x| \to \infty}({{1}\over{| x| }}\int_0^{x}va(v)dv-c(x))=0\). The asymptotic behaviour of the process \(\xi(t)\) is given as \(t\to\infty\).

Reviewer: A.Ya.Olenko (Kyïv)

MSC:

60H10

Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

stochastic differential equation; asymptotic behaviour; Itô equation