PRV property of functions and asymptotic behaviour of solutions to stochastic differential equations.(Ukrainian, English)Zbl 1123.60044

Teor. Jmovirn. Mat. Stat. 72, 10-23 (2005); translation in Theory Probab. Math. Stat. 72, 11-25 (2006).
A solution $$X(t)$$ to the stochastic differential equation $$dX(t)=g(X(t))dt+\sigma(X(t))dw(t)$$ is considered with $$X(0)=1$$, $$\sigma(\cdot)>0$$, $$g(\cdot)>0$$. Let $$F(t)=\inf\{s\geq 0: X(s)=t\}$$, $$L(t)=\sup\{s\geq 0: X(s)=t\}$$, $$T(t)=\text{meas}\{s\geq 0: X(s)\leq t\}$$. Let $$\mu(t)$$ be a solution to the ordinary stochastic equation $$d\mu=g(\mu)\,dt$$ with $$\mu(0)=1$$. Conditions are derived under which $$X(t)/\mu(t)\to 1$$ a.s. as $$t\to\infty$$ on $$A_\infty=\{\lim_{t\to\infty} X(t)=\infty\}$$. It is shown that under these conditions $$\lim_{t\to\infty} F(\mu(t))/t= \lim_{t\to\infty} \mu(F(t))/t=1$$ on $$A_\infty$$ and the same property holds true for $$L(t)$$ and $$T(t)$$. The proofs are based on the theory of pseudo regularly varying functions.

MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34D05 Asymptotic properties of solutions to ordinary differential equations
Full Text: