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

Buldygin, V. V.; Klesov, O. I.; Steinebach, J. G.

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.

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

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

Keywords:

diffusion process; law of large numbers; pseudo regularly varying function