On stochastic processes defined by differential equations

R. Z. Khas'minskii

Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 2, pp. 240-259

Résumé

Let the function $X_\varepsilon(\tau,\omega)$ be the solution of the problem (1.3). The main results of this paper are the following theorems. Theorem 1. {\it If the function $F$ satisfies conditions (1.1), (1.2) and (1.4) the stochastic process $X_\varepsilon(\tau,\omega)$ has the following asymptotic behaviour $$ \sup_{0\le\tau\le\tau_0}\mathbf M|X_\varepsilon(\tau,\omega)-x^0(\tau)|\to0\quad(\varepsilon\to0), $$ where $x^0(\tau)$ is the solution of the problem} (1.5). Theorem 2. {\it If $F$ satisfies conditions (3.1)–(3.4) and $\varepsilon\to0$ $n$-order distributions of the stochastic process $Y^{(\varepsilon)}(\tau,\omega)=\varepsilon^{-1/2}(X^{(\varepsilon)}(\tau,\omega)-x^0(\tau))$ approach those of the Gaussian Markov process} (3.6), (3.7). In addition some applications of these theorems to problems of nonlinear mechanics are considered.