On asymptotic equivalence of solutions of stochastic and ordinary equations
The authors study the asymptotic behavior of the solution of a stochastic differential equation at infinitely by method of qualitative theory of ordinary differential equations (see A. Witner (1946), N. Levinson (1948) or V. A. Yakubovich (1951) for ordinary differential equations). They extend the results of \textit{A. P. Krenevych} [Ukr. Mat. Zh. 58, No. 10, 1368--1384 (2006); translation in Ukr. Math. J. 58, No. 10, 1552--1569 (2006; Zbl 1119.60047)] and A. M. Samoilenko and O. M. Stanzhyts'kyi (2009), where conditions for the asymptotic equivalence of a linear stochastic system and a system of ordinary differential equations were obtained in the case of deterministic system is a system with constant coefficients and all its solutions are bounded on the half axis \(t\geq 0\). In fact, they consider the case of a linear deterministic system has variable coefficients that admits unbounded solution. In the first section, they study some adequate conditions for the asymptotic equivalence of a weakly nonlinear stochastic system and a linear system of ordinary under assumption that the latter is exponentially dichotomic on the axis. The main results of this section are given in Theorem 1 and in Theorem 2. In the second section, they consider the linear case. For this case, the authors associate every nontrivial solution of the stochastic system with a nontrivial solution of the deterministic system. The last section is devoted to applications.
scientific article; zbMATH DE number 6110027
scientific article