Stochastically bounded solutions of linear non-homogeneous stochastic differential equation. (Ukrainian, English) Zbl 1050.60059

Il’chenko, O.

Teor. Jmovirn. Mat. Stat. 68, 37-43 (2003); translation in Theory Probab. Math. Stat. 68, 41-48 (2004).

The author proposes conditions for existence and uniqueness of a stochastically bounded solution to the stochastic differential equation
\[ dx(t)= (bx(t)+f(t))\,dt+ \sum_{k=1}^{m} (\sigma_{k}x(t)+g_{k}(t))\,dw_{k}(t), \]
where \(b,\sigma_{k}, k=1,\ldots,m\), are constants; \(f(t), g_{k}(t), t\in R, k=1,\ldots,m\), are continuous bounded real functions; \(w_{k}(t), t\in R, k=1,\ldots,m\), are one-dimensional independent Wiener processes. If the coefficients \(f(t)\equiv f\), \(g_{k}(t)\equiv g_{k}, k=1,\ldots,m\), do not depend on \(t\), then there exists a unique stationary solution to the considered equation. If the coefficients \(f(t), g_{k}(t), k=1,\ldots,m\), are periodic with period \(T\), then the solution is also periodic with period \(T\).

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60H10

Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

stochastically bounded solutions; non-homogeneous stochastic differential equation; existence and uniqueness theorem