## Stochastic differential equations in Hilbert space: Properties of solutions, limit theorems, asymptotic expansions with respect to a small parameter. I.(English. Ukrainian original)Zbl 0940.60082

Theory Probab. Math. Stat. 58, 123-137 (1999); translation from Teor. Jmovirn. Mat. Stat. 58, 114-127 (1998).
Let $$(\Omega,{\mathcal F},P)$$ be a complete probability space and let $$\{{\mathcal F}_{t},t\geq 0\}$$ be a stochastic basis, that is: $${\mathcal F}_{s}\subset {\mathcal F}_{t}$$ as $$s\leq t$$ and $${\mathcal F}=\bigcap_{\varepsilon>0} {\mathcal F}_{t+\varepsilon}.$$ Let $$H$$ and $$X$$ be two separable Hilbert spaces and let $$H_+\subset H\subset H_-$$ be the closure of the space $$H.$$ Let $$\{ w(t),{\mathcal F}_{t}, t\geq 0\}$$ be a standard Wiener process in $$H_-$$ and let $$E\|w(t+\tau)-w(t)\|^2 =\tau \operatorname{Sp}C$$, where $$C$$ is an operator in $$H_-.$$ Let $$\{ A(s),s\geq 0\}$$ be a family of operators in $$X$$ and let $$B(s,x)$$ be a family of operator-valued functions. The authors consider the stochastic differential equation $x(t)=\xi_0+\int_{t_0}^{t}A(s)x(s) ds + \int_{t_0}^{t}f(s) ds+ \int_{t_0}^{t}B(s,x(s)) dw(s), \quad t\geq t_0\geq 0. \tag{1}$ Conditions of existence and uniqueness of a solution to equation (1) are found. Limit theorems for solutions of equation (1) are found, too.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G44 Martingales with continuous parameter