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

Mishura, Yu. S.; Lavrentiev, O. S.

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.

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

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

Keywords:

stochastic differential equation; Hilbert space; stochastic integral; Wiener process; correlation operator