A limit theorem for stochastic differential equations in \(R^ m\). (English. Russian original) Zbl 0633.60079

Pisanets, S. I.

Theory Probab. Math. Stat. 34, 127-137 (1987); translation from Teor. Veroyatn. Mat. Stat. 34, 111-122 (1986).

Consider a sequence of m-dimensional diffusion type equations \[ d\xi_ t^{(n)}=\alpha_ t^{(n)}(\xi^{(n)})dt+\beta_ t^{(n)}(\xi^{(n)})dw_ t,\quad \xi_ 0^{(n)}=\xi_ 0, \] n\(=0,1,...\), and \(0\leq t\leq T\). Under fairly general assumptions on the diffusion coefficient matrices \(\beta^{(n)}\) and the drift vectors \(\alpha^{(n)}\), the author presents necessary and sufficient conditions in order that \(\xi_ t^{(n)}\to \xi_ t\) in \(L^ 2\)-mean, for each \(0\leq t\leq T.\)
The result here extends an earlier one by the author [Some questions of the theory of stochastic processes. Collect. sci. Work Kiev 1982, 89-99 (1982; Zbl 0537.60049)] in which only the drift coefficient was allowed to depend on n. The precise conditions are too complicated to be stated here, but they include the standard Lipschitz type restrictions.

Reviewer: M.M.Rao

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60F25	\(L^p\)-limit theorems
60J60	Diffusion processes

Keywords:

strong convergence of solution processes

Citations:

Zbl 0537.60049