Continuous dependence of a solution of a system of stochastic differential equations with unit diffusion on the initial conditions. (English. Russian original) Zbl 0622.60061

Fedorenko, I. V.

Theory Probab. Math. Stat. 32, 117-119 (1986); translation from Teor. Veroyatn. Mat. Stat. 32, 105-107 (1985).

It is proved that the following condition suffices for a solution of a system of the form \(dx_ t=A(t,x_ t)dt+dw_ t\) to depend continuously on the initial condition: the ith component \(A^ i(t,x^ 1,...,x^ d)\) of the drift is nondecreasing in the arguments \(x^ 1,...,x^{i- 1},x^{i+1},...,x^ d\) for any i. The proof is based on a multi- dimensional ”comparison theorem”.

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
60G17	Sample path properties
34F05	Ordinary differential equations and systems with randomness

Keywords:

continuous dependence; comparison theorem