On a type of stochastic differential equations driven by countably many Brownian motions

Abstract

Consider the one-dimensional SDE X t=x+∑ i=1 ∞ ∫ 0 t σ i(X s) dW s i+ ∫ 0 t b(X s) ds , where W i is an infinite sequence of independent standard Brownian motions, i=1,2,3,… . We prove two theorems on the existence and uniqueness of solutions with non-Lipschitz coefficients, and give a non-contact property and a strong comparison theorem for solutions.