Stochastic differential equation driven by countably many Brownian motions with non-Lipschitzian coefficients

We study m -dimensional SDE X-t = x(0) + Sigma(infinity)(i=1) integral(t)(0) sigma(i) (X-s) dW(s)(i) + integral(t)(0) b(X-s)ds, where {W-i)(i >= 1) is an infinite sequence of independent standard one-dimensional Brownian motions. Existence and pathwise uniqueness, non-explosion, and a Freidlin-Wentzell large deviations principle of strong solutions to the SDE are established under modulus of continuity of the coefficients is less than vertical bar x -y vertical bar (log(1)/((x - y)))(y), y epsilon [0, 1], which are different from that results constructed recently by Cao and He [2] and Fang and Zhang [6].