Uniqueness of Solutions of Stochastic Differential Equations

We consider the stochastic differential equation dx(t) = dW(t) + f (t, x(t)) dt, x(0) = x(0) for t >= 0, where x(t) is an element of R-d, W is a standard d-dimensional Brownian motion, and f is a bounded Borel function from [0,infinity) x R-d to R-d. We show that, for almost all Brownian paths W(t), there is a unique x(t) satisfying this equation.