M.: Uniqueness of solutions of stochastic differential equations

Abstract. We consider the stochastic differential equation dx(t) = dW (t) + f(t, x(t))dt, x(0) = x0 for t ≥ 0, where x(t) ∈ Rd, W is a standard d-dimensional Brownian motion, and f is a bounded Borel function from [0,∞) × Rd → Rd to Rd. We show that, for almost all Brownian paths W (t), there is a unique x(t) satisfying this equation. 1