Individual path uniqueness of solutions of stochastic differential equations

Abstract

We consider the stochastic differential equation dx(t) = f(t, x(t))dt + b(t, x(t))dW(t), x(0) = x 0 for t ≥ 0, where x(t) ∈ ℝd, W is a standard d-dimensional Brownian motion, f is a bounded Borel function from [0, ∞) × ℝd to ℝd, and b is an invertible matrix-valued function satisfying some regularity conditions. We show that, for almost all Brownian paths W(t), there is a unique x(t) satisfying this equation, interpreted in a "rough path" sense. © 2011 Springer-Verlag Berlin Heidelberg.