Multiplicative functional for reflected Brownian motion via   deterministic ODE
  We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof is based on a study of the deterministic version of the problem and the excursion theory. 
Krzysztof Burdzy and John M. Lee