Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral

A numerical simulation algorithm that is exact for any time step Delta t > 0 is derived for the Ornsiein-Uhlenbeck process X(t) and its time integral Y(t). The algorithm allows one to make efficient, unapproximated simulations of, for instance, the velocity and position components of a particle undergoing Brownian motion, and the electric current and transported charge in a simple R-L circuit, provided appropriate valuer; are assigned to the Ornstein-Uhlenbeck relaxation time tau and diffusion constant c. A simple Taylor expansion in Delta t of the exact simulation formulas shows how the first-order simulation formulas, which are implicit ir, the Langevin equation for X(t) and the defining equation for Y(t), are modified in second order. The exact simulation algorithm is used here to illustrate the zero-tau limit theorem.