  A stochastic leap-frog algorithm for the numerical integration of Brownian
motion stochastic differential equations with multiplicative noise is proposed
and tested. The algorithm has a second-order convergence of moments in a finite
time interval and requires the sampling of only one uniformly distributed
random variable per time step. The noise may be white or colored. We apply the
algorithm to a study of the approach towards equilibrium of an oscillator
coupled nonlinearly to a heat bath and investigate the effect of the
multiplicative noise (arising from the nonlinear coupling) on the relaxation
time. This allows us to test the regime of validity of the energy-envelope
approximation method.
