The Lyapunov Exponent of the Euler Scheme for Stochastic Differential Equations

In this paper we review some results about the approximation of the upper Lyapunov exponents λ of linear and nonlinear diffusion processes X. The stochastic differential system solved by X is discretized by the Euler scheme. Under appropriate assumptions, the upper Lyapunov exponent $$ \bar \lambda $$ of the resulting approximate process $$ \bar X $$ is well defined and can be efficiently computed by simulating one single trajectory of $$ \bar X $$ during a time long enough. We describe the mathematical technique which leads to estimates on the convergence rate of $$ \bar \lambda $$ to λ. We start by an elementary example, then we deal with linear systems, and finally we consider nonlinear systems.