Approximation of Upper Lyapunov Exponents of Bilinear Stochastic Differential Systems

A bilinear Stochastic Differential System is considered whose solution starting from x is denoted by $(X(t,x))$; it is supposed that its (upper) Lyapunov exponent λ exists. The purpose of the paper is to propose an efficient algorithm to approximate λ. It is shown that, for Markov chains ( Xₚʰ (x),p ∈ N) defined by several approximation schemes, the Lyapunov exponents λ ʰ are well defined and can be computed; estimates are given in terms of the discretization step h of the theoretical error | λ - λ ʰ |. Then these results are applied to an engineering problem: the stability of the motion of helicopter rotor blades in a turbulent wind.