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Approximation of upper Lyapunov exponents of bilinear stochastic differential systems. (English) Zbl 0738.65106

This study is concerned with the numerical computation of the Lyapunov exponent \(\lambda\) of the solution \(X(t)\) of a multivariate stochastic differential equation written in Stratonovitch form whose drift and diffusion coefficients are linear. Under explicit assumptions three discretization methods are given which yield discrete time processes \(\tilde X(nh)\) such that the limit \(\lambda^ h=\lim_{n\uparrow\infty}(1/nh)\hbox{Log}|\tilde X(nh)|\) exists a.s. and satisfies order of approximation of \(\lambda\) by \(\lambda^ h\) in \(O(h)\) or \(O(h^ 2)\).
In addition to the mathematical proofs of the estimates, a detailed discussion of the performances of the algorithms on an explicit case and on an engineering problem shows the efficiency and the superiority of one of these methods. Finally the problem of a stopping criterion in the almost sure limit giving \(\lambda^ h\) is attacked through a central limit result.

MSC:

65C99 Probabilistic methods, stochastic differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
93E15 Stochastic stability in control theory
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