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Estimates and exact expressions for Lyapunov exponents of stochastic linear differential equations. (English) Zbl 0669.34060
The paper deals with the asymptotic behaviour of the solutions of a stochastic linear differential equation of the form \(\dot x(t)=A_{j_ t}x(t)\), \(t\geq 0\), \(x(0)=x_ 0\in R^ d\), where \(j_ t\) is a stationary, ergodic, Markov process with a finite state space \(\{1,...,N\}\), and \(A_ 1,...,A_ N\) are \(d\times d\) constant matrices. Both sample path Lyapunov exponent \(\lambda\) and p-moment Lyapunov exponents g(p) (for positive p) are considered. It is proved that g(p) does not depend on \(x_ 0\), is a convex function of p, \(g(0)=0\), \(g(p)/p\) is nondecreasing, and \(g(p)\) and \(\lambda\) are simply related. For g(2), using a Lyapunov function approach, an exact expression is obtained as the maximal real part of the eigenvalues of a certain \(Nd^ 2\) dimensional matrix. A similar method allows to compute upper and lower bounds for \(g(p)\). Therefore, exploiting the relation between \(\lambda\) and the \(g(p)\)’s, upper and lower bounds for \(\lambda\) are obtained.
Reviewer: F.Flandoli

34F05 Ordinary differential equations and systems with randomness
93E15 Stochastic stability in control theory
34D20 Stability of solutions to ordinary differential equations
Full Text: DOI
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