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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

##### MSC:
 34F05 Ordinary differential equations and systems with randomness 93E15 Stochastic stability in control theory 34D20 Stability of solutions to ordinary differential equations
##### Keywords:
Markov process; Lyapunov exponent; Lyapunov function
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##### References:
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