Estimates and exact expressions for lyapunov exponents of stochastic linear differential equations

Abstract

We consider the asymptotic behavior of the solutions of a stochastic linear differential equation driven by a finite states Markov process. We consider the sample path Lyapunov exponent λ and the p-moment Lyapunov exponents g(p) for positive p. We derive relations between X and g{p\ which are extensions to our situation of results of Arnold [1] in a different context. Using a Lyapunov function approach, an exact expression forg(2) and estimates for g(p) are obtained, thus leading to upper and lower bounds for λ

Keywords:

• Linear stochastic differential equations
• Lyapunov exponent
• p-moment Lyapunov exponent
• finite state Markov process