On the relationship between the sample path and moment lyapunov exponents for jump linear systems

In this note, we study the relationship between the sample and moment Lyapunov exponents for jump linear systems. Using a large deviation theorem, a modified version of Arnold’s formula for connecting sample path and moment Lyapunov exponents for continuous-time linear stochastic systems is extended to discrete-time jump linear systems. Sample path stability properties of linear stochastic systems are determined by the top Lyapunov exponent and relating sample and moment Lyapunov exponents may be useful for developing computationally efficient methods for determining the almost-sure (sample path) stability of linear stochastic systems.