Estimate on the Pathwise Lyapunov Exponent of Linear Stochastic Differential Equations with Constant Coefficients

Abstract

In this article, we study the problem of estimating the pathwise Lyapunov exponent for linear stochastic systems with multiplicative noise and constant coefficients. We present a Lyapunov type matrix inequality that is closely related to this problem, and show under what conditions we can solve the matrix inequality. From this we can deduce an upper bound for the Lyapunov exponent. In the converse direction, it is shown that a necessary condition for the stochastic system to be pathwise asymptotically stable can be formulated in terms of controllability properties of the matrices involved.

Keywords:

• Linear stochastic differential equations
• Lyapunov exponent
• Matrix inequalities
• Stability

Mathematics Subject Classification:

• 93E15
• 15A24
• 60H10

J. B. and O. V. G. acknowledge the support by a “VIDI subsidie” (639.032.510) of the Netherlands Organisation for Scientific Research (NWO). All of the authors thank the anonymous referee for valuable comments and improvements.