Stability Properties of Systems of Linear Stochastic Differential Equations with Random Coefficients

Adrian N Bishop (1) , Pierre del Moral (2, 3, 4)

1 UTS - University of Technology Sydney
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 CQFD - Quality control and dynamic reliability
4 IMB - Institut de Mathématiques de Bordeaux

This work is concerned with the stability properties of linear stochastic differential equationswith random (drift and diffusion) coefficient matrices, and the stability of a corresponding ran-dom transition matrix (or exponential semigroup). We consider a class of random matrix driftcoefficients that involves random perturbations of an exponentially stable flow of deterministic(time-varying) drift matrices. In contrast with more conventional studies, our analysis is notbased on the existence of Lyapunov functions, and it does notrely on any ergodic properties.These approaches are often difficult to apply in practice whenthe drift/diffusion coefficients arerandom. We present rather weak and easily checked perturbation-type conditions for the asymp-totic stability of time-varying and random linear stochastic differential equations. We providenew log-Lyapunov estimates and exponential contraction inequalities on any time horizon assoon as the fluctuation parameter is sufficiently small. Theseseem to be the first results of thistype for this class of linear stochastic differential equations with random coefficient matrices.

Domaines: Probabilités [math.PR]

Adrian N Bishop, Pierre del Moral. Stability Properties of Systems of Linear Stochastic Differential Equations with Random Coefficients. SIAM Journal on Control and Optimization, 2019, 57 (2), pp.1023-1042. ⟨10.1137/18M1182759⟩. ⟨hal-02429241⟩