Instability and stability of solutions of systems of nonlinear stochastic difference equations with diagonal noise

Abstract

We prove results about almost sure instability and stability of the equilibrium of a system of nonlinear stochastic difference equation with a small parameter h. The structure of the system is motivated by the Euler–Maruyama discretization of an Itô stochastic differential equation where h is a discretization parameter. Our analysis relies upon a variant of the discrete Itô formula, which is derived for this system.

Keywords::

• system of stochastic nonlinear equations
• stability
• instability
• discrete Itô formula

AMS Subject Classification::

• 39A50
• 34F05
• 39A30
• 93E15
• 37H10

Acknowledgements

The authors thank Dr G. Berkolaiko for his useful discussions, ARC grant of Australia DP120100928 for support of this work and anonymous referee for valuable remarks.