Abstract
In this paper, we show that the mean square of the solution of a scalar autonomous linear stochastic difference equation of finite order can be written in terms of the solution of an auxiliary deterministic Volterra summation equation. The dynamics of the covariance of the process can also be written in terms of this deterministic equation. As an application, we determine necessary and sufficient conditions for the mean square stability of the equilibrium solution in the case that the underlying deterministic equation is of first order, and determine the exact real exponential rate of growth or decay in the mean–square.
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Acknowledgements
The authors wish to thank the conference organisers for the opportunity to present their work. We are also grateful to the anonymous referee for their careful reading of our manuscript, and for suggesting the inclusion of further relevant literature.
EL is partially supported by Science Foundation Ireland (16/IA/4443).
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Appleby, J.A.D., Lawless, E. (2023). On the Dynamics and Asymptotic Behaviour of the Mean Square of Scalar Linear Stochastic Difference Equations. In: Elaydi, S., Kulenović, M.R.S., Kalabušić, S. (eds) Advances in Discrete Dynamical Systems, Difference Equations and Applications. ICDEA 2021. Springer Proceedings in Mathematics & Statistics, vol 416. Springer, Cham. https://doi.org/10.1007/978-3-031-25225-9_2
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