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Characterisation of the asymptotic behaviour of scalar linear differential equations with respect to a fading stochastic perturbation

Pages: 79 - 90, Issue Special, September 2011

 Abstract        Full Text (345.6K)              

John A. D. Appleby - Edgeworth Centre for Financial Mathematics, School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland (email)
Jian Cheng - Edgeworth Centre for Financial Mathematics, School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland (email)
Alexandra Rodkina - The University of the West Indies, Mona Campus, Department of Mathematics, Mona, Kingston 7, Jamaica (email)

Abstract: In this paper, we characterise the global stability, boundedness, and unboundedness of solutions of a scalar linear stochastic di erential equation, where the di usion coecient is independent of the state. The di erential equation is a perturbed version of a linear deterministic equation with a globally stable equilibrium at zero. We give conditions on the rate of decay of the noise intensity under which all solutions either tend to the equilibrium, are bounded but tend to zero with probability zero, or are unbounded on the real line. We also show that no other types of asymptotic behaviour are possible.

Keywords:  Stochastic di erential equation, asymptotic stability, globally bounded, unbounded, recurrent, simulated annealing, fading stochastic perturbations.
Mathematics Subject Classification:  Primary: 60H10, 93E15; Secondary: 93D09, 93D20.

Received: July 2010;      Revised: April 2011;      Published: October 2011.