Exponential growth rate for a singular linear stochastic delay differential equation

Michael Scheutzow

doi:10.3934/dcdsb.2013.18.1683

Article Contents

2013, Volume 18, Issue 6: 1683-1696. Doi: 10.3934/dcdsb.2013.18.1683

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Exponential growth rate for a singular linear stochastic delay differential equation

Michael Scheutzow^1,

1.
Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin

Received: December 31, 2011

Revised: February 29, 2012

Published: July 2013

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Abstract

We establish the existence of a deterministic exponential growth rate for the norm (on an appropriate function space) of the solution of the linear scalar stochastic delay equation $d X(t) = X(t-1) d W(t)$ which does not depend on the initial condition as long as it is not identically zero. Due to the singular nature of the equation this property does not follow from available results on stochastic delay differential equations. The key technique is to establish existence and uniqueness of an invariant measure of the projection of the solution onto the unit sphere in the chosen function space via asymptotic coupling and to prove a Furstenberg-Hasminskii-type formula (like in the finite dimensional case).

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Mathematics Subject Classification: Primary: 34K50; Secondary: 60H10.

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