Exponential growth rate for a singular linear stochastic delay differential equation

2012/01/12 by Michael Scheutzow, Scheutzow, Michael · 1 citation
Mathematics · #34K50 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:34K50

paper · pdf · doi:10.48550/arxiv.1201.2599

15 pages

arxiv created 2012/01/12 · arxiv updated 2012/01/13

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 dX(t) = X(t-1) dW(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).

Cited by

• Approximation to Deep Q-Network by Stochastic Delay Differential Equations

Related

• Exact growth rates of solutions of delay--dominated differential equations with regularly varying coefficients
• Stochastic Delay Differential Equations have blow-up solutions if and only if their instantaneous counterparts have them
• Strong Uniqueness of Singular Stochastic Delay Equations
• Subexponential Growth Rates in Functional Differential Equations
• Stationarity of Stochastic Linear Equations with Additive Noise and Delays in the Unbounded Drift Terms