Geometric Brownian Motion with delay: mean square characterisation

Markus Riedle , John A. D. Appleby , Xuerong Mao

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 26, 9

MSC 2000

60H20 Stochastic integral equations

60H10 Stochastic ordinary differential equations

34K20 Stability theory

34K50 Stochastic delay equations

Abstract
A geometric Brownian motion with delay is the solution of a stochastic differential equation where the drift and diffusion coefficient depend linearly on the past of the solution, i.e. a linear stochastic functional differential equation. In this work the asymptotic behavior in mean square of a geometric Brownian motion with delay is completely characterized by a sufficient and necessary condition in terms of the drift and diffusion coefficients.