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Geometric Brownian motion with delay : mean square characterisation

Appleby, John A. D. and Mao, Xuerong and Riedle, Markus (2009) Geometric Brownian motion with delay : mean square characterisation. Proceedings of the American Mathematical Society, 137 (1). pp. 339-348. ISSN 0002-9939

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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 coefficient.