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Approximation of stationary solutions of Gaussian driven Stochastic Differential Equations

By Serge Cohen and Fabien Panloup

Abstract

We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. Then, we end the paper by some specific properties of this stationary solution. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent to the future of the underlying innovation process of the Gaussian driving process

Topics: Euler scheme, stochastic differential equation, Gaussian process, stationary process, Euler scheme., 60G10, 60G15, 60H35., [MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Publisher: 'Elsevier BV'
Year: 2011
DOI identifier: 10.1016/j.spa.2011.08.001
OAI identifier: oai:HAL:hal-00441180v2

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