Approximation of stationary solutions of Gaussian driven stochastic differential equations |
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Authors: | Serge Cohen Fabien Panloup |
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Institution: | aInstitut de Mathématiques de Toulouse, Université de Toulouse, 118 route de Narbonne F-31062 Toulouse Cedex 9, France;bLaboratoire de Statistiques et Probabilités, Université de Toulouse&INSA Toulouse, 135, Avenue de Rangueil, 31077 Toulouse Cedex 4, France |
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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 of the future of the underlying innovation process of the Gaussian driving process. |
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Keywords: | MSC: 60G10 60G15 60H35 |
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