Stationary solutions of the stochastic differential equation with Lévy noise
For a given bivariate Lévy process (Ut,Lt)t>=0, necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation are obtained. Neither strict positivity of the stochastic exponential of U nor independence of V0 and (U,L) is assumed and non-causal solutions may appear. The form of the stationary solution is determined and shown to be unique in distribution, provided it exists. For non-causal solutions, a sufficient condition for U and L to remain semimartingales with respect to the corresponding expanded filtration is given.
Year of publication: |
2011
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Authors: | Behme, Anita ; Lindner, Alexander ; Maller, Ross |
Published in: |
Stochastic Processes and their Applications. - Elsevier, ISSN 0304-4149. - Vol. 121.2011, 1, p. 91-108
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Publisher: |
Elsevier |
Keywords: | Stochastic differential equation Lévy process Generalized Ornstein-Uhlenbeck process Stochastic exponential Stationarity Non-causal Filtration expansion |
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