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Pré-Publication, Document De Travail Année : 2007

Nonlinear SDEs driven by Lévy processes and related PDEs

Résumé

In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a Lévy process. We also suppose that the coefficient multiplying the increments of this process is merely Lipschitz continuous and not necessarily linear in the time-marginals of the solution as is the case in the classical McKean-Vlasov model. We first study existence, uniqueness and particle approximations for these stochastic differential equations. When the driving process is a pure jump Lévy process with a smooth but unbounded Lévy measure, we develop a stochastic calculus of variations to prove that the time-marginals of the solutions are absolutely continuous with respect to the Lebesgue measure. In the case of a symmetric stable driving process, we deduce the existence of a function solution to a nonlinear integro-differential equation involving the fractional Laplacian.
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Dates et versions

hal-00163798 , version 1 (18-07-2007)

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Benjamin Jourdain, Sylvie Méléard, Wojbor Woyczynski. Nonlinear SDEs driven by Lévy processes and related PDEs. 2007. ⟨hal-00163798⟩
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