Cari colleghi,
annuncio il seguente seminario che si terrà lunedì 25 marzo, alle 14:00, nella Sala Seminari presso il dipartimento di Matematica dell'università degli studi di Pisa.
Dear colleagues,
I would like to announce the following talk that will be held on Monday, 25 March 2019, at 2 PM, in Sala Seminari, Department of Mathematics, University of Pisa.
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Speaker: Stefano Pagliarani (Università degli studi di Udine)
Title: Contraction methods for a class of McKean-Vlasov (mean-field) SDEs with jumps.
Abstract: We consider two prototype classes of McKean-Vlasov (mean-field) SDEs with jumps. In the first case, the coefficient is assumed to be affine in the state-variable, only measurable in the law, and the dynamics allow for Lévy jumps. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK-V SDE, and derive a discretized Picard iteration method that approximates the law of the solution. Numerical illustrations show the effectiveness of the method, which appears to be appropriate to handle multi-dimensional settings. We finally describe possible extensions and generalizations to more general settings. The second class of MKV SDEs that we consider allows for self-exicitng jumps, which amounts to having jumps through hitting the boundary in the equivalent large particle system. The corresponding PDE problem is a particular instance of free-boundary problems known as supercooled Stefan-like problems. We write a Volterra equation for the free boundary and prove contraction results that are useful to determine the properties of the solution and to provide convergent numerical schemes.