[Forum SIS] UNIBO Statistics Seminars

[Silvia Cagnone]

We are glad to announce the following  Statistics Seminar:



Thursday, November 16, 2.30 pm


Room 2, Piazza Scaravilli 2, Bologna


Stefano Pagliarani (DEAMS, University of Trieste)

Analytical approximations for a class of non-Linear SDEs of McKean-Vlasov type



Abstract:  We provide analytical approximations for the law of the solutions to a certain class of scalar McKean-Vlasov stochastic differential equations (MKV-SDEs) with random initial datum. "Propagation of chaos" results (see Sznitman 1991) connect this class of SDEs with the macroscopic limiting behavior of a particle, evolving within a mean-field interaction particle system, as the total number of particles tends to infinity. Here we assume the mean-field interaction only acting on the drift of each particle, this giving rise to a MKV-SDE where the drift coefficient depends on the law of the unknown solution. By perturbing the non-linear forward Kolmogorov equation associated to the MKV-SDE, we perform a two-steps approximating procedure that decouples the McKean-Vlasov interaction from the standard dependence on the state-variables. The first step yields an expansion for the marginal distribution at a given time, whereas the second yields an expansion for the transition density. Both the approximating series turn out to be asymptotically convergent in the limit of short times and small noise. Concise numerical tests are presented to illustrate the accuracy of the resulting approximation formulas. The latter are expressed in semi-closed form and can be then regarded as a viable alternative to the numerical simulation of the large-particle system. Moreover, these results pave the way for further extensions of this approach to more general dynamics and to high-dimensional settings. Finally, some recent techniques to numerically approximate a specific class of MKV-SDEs with jumps are briefly outlined.



Contact person: Enrico Bernardi


The schedule of the statistics seminars are available at http://www.stat.unibo.it/it/dipartimento/seminari-di-statistica-2017




Silvia Cagnone
Department of Statistical Sciences "Paolo Fortunati"
University of Bologna
Via delle Belle Arti 41 - 40126  Bologna,  ITALY
Tel: +39 051 2098213  Fax: +39 051 232153
http://www2.stat.unibo.it/cagnone/

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