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MEAN-FIELD BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS: A LIMIT APPROACH
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0002-6608-0715
2009 (English)In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 37, no 4, 1524-1565 p.Article in journal (Refereed) Published
Abstract [en]

Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution (Y, Z) of a mean-field backward stochastic differential equation driven by a forward stochastic differential of McKean-Vlasov type with solution X we study a special approximation by the solution (X-N, Y-N, Z(N)) of some decoupled forward-backward equation which coefficients are governed by N independent copies of (X-N, Y-N, Z(N)). We show that the convergence speed of this approximation is of order 1/root N. Moreover, our special choice of the approximation allows to characterize the limit behavior of root N(X-N - X, Y-N - Y, Z(N) - Z). We prove that this triplet converges in law to the solution of some forward-backward. stochastic differential equation of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field.

Place, publisher, year, edition, pages
2009. Vol. 37, no 4, 1524-1565 p.
Keyword [en]
Backward stochastic differential equation, mean-field approach, McKean-Vlasov equation, mean-field BSDE, tightness, weak convergence, mckean-vlasov, particle method
Identifiers
URN: urn:nbn:se:kth:diva-18662DOI: 10.1214/08-aop442ISI: 000268692700010Scopus ID: 2-s2.0-69249229620OAI: oai:DiVA.org:kth-18662DiVA: diva2:336709
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved

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Djehiche, Boualem

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