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Stochastic Fokker-Planck equations for conditional McKean-Vlasov jump diffusions and applications to optimal control
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0003-1662-0215
Department of Mathematics, University of Oslo, Oslo, Norway.
2023 (English)In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 61, no 3, p. 1472-1493Article in journal (Refereed) Published
Abstract [en]

The purpose of this paper is to study optimal control of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffu-sions, for short). To this end, we first prove a stochastic Fokker-Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate a Hamilton-Jacobi-Bellman equation for the optimal control of conditional McKean-Vlasov jump diffusions. Then we study the situation when the law is absolutely continuous with respect to Lebesgue measure. In that case the Fokker-Planck equation reduces to a stochastic par-tial differential equation for the Radon-Nikodym derivative of the conditional law. Finally we apply these results to solve explicitly the linear-quadratic optimal control problem of conditional stochastic McKean-Vlasov jump diffusions, and optimal consumption from a cash flow.

Place, publisher, year, edition, pages
Society for Industrial & Applied Mathematics (SIAM) , 2023. Vol. 61, no 3, p. 1472-1493
Keywords [en]
jump diffusion, common noise, conditional McKean-Vlasov differential equation, stochastic Fokker-Planck equation, optimal control, HJB equation
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-334288DOI: 10.1137/21M1461034ISI: 001031998600016Scopus ID: 2-s2.0-85163551831OAI: oai:DiVA.org:kth-334288DiVA, id: diva2:1789189
Note

QC 20231122

Available from: 2023-08-18 Created: 2023-08-18 Last updated: 2023-11-22Bibliographically approved

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Agram, NaciraØksendal, Bernt

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