2015
DOI: 10.1214/14-aop946
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Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics

Abstract: ABSTRACT. The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of the McKean Vlasov type. Motivated by the recent interest in mean field games, we highlight the connection and the differences between the two sets of problems. We prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control. We also provide examples for which our sufficient conditions for existence o… Show more

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Cited by 237 publications
(291 citation statements)
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“…We next need an Itô's formula along a flow of conditional measures proved in [19] (see also [12] and [13]). Let (Ω, F, P) be a probability space of the form (Ω,…”
Section: If the Liftũ Is Fréchet Differentiable (Resp Fréchet Differmentioning
confidence: 99%
“…We next need an Itô's formula along a flow of conditional measures proved in [19] (see also [12] and [13]). Let (Ω, F, P) be a probability space of the form (Ω,…”
Section: If the Liftũ Is Fréchet Differentiable (Resp Fréchet Differmentioning
confidence: 99%
“…Once properly extended to cover the present situation, (see [6] for the necessary condition in the unconditional case, and the appendix for the sufficient condition) the necessary part of the Pontryagin stochastic maximum principle says that, if the control u 0 = (u 0 t ) t is optimal, then the Hamiltonian (12) is minimized along the trajectory of (X 0 t , X t , P t , Q t ). So given assumption (M0) and the sufficient condition of the stochastic maximum principle proven in the appendix at the end of the paper,û…”
Section: Mean Field Games With Major and Minor Players: The General Casementioning
confidence: 99%
“…The notion of differentiability we use was introduced by P.-L. Lions in his lectures at the Collège de France, recorded in a set of notes by Cardaliaguet [9]. The underlying idea is very well exposed in [11], which we draw on here.…”
Section: Differentiation In P 2 (R N )mentioning
confidence: 99%
“…As MVSDEs can be interpreted as limiting equations for large systems, they are widely used as models in statistical physics [7,31] as well as in the study of largescale social interactions within the theory of mean-field games [10,11,19,20,[26][27][28]. Recently, these equations have also appeared in the mathematical finance literature in the specification and calibration of multi-factor stochastic volatility and hybrid models [5,17].…”
Section: Introductionmentioning
confidence: 99%