Forward-Backward Stochastic Differential Equations and Controlled McKean Vlasov Dynamics

By René A. Carmona, François Delarue. 2013

325 citations recorded by the source corpus.

The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of 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 of an optimal solution are satisfied. Finally we show that our solution to the control problem provides approximate equilibria for large stochastic controlled systems with mean-field interactions when subject to a common policy.

Subfields: Demography, Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/631f16ee7474c21592e33bd3d4f4e8bcdcfa5d09

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Citations in the corpus

A Probabilistic Approach to Mean Field Games with Major and Minor Players (2014, background): “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 = (ut )t is optimal, then the Hamiltonia”
A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria (2014, methodology): “Secondly we use results from [5, 7] to pass from a small time to an arbitrary time horizon and there justify the existence of a unique global solution to (1.”
Distribution dependent SDEs driven by fractional Brownian motions (2021, background): “mention, for examples, the works in [10, 11, 18] for large scale social interactions within the memory of mean-field games, [8, 13, 23] for value functions and related PDEs, [19, 20, 32, 36] for (shift) Harnack type inequalities and gradient estimates, and the references therein.”
McKean–Vlasov optimal control: The dynamic programming principle (2019, methodology): “Motivated by the notion of weak solution of classical SDEs, and similarly to the ideas used by El Karoui and Tan [37], and Carmona, Delarue, and Lacker [28] in a mean–field games context, our first task is to provide an appropriate ‘relaxation‘ of the problem.”
Bellman equation and viscosity solutions for mean-field stochastic control problem (2015, unknown): “It has been used in [13] for diﬀerentiating the Hamiltonian function appearing in stochastic Pontryagin principle for controlled McKean − Vlasov dynamics.”
Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications (2016, background): “Optimal control of McKean-Vlasov equation is a rather new topic in the area of stochastic control and applied probability, and addressed, e.g., in [4], [11], [8], [15], [31].”
Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics (2016, unknown): “In the literature McKean-Vlasov control problem is tackled by two different approaches: On the one hand, the stochastic Pontryagin maximum principle allows one to characterize solutions to the controlled McKean-Vlasov systems in terms of an adjoint backward stochastic differential equation (BSDE) co”
A mean-field stochastic control problem with partial observations (2017, background): “Following the argument of Lasry and Lions [20] (see also [8, 9, 11, 12, 17]), if we assume that the game is symmetric, i.”
Mean-field stochastic differential equations and associated PDEs (2014, methodology): “Among the different notions of differentiability of a function f defined over P2(Rd), we adopt for our approach that introduced by Lions [17] in his lectures at Collège de France in Paris and revised in the notes by Cardaliaguet [6]; we refer the reader also, for instance, to Carmona and Delarue [9]”
Smoothing properties of McKean–Vlasov SDEs (2017, background): “The underlying idea is very well exposed in [11], which we draw on here.”
Mean field games master equations with nonseparable Hamiltonians and displacement monotonicity (2021, background): “In the context of control problems of McKeanVlasov type, displacement convexity assumptions appeared first in [16] and [19].”
A Pontryagin Maximum Principle in Wasserstein spaces for constrained optimal control problems (2018, unknown): “In particular in [20], a PMP was derived for controlled McKean-Vlasov dynamics describing such an optimal control problem from a probabilistic point of view.”