Linear-Quadratic Mean Field Games

Bensoussan, Alain; Sung, K. C. J.; Yam, Philip; Yung, S. P.

doi:10.1007/s10957-015-0819-4

Cited by 225 publications

(229 citation statements)

References 46 publications

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“…The mean variance portfolio optimization example discussed in [2] and the solution proposed in [3] and [8] of the optimal control of linear-quadratic (LQ) McKean-Vlasov dynamics are based on the general form of the Pontryagin principle proven in this section as applied to the scalar interactions considered in this subsection.…”

Section: Scalar Interactionsmentioning

confidence: 99%

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Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics

Carmona¹,

Delarue²

2015

Ann. Probab.

237

286

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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 of an optimal solution are satisfied. Finally we show that our solution to the control problem provides approximate equilibria for large stochastic games with mean field interactions.

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Section: Scalar Interactionsmentioning

confidence: 99%

“…[3][4] in Section 4 (with respect to some constant L). In particular, there exists a constantL such that…”

Section: Technical Assumptionsmentioning

confidence: 99%

Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics

Carmona¹,

Delarue²

2015

Ann. Probab.

237

286

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“…We apply our verification theorem to the important class of linear quadratic (LQ) McKean-Vlasov control problems, addressed e.g. in [38] and [7] by maximum principle and adjoint equations, and that we solve by a different approach where it turns out that derivations in the space of probability measures are quite tractable and lead to explicit classical solutions for the Bellman equation. We illustrate these results with two examples arising from finance: the mean-variance portfolio selection and an inter-bank systemic risk model, and retrieve the results obtained in [29], [20] and [15] by different methods.…”

Section: Introductionmentioning

confidence: 99%

Bellman equation and viscosity solutions for mean-field stochastic control problem

2018

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We consider the stochastic optimal control problem of McKean-Vlasov stochastic differential equation where the coefficients may depend upon the joint law of the state and control. By using feedback controls, we reformulate the problem into a deterministic control problem with only the marginal distribution of the process as controlled state variable, and prove that dynamic programming principle holds in its general form. Then, by relying on the notion of differentiability with respect to probability measures recently introduced by P.L. Lions in [32], and a special Itô formula for flows of probability measures, we derive the (dynamic programming) Bellman equation for mean-field stochastic control problem, and prove a verification theorem in our McKeanVlasov framework. We give explicit solutions to the Bellman equation for the linear quadratic mean-field control problem, with applications to the mean-variance portfolio selection and a systemic risk model. We also consider a notion of lifted viscosity solutions for the Bellman equation, and show the viscosity property and uniqueness of the value function to the McKean-Vlasov control problem. Finally, we consider the case of McKean-Vlasov control problem with open-loop controls and discuss the associated dynamic programming equation that we compare with the case of closed-loop controls.MSC Classification: 93E20, 60H30, 60K35.

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“…Since the dynamics under the optimal control of the linear-quadratic mean-field game with ambiguity aversion can be associated with a version of the linear-quadratic meanfield game without ambiguity aversion, the -Nash equilibrium holds with the same arguments as in Bensoussan et al (2016). We note that the existence, uniqueness and validity of the -Nash equilibrium is contingent on Assumption 4.1.2, so that the condition provides a sufficient bound for the disturbance attenuation parameter Φ.…”

Section: -Nash Equilibriummentioning

confidence: 96%

Mean-Field Games and Ambiguity Aversion

Huang

2017

SSRN Journal

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This thesis focuses on incorporating the idea of ambiguity aversion into mean-field games.Intuitively, mean-field games describes the dynamics of a system with an infinite population. It is useful in approximating systems with a large population, for which an exact result is computationally intractable. By adding in ambiguity aversion, the mean-field game reflects how players in the population should act if they wish to protect themselves from model misspecification.Two applications of mean-field games are considered through two distinct approaches.The broker execution problem is investigated in a multi-agent framework containing (i) a major agent who is liquidating a large number of shares, (ii) a number of minor agents (high-frequency traders (HFTs)) who search for statistical arbitrage strategies, and (iii) noise traders who buy and sell for exogenous reasons. All optimizing agents (the broker and HFTs) trade against noise traders as well as one another. We use a mean-field game approach to solve the problem and obtain a set of decentralized feedback trading strategies for the major and minor agents. Furthermore, the mean-field game strategies have an N -Nash equilibrium property where N → 0 as N → ∞.The second application focuses on interbank borrowing and lending, which may induce systemic risk into financial markets. A simple model of this is to assume that log-monetary reserves are coupled, and that banks can also borrow/lend from/to a central bank. When all banks optimize their cost of borrowing and lending, this leads to a stochastic game. We account for model uncertainty by recasting the problem as a robust stochastic game and succeed in providing a strategy which leads to a Nash equilibria for ii both the finite game and the mean-field game limit. We prove that an -Nash equilibrium exists and show that when firms are ambiguity-averse, default probabilities can be reduced relative to their ambiguity-neutral counterparts.Finally, this thesis develops a modified stochastic maximum principle for min-max problems, and derives new existence and uniqueness results for a mean-field game with ambiguity averse players. An -Nash equilibrium is shown to exist for this class of mean-field games, and the mean-field game equations are explicitly derived in the linearquadratic framework.iii

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Linear-Quadratic Mean Field Games

Cited by 225 publications

References 46 publications

Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics

Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics

Bellman equation and viscosity solutions for mean-field stochastic control problem

Mean-Field Games and Ambiguity Aversion

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