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26
Mean-field backward stochastic differential equations and related patial differential equations
, 2007
"... In [5] the authors obtained Mean-Field backward stochastic differential equations (BSDE) associated with a Mean-field stochastic differential equation (SDE) in a natural way as limit of some highly dimensional system of forward and backward SDEs, corresponding to a large number of “particles” (or “a ..."
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Cited by 181 (14 self)
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In [5] the authors obtained Mean-Field backward stochastic differential equations (BSDE) associated with a Mean-field stochastic differential equation (SDE) in a natural way as limit of some highly dimensional system of forward and backward SDEs, corresponding to a large number of “particles” (or “agents”). The objective of the present paper is to deepen the investigation of such Mean-Field BSDEs by studying them in a more general framework, with general driver, and to discuss comparison results for them. In a second step we are interested in partial differential equations (PDE) whose solutions can be stochastically interpreted in terms of Mean-Field BSDEs. For this we study a Mean-Field BSDE in a Markovian framework, associated with a Mean-Field forward equation. By combining classical BSDE methods, in particular that of “backward semigroups” introduced by Peng [14], with specific arguments for Mean-Field BSDEs we prove that this Mean-Field BSDE describes the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated to Mean-Field BSDEs
Stochastic linear quadratic optimal control problems
- Appl. Math. Optim
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Linear-Quadratic Mean Field Games
, 2014
"... As an organic combination of mean field theory in statistical physics and (non-zero sum) stochastic differential games, Mean Field Games (MFGs) has become a very popular research topic in the fields ranging from physical and social sciences to engineering applications, see for example the earlier st ..."
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Cited by 16 (0 self)
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As an organic combination of mean field theory in statistical physics and (non-zero sum) stochastic differential games, Mean Field Games (MFGs) has become a very popular research topic in the fields ranging from physical and social sciences to engineering applications, see for example the earlier studies by Huang, Caines and Malhame ́ (2003), and that by Lasry and Lions (2006a, b and 2007). In this paper, we provide a comprehensive study of a general class of mean field games in the linear quadratic framework. We adopt the adjoint equation approach to investigate the existence and uniqueness of equilibrium strategies of these Linear-Quadratic Mean Field Games (LQMFGs). Due to the linearity of the adjoint equations, the optimal mean field term satisfies a forward-backward ordinary differential equation. For the one dimensional case, we show that the equilibrium strategy always exists uniquely. For dimension greater than one, by choosing a suitable norm and then applying the Banach Fixed Point Theorem, a sufficient condition for the unique existence of the equilibrium strategy is provided, which is independent of the coefficients of controls and is always satisfied whenever those of the
FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND CONTROLLED MCKEAN VLASOV DYNAMICS
"... 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 o ..."
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Cited by 10 (5 self)
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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. 1.
Control of McKean-Vlasov Dynamics versus Mean Field Games
, 2012
"... We discuss and compare two methods of investigations for the asymptotic regime of stochastic differential games with a finite number of players as the number of players tends to the infinity. These two methods differ in the order in which optimization and passage to the limit are performed. When opt ..."
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Cited by 10 (2 self)
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We discuss and compare two methods of investigations for the asymptotic regime of stochastic differential games with a finite number of players as the number of players tends to the infinity. These two methods differ in the order in which optimization and passage to the limit are performed. When optimizing first, the asymptotic problem is usually referred to as a mean-field game. Otherwise, it reads as an optimization problem over controlled dynamics of McKean-Vlasov type. Both problems lead to the analysis of forward-backward stochastic differential equations, the coefficients of which depend on the marginal distributions of the solutions. We explain the difference between the nature and solutions to the two approaches by investigating the corresponding forward-backward systems. General results are stated and specific examples are treated, especially when cost functionals are of linear-quadratic type.
Linear-quadratic-Gaussian mixed games with . . .
, 2012
"... We consider a mean field linear-quadratic-Gaussian game with a major player and a large number of minor players parametrized by a continuum set. The mean field generated by the minor players is approximated by a random process depending only on the initial state and the Brownian motion of the major ..."
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Cited by 7 (1 self)
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We consider a mean field linear-quadratic-Gaussian game with a major player and a large number of minor players parametrized by a continuum set. The mean field generated by the minor players is approximated by a random process depending only on the initial state and the Brownian motion of the major player, and this leads to two limiting optimal control problems with random coefficients, which are solved subject to a consistency requirement on the mean field approximation. The set of decentralized strategies constructed from the limiting control problems has an ε-Nash equilibrium property when applied to the large but finite population model.
