Mean field approach to stochastic control with partial information

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2021085 ◽

2021 ◽

Author(s):

Alain Bensoussan ◽

Phillip Yam

Keyword(s):

Partial Information ◽

Mean Field Theory ◽

Mean Field ◽

Initial Distribution ◽

Linear Quadratic ◽

Gaussian Case ◽

Hamilton Jacobi Bellman ◽

Non Gaussian ◽

Initial Distributions ◽

Quadratic Case

In our present article, we follow our way of developing mean field type control theory in our earlier works [4], by first introducing the Bellman and then master equations, the system of Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) equations, and then tackling them by looking for the semi-explicit solution for the linear quadratic case, especially with an arbitrary initial distribution; such a problem, being left open for long, has not been specifically dealt with in the earlier literature, such as [3, 13], which only tackled the linear quadratic setting with Gaussian initial distributions. Thanks to the effective mean-field theory, we propose a solution to this long standing problem of the general non-Gaussian case. Besides, our problem considered here can be reduced to the model in [2], which is fundamentally different from our present proposed framework.

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Backward Mean-Field Linear-Quadratic-Gaussian (LQG) Games: Full and Partial Information

IEEE Transactions on Automatic Control ◽

10.1109/tac.2016.2519501 ◽

2016 ◽

Vol 61 (12) ◽

pp. 3784-3796 ◽

Cited By ~ 12

Author(s):

Jianhui Huang ◽

Shujun Wang ◽

Zhen Wu

Keyword(s):

Partial Information ◽

Mean Field ◽

Linear Quadratic ◽

Linear Quadratic Gaussian

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Fractal heterogeneities in sonic logs and low-frequency scattering attenuation

Geophysics ◽

10.1190/1.3062859 ◽

2009 ◽

Vol 74 (2) ◽

pp. WA77-WA92 ◽

Cited By ~ 17

Author(s):

Thomas J. Browaeys ◽

Sergey Fomel

Keyword(s):

Mean Field Theory ◽

Mean Field ◽

Analytical Formula ◽

Low Frequency ◽

Compressional Waves ◽

Scattering Attenuation ◽

Spatial Frequencies ◽

Fractal Properties ◽

Non Gaussian ◽

Sonic Logs

Cycles in sedimentary strata exist at different scales and can be described by fractal statistics. We use von Kármán’s autocorrelation function to model heterogeneities in sonic logs from a clastic reservoir and propose a nonlinear parameter estimation. Our method is validated using synthetic signals. When applied to real sonic logs, it extracts the fractal properties of high spatial frequencies and one dominant cycle between 2.5 and [Formula: see text]. Results demonstrate non-Gaussian and antipersistent statistics of sedimentary layers. We have derived an analytical formula for the scattering attenuation of scalar waves by 3D isotropic fractal heterogeneities using the mean field theory. Penetration of waves exhibits a high-frequency cutoff sensitive to heterogeneity size. Therefore, shear waves can be attenuated more than compressional waves because of their shorter wavelength.

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Mean Field Game with Delay: A Toy Model

Risks ◽

10.3390/risks6030090 ◽

2018 ◽

Vol 6 (3) ◽

pp. 90

Author(s):

Jean-Pierre Fouque ◽

Zhaoyu Zhang

Keyword(s):

Master Equation ◽

Dimensional Space ◽

Mean Field ◽

Kolmogorov Equation ◽

Linear Quadratic ◽

Quadratic Mean ◽

Mean Field Game ◽

Infinite Dimensional ◽

Player Game ◽

Hamilton Jacobi Bellman

We study a toy model of linear-quadratic mean field game with delay. We “lift” the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We identify the corresponding master equation. A solution to this master equation is computed, and we show that it provides an approximation to a Nash equilibrium of the finite player game.

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Nonzero Sum Differential Game of Mean-Field BSDEs with Jumps under Partial Information

Mathematical Problems in Engineering ◽

10.1155/2014/561382 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12

Author(s):

Xiaolan Chen ◽

Qingfeng Zhu

Keyword(s):

Nash Equilibrium ◽

Equilibrium Point ◽

Differential Game ◽

Partial Information ◽

Mean Field ◽

Random Measure ◽

Necessary Condition ◽

Linear Quadratic ◽

Nash Equilibrium Point ◽

State Process

This paper is concerned with a kind of nonzero sum differential game of mean-field backward stochastic differential equations with jump (MF-BSDEJ), in which the coefficient contains not only the state process but also its marginal distribution. Moreover, the cost functional is also of mean-field type. It is required that the control is adapted to a subfiltration of the filtration generated by the underlying Brownian motion and Poisson random measure. We establish a necessary condition in the form of maximum principle with Pontryagin’s type for open-loop Nash equilibrium point of this type of partial information game and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a partial information linear-quadratic (LQ) game.

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On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2019057 ◽

2020 ◽

Vol 26 ◽

pp. 41

Author(s):

Tianxiao Wang

Keyword(s):

Optimal Control ◽

Closed Loop ◽

Optimal Control Problems ◽

Mean Field ◽

Open Loop ◽

Time Inconsistency ◽

Control Problems ◽

Linear Quadratic ◽

Linear Quadratic Optimal Control ◽

Equilibrium Strategies

This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

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