scholarly journals Two-scale homogenization of a stationary mean-field game

2020 ◽  
Vol 26 ◽  
pp. 17
Author(s):  
Rita Ferreira ◽  
Diogo Gomes ◽  
Xianjin Yang
Keyword(s):  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Mean Field ◽  
Main Tool ◽  
Homogenization Theory ◽  
Mean Field Game ◽  
First Order ◽  
Limit Problems ◽  
Uniqueness Of The Solution ◽  
Oscillating Potential

In this paper, we characterize the asymptotic behavior of a first-order stationary mean-field game (MFG) with a logarithm coupling, a quadratic Hamiltonian, and a periodically oscillating potential. This study falls into the realm of the homogenization theory, and our main tool is the two-scale convergence. Using this convergence, we rigorously derive the two-scale homogenized and the homogenized MFG problems, which encode the so-called macroscopic or effective behavior of the original oscillating MFG. Moreover, we prove existence and uniqueness of the solution to these limit problems.

The Second-Order Master Equation

2019 ◽  
pp. 128-158
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Master Equation ◽  
Mean Field ◽  
Second Order ◽  
Well Posedness ◽  
Mean Field Game ◽  
First Order ◽  
Common Noise ◽  
System P ◽  
The Mean

This chapter investigates the second-order master equation with common noise, which requires the well-posedness of the mean field game (MFG) system. It also defines and analyzes the solution of the master equation. The chapter explains the forward component of the MFG system that is recognized as the characteristics of the master equation. The regularity of the solution of the master equation is explored through the tangent process that solves the linearized MFG system. It also analyzes first-order differentiability and second-order differentiability in the direction of the measure on the same model as for the first-order derivatives. This chapter concludes with further description of the derivation of the master equation and well-posedness of the stochastic MFG system.

scholarly journals Viability analysis of the first-order mean field games

2020 ◽  
Vol 26 ◽  
pp. 33
Author(s):  
Yurii Averboukh
Keyword(s):  
Mean Field ◽  
Initial Distribution ◽  
Initial Time ◽  
Mean Field Games ◽  
Sufficient Condition ◽  
Mean Field Game ◽  
First Order ◽  
Viability Analysis ◽  
A Value ◽  
Object Of Study

In the paper, we examine the dependence of the solution of the deterministic mean field game on the initial distribution of players. The main object of study is the mapping which assigns to the initial time and the initial distribution of players the set of expected rewards of the representative player corresponding to solutions of mean field game. This mapping can be regarded as a value multifunction. We obtain the sufficient condition for a multifunction to be a value multifunction. It states that if a multifunction is viable with respect to the dynamics generated by the original mean field game, then it is a value multifunction. Furthermore, the infinitesimal variant of this condition is derived.

Stochastic elastic equation driven by multiplicative multi-parameter fractional noise

2017 ◽  
Vol 17 (02) ◽  
pp. 1750012 ◽  
Author(s):  
Yinghan Zhang ◽  
Xiaoyuan Yang
Keyword(s):  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Chaos Expansion ◽  
Distribution Space ◽  
Wiener Chaos ◽  
Fractional Noise ◽  
Undetermined Coefficient ◽  
Uniqueness Of The Solution ◽  
Wiener Chaos Expansion

In this paper, we consider the stochastic elastic equation driven by multiplicative multiparameter fractional noise. By using the Wiener chaos expansion and undetermined coefficient methods, we obtain the existence and uniqueness of the solution in a distribution space. The asymptotic behavior and the Hölder index of the solution are also estimated.

scholarly journals A Mean Field Game of Optimal Portfolio Liquidation

2021 ◽  
Author(s):  
Guanxing Fu ◽  
Paulwin Graewe ◽  
Ulrich Horst ◽  
Alexandre Popier
Keyword(s):  
Differential Equation ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equation ◽  
Mean Field ◽  
Uniqueness Result ◽  
Terminal Condition ◽  
Optimal Portfolio ◽  
Linear Quadratic ◽  
Mean Field Game ◽  
Terminal Values

We consider a mean field game (MFG) of optimal portfolio liquidation under asymmetric information. We prove that the solution to the MFG can be characterized in terms of a forward-backward stochastic differential equation (FBSDE) with a possibly singular terminal condition on the backward component or, equivalently, in terms of an FBSDE with a finite terminal value yet a singular driver. Extending the method of continuation to linear-quadratic FBSDEs with a singular driver, we prove that the MFG has a unique solution. Our existence and uniqueness result allows proving that the MFG with a possibly singular terminal condition can be approximated by a sequence of MFGs with finite terminal values.

Asymptotic behavior of the Timoshenko-type system with nonlinear boundary control

2019 ◽  
Vol 10 (2) ◽  
pp. 171-182
Author(s):  
Mohamed Ali Ayadi ◽  
Ahmed Bchatnia
Keyword(s):  
Asymptotic Behavior ◽  
Wide Class ◽  
Existence And Uniqueness ◽  
Relaxation Function ◽  
Nonlinear Boundary ◽  
Type System ◽  
Timoshenko Type ◽  
Uniqueness Of The Solution ◽  
Boundary Dissipation ◽  
Nonlinear Boundary Control

AbstractIn this paper, we consider the Timoshenko-type system with nonlinear boundary dissipation. We prove the existence and uniqueness of the solution and we establish an explicit and general decay result for a wide class of the relaxation function, which depends on the length of the beam.

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