264 research outputs found
'Phase diagram' of a mean field game
Mean field games were introduced by J-M.Lasry and P-L. Lions in the
mathematical community, and independently by M. Huang and co-workers in the
engineering community, to deal with optimization problems when the number of
agents becomes very large. In this article we study in detail a particular
example called the 'seminar problem' introduced by O.Gu\'eant, J-M Lasry, and
P-L. Lions in 2010. This model contains the main ingredients of any mean field
game but has the particular feature that all agent are coupled only through a
simple random event (the seminar starting time) that they all contribute to
form. In the mean field limit, this event becomes deterministic and its value
can be fixed through a self consistent procedure. This allows for a rather
thorough understanding of the solutions of the problem, through both exact
results and a detailed analysis of various limiting regimes. For a sensible
class of initial configurations, distinct behaviors can be associated to
different domains in the parameter space . For this reason, the 'seminar
problem' appears to be an interesting toy model on which both intuition and
technical approaches can be tested as a preliminary study toward more complex
mean field game models
Convergence, Fluctuations and Large Deviations for finite state Mean Field Games via the Master Equation
We show the convergence of finite state symmetric N-player differential
games, where players control their transition rates from state to state, to a
limiting dynamics given by a finite state Mean Field Game system made of two
coupled forward-backward ODEs. We exploit the so-called Master Equation, which
in this finite-dimensional framework is a first order PDE in the simplex of
probability measures, obtaining the convergence of the feedback Nash
equilibria, the value functions and the optimal trajectories. The convergence
argument requires only the regularity of a solution to the Master equation.
Moreover, we employ the convergence method to prove a Central Limit Theorem and
a Large Deviation Principle for the evolution of the N-player empirical
measures. The well-posedness and regularity of solution to the Master Equation
are also studied
A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow
In this paper we present a Semi-Lagrangian scheme for a regularized version
of the Hughes model for pedestrian flow. Hughes originally proposed a coupled
nonlinear PDE system describing the evolution of a large pedestrian group
trying to exit a domain as fast as possible. The original model corresponds to
a system of a conservation law for the pedestrian density and an Eikonal
equation to determine the weighted distance to the exit. We consider this model
in presence of small diffusion and discuss the numerical analysis of the
proposed Semi-Lagrangian scheme. Furthermore we illustrate the effect of small
diffusion on the exit time with various numerical experiments
The Master Equation for Large Population Equilibriums
We use a simple N-player stochastic game with idiosyncratic and common noises
to introduce the concept of Master Equation originally proposed by Lions in his
lectures at the Coll\`ege de France. Controlling the limit N tends to the
infinity of the explicit solution of the N-player game, we highlight the
stochastic nature of the limit distributions of the states of the players due
to the fact that the random environment does not average out in the limit, and
we recast the Mean Field Game (MFG) paradigm in a set of coupled Stochastic
Partial Differential Equations (SPDEs). The first one is a forward stochastic
Kolmogorov equation giving the evolution of the conditional distributions of
the states of the players given the common noise. The second is a form of
stochastic Hamilton Jacobi Bellman (HJB) equation providing the solution of the
optimization problem when the flow of conditional distributions is given. Being
highly coupled, the system reads as an infinite dimensional Forward Backward
Stochastic Differential Equation (FBSDE). Uniqueness of a solution and its
Markov property lead to the representation of the solution of the backward
equation (i.e. the value function of the stochastic HJB equation) as a
deterministic function of the solution of the forward Kolmogorov equation,
function which is usually called the decoupling field of the FBSDE. The
(infinite dimensional) PDE satisfied by this decoupling field is identified
with the \textit{master equation}. We also show that this equation can be
derived for other large populations equilibriums like those given by the
optimal control of McKean-Vlasov stochastic differential equations. The paper
is written more in the style of a review than a technical paper, and we spend
more time and energy motivating and explaining the probabilistic interpretation
of the Master Equation, than identifying the most general set of assumptions
under which our claims are true
Controlled diffusion processes
This article gives an overview of the developments in controlled diffusion
processes, emphasizing key results regarding existence of optimal controls and
their characterization via dynamic programming for a variety of cost criteria
and structural assumptions. Stochastic maximum principle and control under
partial observations (equivalently, control of nonlinear filters) are also
discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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