1,188 research outputs found
Existence and uniqueness for Mean Field Games with state constraints
In this paper, we study deterministic mean field games for agents who operate
in a bounded domain. In this case, the existence and uniqueness of Nash
equilibria cannot be deduced as for unrestricted state space because, for a
large set of initial conditions, the uniqueness of the solution to the
associated minimization problem is no longer guaranteed. We attack the problem
by interpreting equilibria as measures in a space of arcs. In such a relaxed
environment the existence of solutions follows by set-valued fixed point
arguments. Then, we give a uniqueness result for such equilibria under a
classical monotonicity assumption
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
A convex duality method for optimal liquidation with participation constraints
In spite of the growing consideration for optimal execution in the financial
mathematics literature, numerical approximations of optimal trading curves are
almost never discussed. In this article, we present a numerical method to
approximate the optimal strategy of a trader willing to unwind a large
portfolio. The method we propose is very general as it can be applied to
multi-asset portfolios with any form of execution costs, including a bid-ask
spread component, even when participation constraints are imposed. Our method,
based on convex duality, only requires Hamiltonian functions to have
regularity while classical methods require additional regularity and cannot be
applied to all cases found in practice
Clickers or flashcards is there really a difference? /
Titre de la page Web (visionnée le 12 oct. 2007)Paraît aussi en version papierBibliogr
Optimal Real-Time Bidding Strategies
The ad-trading desks of media-buying agencies are increasingly relying on
complex algorithms for purchasing advertising inventory. In particular,
Real-Time Bidding (RTB) algorithms respond to many auctions -- usually Vickrey
auctions -- throughout the day for buying ad-inventory with the aim of
maximizing one or several key performance indicators (KPI). The optimization
problems faced by companies building bidding strategies are new and interesting
for the community of applied mathematicians. In this article, we introduce a
stochastic optimal control model that addresses the question of the optimal
bidding strategy in various realistic contexts: the maximization of the
inventory bought with a given amount of cash in the framework of audience
strategies, the maximization of the number of conversions/acquisitions with a
given amount of cash, etc. In our model, the sequence of auctions is modeled by
a Poisson process and the \textit{price to beat} for each auction is modeled by
a random variable following almost any probability distribution. We show that
the optimal bids are characterized by a Hamilton-Jacobi-Bellman equation, and
that almost-closed form solutions can be found by using a fluid limit.
Numerical examples are also carried out
Une mise en oeuvre au cégep de la méthode d'apprentissage par les pairs de Harvard
Titre de l'écran-titre (visionné le 23 avril 2009)
Ecological intuition versus economic "reason"
This article discusses the discount rate to be used in projects that aimed at improving the environment. The model has two different goods, one is the usual consumption good whose production may increase exponentially, the other is an environmental good whose quality remains limited. The stylized world we describe is fully determined by four parameters, reflecting basic preferences "ecological" and intergenerational concerns and feasibility constraints. We define an ecological discount rate and examine its connections with the usual interest rate and the optimized growth rate. We discuss, in this simple world, a variety of forms of the precautionary principle.discount rate ; ecological discount rate ; environmental goods ; relative prices ; irreversible damage ; precautionnary principle
Mean Field Games and Applications.
This text is inspired from a “Cours Bachelier” held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three authors and upon unpublished materials they developed. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class.Mean Field Games;
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