432 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
Instantaneous self-fulfilling of long-term prophecies on the probabilistic distribution of financial asset values
Our goal here is to present various examples of situations where a “large” investor (i.e. an investor whose “size” challenges the liquidity or the depth of the market) sees his long-term guesses on some important financial parameters instantaneously confirmed by the market dynamics as a consequence of his trading strategy, itself based upon his guesses. These examples are worked out in the context of a model (i.e. a quantitative framework) which attempts to provide a rigorous basis for the qualitative intuitions of many practitioners. Our results may be viewed as some kind of reverse Black–Scholes paradigm where modifications of option prices affect today's real volatility.ou
Conservation laws arising in the study of forward-forward Mean-Field Games
We consider forward-forward Mean Field Game (MFG) models that arise in
numerical approximations of stationary MFGs. First, we establish a link between
these models and a class of hyperbolic conservation laws as well as certain
nonlinear wave equations. Second, we investigate existence and long-time
behavior of solutions for such models
Widespread parainflammation in human cancer.
BackgroundChronic inflammation has been recognized as one of the hallmarks of cancer. We recently showed that parainflammation, a unique variant of inflammation between homeostasis and chronic inflammation, strongly promotes mouse gut tumorigenesis upon p53 loss. Here we explore the prevalence of parainflammation in human cancer and determine its relationship to certain molecular and clinical parameters affecting treatment and prognosis.ResultsWe generated a transcriptome signature to identify parainflammation in many primary human tumors and carcinoma cell lines as distinct from their normal tissue counterparts and the tumor microenvironment and show that parainflammation-positive tumors are enriched for p53 mutations and associated with poor prognosis. Non-steroidal anti-inflammatory drug (NSAID) treatment suppresses parainflammation in both murine and human cancers, possibly explaining a protective effect of NSAIDs against cancer.ConclusionsWe conclude that parainflammation, a low-grade form of inflammation, is widely prevalent in human cancer, particularly in cancer types commonly harboring p53 mutations. Our data suggest that parainflammation may be a driver for p53 mutagenesis and a guide for cancer prevention by NSAID treatment
Smooth Approximation of Lipschitz functions on Riemannian manifolds
We show that for every Lipschitz function defined on a separable
Riemannian manifold (possibly of infinite dimension), for every continuous
, and for every positive number , there exists
a smooth Lipschitz function such that
for every and
. Consequently, every separable
Riemannian manifold is uniformly bumpable. We also present some applications of
this result, such as a general version for separable Riemannian manifolds of
Deville-Godefroy-Zizler's smooth variational principle.Comment: 10 page
Evolutionary game of coalition building under external pressure
We study the fragmentation-coagulation (or merging and splitting)
evolutionary control model as introduced recently by one of the authors, where
small players can form coalitions to resist to the pressure exerted by the
principal. It is a Markov chain in continuous time and the players have a
common reward to optimize. We study the behavior as grows and show that the
problem converges to a (one player) deterministic optimization problem in
continuous time, in the infinite dimensional state space
Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian
We consider Hamilton Jacobi Bellman equations in an inifinite dimensional
Hilbert space, with quadratic (respectively superquadratic) hamiltonian and
with continuous (respectively lipschitz continuous) final conditions. This
allows to study stochastic optimal control problems for suitable controlled
Ornstein Uhlenbeck process with unbounded control processes
Allocating HIV Prevention Funds in the United States: Recommendations from an Optimization Model
The Centers for Disease Control and Prevention (CDC) had an annual budget of approximately $327 million to fund health departments and community-based organizations for core HIV testing and prevention programs domestically between 2001 and 2006. Annual HIV incidence has been relatively stable since the year 2000 [1] and was estimated at 48,600 cases in 2006 and 48,100 in 2009 [2]. Using estimates on HIV incidence, prevalence, prevention program costs and benefits, and current spending, we created an HIV resource allocation model that can generate a mathematically optimal allocation of the Division of HIV/AIDS Prevention’s extramural budget for HIV testing, and counseling and education programs. The model’s data inputs and methods were reviewed by subject matter experts internal and external to the CDC via an extensive validation process. The model projects the HIV epidemic for the United States under different allocation strategies under a fixed budget. Our objective is to support national HIV prevention planning efforts and inform the decision-making process for HIV resource allocation. Model results can be summarized into three main recommendations. First, more funds should be allocated to testing and these should further target men who have sex with men and injecting drug users. Second, counseling and education interventions ought to provide a greater focus on HIV positive persons who are aware of their status. And lastly, interventions should target those at high risk for transmitting or acquiring HIV, rather than lower-risk members of the general population. The main conclusions of the HIV resource allocation model have played a role in the introduction of new programs and provide valuable guidance to target resources and improve the impact of HIV prevention efforts in the United States
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