166 research outputs found
Liouville properties and critical value of fully nonlinear elliptic operators
We prove some Liouville properties for sub- and supersolutions of fully
nonlinear degenerate elliptic equations in the whole space. Our assumptions
allow the coefficients of the first order terms to be large at infinity,
provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We
give two applications. The first is a stabilization property for large times of
solutions to fully nonlinear parabolic equations. The second is the solvability
of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique
critical value of the operator.Comment: 18 pp, to appear in J. Differential Equation
Comparison Principles for subelliptic equations of Monge-Ampere type
We present two comparison principles for viscosity sub- and supersolutions of
Monge-Ampere-type equations associated to a family of vector fields. In
particular, we obtain the uniqueness of a viscosity solution to the Dirichlet
problem for the equation of prescribed horizontal Gauss curvature in a Carnot
group
Linear-Quadratic -person and Mean-Field Games with Ergodic Cost
We consider stochastic differential games with players, linear-Gaussian
dynamics in arbitrary state-space dimension, and long-time-average cost with
quadratic running cost. Admissible controls are feedbacks for which the system
is ergodic. We first study the existence of affine Nash equilibria by means of
an associated system of Hamilton-Jacobi-Bellman and
Kolmogorov-Fokker-Planck partial differential equations. We give necessary and
sufficient conditions for the existence and uniqueness of quadratic-Gaussian
solutions in terms of the solvability of suitable algebraic Riccati and
Sylvester equations. Under a symmetry condition on the running costs and for
nearly identical players we study the large population limit, tending to
infinity, and find a unique quadratic-Gaussian solution of the pair of Mean
Field Game HJB-KFP equations. Examples of explicit solutions are given, in
particular for consensus problems.Comment: 31 page
Mean Field Games models of segregation
This paper introduces and analyses some models in the framework of Mean Field
Games describing interactions between two populations motivated by the studies
on urban settlements and residential choice by Thomas Schelling. For static
games, a large population limit is proved. For differential games with noise,
the existence of solutions is established for the systems of partial
differential equations of Mean Field Game theory, in the stationary and in the
evolutive case. Numerical methods are proposed, with several simulations. In
the examples and in the numerical results, particular emphasis is put on the
phenomenon of segregation between the populations.Comment: 35 pages, 10 figure
Large deviations for some fast stochastic volatility models by viscosity methods
We consider the short time behaviour of stochastic systems affected by a
stochastic volatility evolving at a faster time scale. We study the asymptotics
of a logarithmic functional of the process by methods of the theory of
homogenisation and singular perturbations for fully nonlinear PDEs. We point
out three regimes depending on how fast the volatility oscillates relative to
the horizon length. We prove a large deviation principle for each regime and
apply it to the asymptotics of option prices near maturity
Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations
We study Hamilton-Jacobi equations with upper semicontinuous initial data without convexity assumptions on the Hamiltonian. We analyse the behavior of generalized u.s.c. solutions at the initial time t = 0, and find necessary and sufficient conditions on the Hamiltonian such that the solution attains the initial data along a sequence (right accessibility)
- âŠ