On the homogenization of some non-coercive Hamilton--Jacobi--Isaacs equations
Martino Bardi Gabriele Terrone
Communications on Pure & Applied Analysis 2013, 12(1): 207-236 doi: 10.3934/cpaa.2013.12.207
We study the homogenization of Hamilton-Jacobi equations with oscillating initial data and non-coercive Hamiltonian, mostly of the Bellman-Isaacs form arising in optimal control and differential games. We describe classes of equations for which pointwise homogenization fails for some data. We prove locally uniform homogenization for various Hamiltonians with some partial coercivity and some related restrictions on the oscillating variables, mostly motivated by the applications to differential games, in particular of pursuit-evasion type. The effective initial data are computed under some assumptions of asymptotic controllability of the underlying control system with two competing players.
keywords: oscillating initial data. Homogenization Isaacs equations viscosity solutions Hamilton-Jacobi equations differential games
Explicit solutions of some linear-quadratic mean field games
Martino Bardi
Networks & Heterogeneous Media 2012, 7(2): 243-261 doi: 10.3934/nhm.2012.7.243
We consider $N$-person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with long-time-average integral cost functional. We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Planck equations and find explicit Nash equilibria in the form of linear feedbacks. Next we compute the limit as the number $N$ of players goes to infinity, assuming they are almost identical and with suitable scalings of the parameters. This provides a quadratic-Gaussian solution to a system of two differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games [22]. Under a natural normalization the uniqueness of this solution depends on the sign of a single parameter. We also discuss some singular limits, such as vanishing noise, cheap control, vanishing discount. Finally, we compare the L-Q model with other Mean Field models of population distribution.
keywords: differential games models of population distribution. stochastic control linear-quadratic problems Mean field games
On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations
Martino Bardi Paola Mannucci
Communications on Pure & Applied Analysis 2006, 5(4): 709-731 doi: 10.3934/cpaa.2006.5.709
We prove some comparison principles for viscosity solutions of fully nonlinear degenerate elliptic equations that satisfy some conditions of partial non-degeneracy instead of the usual uniform ellipticity or strict monotonicity. These results are applied to the well-posedness of the Dirichlet problem under suitable conditions at the characteristic points of the boundary. The examples motivating the theory are operators of the form of sum of squares of vector fields plus a nonlinear first order Hamiltonian and the Pucci operator over the Heisenberg group.
keywords: comparison principle Pucci operators. Viscosity solution subelliptic equation Heisenberg group degenerate elliptic equation
Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations
Martino Bardi Yoshikazu Giga
Communications on Pure & Applied Analysis 2003, 2(4): 447-459 doi: 10.3934/cpaa.2003.2.447
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).
keywords: Hamilton-Jacobi equations differential games. semicontinuous viscosity solutions Cauchy problem right accessibility
Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximations
Martino Bardi Shigeaki Koike Pierpaolo Soravia
Discrete & Continuous Dynamical Systems - A 2000, 6(2): 361-380 doi: 10.3934/dcds.2000.6.361
In this paper we study the boundary value problem for the Hamilton-Jacobi-Isaacs equation of pursuit-evasion differential games with state constraints. We prove existence of a continuous viscosity solution and a comparison theorem that we apply to establish uniqueness of such a solution and its uniform approximation by solutions of discretized equations.
keywords: comparison principle state-constraint problem Pursuit-evasion game discrete-time approximation. viscosity solution
Large deviations for some fast stochastic volatility models by viscosity methods
Martino Bardi Annalisa Cesaroni Daria Ghilli
Discrete & Continuous Dynamical Systems - A 2015, 35(9): 3965-3988 doi: 10.3934/dcds.2015.35.3965
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 homogenization 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.
keywords: viscosity solutions Singular perturbations stochastic volatility large deviations.

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