Guy Barles

Backward stochastic differential equations and integral-partial differential equations

We consider a backward stochastic differential equation, whose data (the final condition and the coefficient) are given functions of a jump-diffusion process. We prove that under mild conditions the solution of the BSDE provides a viscosity solution of a system of parabolic integral-partial differential equations. Under an additional assumption, that system of equations is proved to have a unique solution, in a given class of continuous functions

A Simple Proof of Convergence for an Approximation Scheme for Computing Motions by Mean Curvature

We prove the convergence of an approximation scheme recently proposed by Bence, Merriman, and Osher for computing motions of hypersurfaces by mean curvature. Our proof is based on viscosity solutions methods.

EXISTENCE AND COMPARISON RESULTS FOR FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS WITHOUT ZEROTH-ORDER TERM*

*This work was partially supported by the TMR Programme “Viscosity Solutions and their Applications.”

Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations

. We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton-Jacobi-Bellman equations by introducing a new notion of consistency. Our results are robust and general - they improve and extend earlier results by Krylov, Barles, and Jakobsen. We apply our general results to various schemes including Crank-Nicholson type finite difference schemes, splitting methods, and the classical approximation by piecewise constant controls. In the first two cases our results are new, and in the last two cases the results are obtained by a new method which we develop here.

SPACE-TIME PERIODIC SOLUTIONS AND LONG-TIME BEHAVIOR OF SOLUTIONS TO QUASI-LINEAR PARABOLIC EQUATIONS ∗

This paper considers (i) the existence of space-time periodic solutions of quasi-linear parabolic equations and (ii) the convergence, as

Error Bounds for Monotone Approximation Schemes for Hamilton-Jacobi-Bellman Equations

t\to\infty

The Dirichlet Problem for Semilinear Second-Order Degenerate Elliptic Equations and Applications to Stochastic Exit Time Control Problems

, of space periodic solutions of the initial value problem of quasi-linear parabolic equations to the space-time periodic solutions.

Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations

We obtain error bounds for monotone approximation schemes of Hamilton--Jacobi--Bellman equations. These bounds improve previous results of Krylov and the authors. The key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate, (almost) smooth supersolutions for the Hamilton--Jacobi--Bellman equation.

Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions

The aim of this article is to study the Dirichlet problem for second-order semilinear degenerate elliptic PDEs and the connections of these problems with stochastic exit time control problems.

Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result

This paper is concerned with the Holder regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth conditions on the equation. These results are concerned with a large class of integro-differential operators whose singular measures depend on x and also a large class of equations, including Bellman-Isaacs Equations.