Espen R. Jakobsen

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.

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

ON THE RATE OF CONVERGENCE OF APPROXIMATION SCHEMES FOR BELLMAN EQUATIONS ASSOCIATED WITH OPTIMAL STOPPING TIME PROBLEMS

We provide estimates on the rate of convergence for approximation schemes for Bellman equations associated with optimal stopping of controlled diffusion processes. These results extend (and slightly improve) the recent results by Barles & Jakobsen to the more difficult time-dependent case. The added difficulties are due to the presence of boundary conditions (initial conditions!) and the new structure of the equation which is now a parabolic variational inequality. The method presented is purely analytic and rather general and is based on earlier work by Krylov and Barles & Jakobsen. As applications we consider so-called control schemes based on the dynamic programming principle and finite difference methods (though not in the most general case). In the optimal stopping case these methods are similar to the Brennan & Schwartz scheme. A simple observation allows us to obtain the optimal rate 1/2 for the finite difference methods, and this is an improvement over previous results by Krylov and Barles & Jakobsen. Finally, we present an idea that allows us to improve all the above-mentioned results in the linear case. In particular, we are able to handle finite difference methods with variable diffusion coefficients without the reduction of order of convergence observed by Krylov in the nonlinear case.

Entropy solution theory for fractional degenerate convection-diffusion equations

Abstract We study a class of degenerate convection–diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection–diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Levy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.

Error estimates for approximate solutions to Bellman equations associated with controlled jump-diffusions

We derive error estimates for approximate (viscosity) solutions of Bellman equations associated to controlled jump-diffusion processes, which are fully nonlinear integro-partial differential equations. Two main results are obtained: (i) error bounds for a class of monotone approximation schemes, which under some assumptions includes finite difference schemes, and (ii) bounds on the error induced when the original Lévy measure is replaced by a finite measure with compact support, an approximation process that is commonly used when designing numerical schemes for integro-partial differential equations. Our proofs use and extend techniques introduced by Krylov and Barles-Jakobsen.

On the Convergence Rate of Operator Splitting for Hamilton--Jacobi Equations with Source Terms

We establish a rate of convergence for a semidiscrete operator splitting method applied to Hamilton--Jacobi equations with source terms. The method is based on sequentially solving a Hamilton--Jacobi equation and an ordinary differential equation. The Hamilton--Jacobi equation is solved exactly while the ordinary differential equation is solved exactly or by an explicit Euler method. We prove that the

The discontinuous Galerkin method for fractal conservation laws

L^{\infty}

The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

error associated with the operator splitting method is bounded by

On Neumann type problems for nonlocal equations set in a half space

\mathcal{O}(\Delta t)

Continuous Dependence Estimates for Nonlinear Fractional Convection-diffusion Equations

, where