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

Abstract

Abstract. 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.

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

“…This is a difficult problem that remained open for a long time before the works of Krylov [32,33,34] and Barles & Jakobsen [8,9,10]. The methods developed in these works involve the use of carefully chosen smooth approximations of the viscosity solution of the underlying equation.…”

“…The methods developed in these works involve the use of carefully chosen smooth approximations of the viscosity solution of the underlying equation. In some recent developments [9,10], Barles & Jakobsen used solutions of certain switching systems to generate suitable approximations of the viscosity solution of the Bellman equation associated with the optimal control of diffusion processes. In a future work we will adapt this approach to the nonlocal Bellman equation of controlled jump-diffusion processes, which is drawing a lot of interests these days due to its applications in mathematical finance (see for example [3], [2], [14], [15], [19] and the references therein).…”

“…In a future work we will adapt this approach to the nonlocal Bellman equation of controlled jump-diffusion processes, which is drawing a lot of interests these days due to its applications in mathematical finance (see for example [3], [2], [14], [15], [19] and the references therein). To derive error estimates like those in [9,10] for the nonlocal Bellman equation we need to have at our disposal a viscosity solution theory for switching systems of the type (1.1).…”

The Rate of Convergence of Finite-Difference Approximations for Bellman Equations with Lipschitz Coefficients

“…We refer to [2] and [3] and the references therein for discussion of what is achieved in this direction.…”

“…In contrast, until quite recently there were no results about the rate of convergence of finite-difference approximations for degenerate Bellman equations. The first result appeared only in 1997 for elliptic Bellman equations with constant "coefficients" (see [8]) and they were later extended to variable coefficients and parabolic equations in [2], [3], [9], and [10]. Surprisingly, as far as we know until now these are the only published result on the rate of convergence of finite-difference approximations even if Bellman equation becomes a linear second order degenerate equation.…”