Error bounds for monotone approximation schemes for parabolic 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

“…The first part of the above Lemma follows exactly in the same way as Lemma A.2 in [10]. Once we have the first part, then for the supersolution it says that at the point (t 0 , x 0 , y 0 ) we can ignore the termû i0 − M i0û , and then the second part follows as a consequence of Theorem 2.2 of [30].…”

“…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).…”

Robust Preferences and Robust Portfolio Choice

“…For numerical solutions of the HJB equations in this section one can use, e.g., a multigrid Howard algorithm as explained in AKIAN [1990] and KUSHNER and DUPUIS [2001]. For convergence results of such numerical schemes see KUSHNER and DUPUIS [2001], KRYLOV [2000], BARLES and JAKOBSEN [2005], and the references therein.…”