High-order filtered schemes for time-dependent second order HJB equations

¹ Laboratoire Jacques-Louis Lions, Université Paris-Diderot, 5 Rue Thomas Mann, 75205 Paris, Cedex 13, France
² Laboratoire UMA, Ensta ParisTech, Palaiseau, France
boka@math.univ-paris-diderot.fr
³ Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG, U.K.
athena.picarelli@maths.ox.ac.uk, christoph.reisinger@maths.ox.ac.uk

Received: 15 November 2016
Revised: 27 May 2017
Accepted: 21 August 2017

Abstract

In this paper, we present and analyse a class of “filtered” numerical schemes for second order Hamilton–Jacobi–Bellman (HJB) equations. Our approach follows the ideas recently introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Ampère partial differential equation, SIAM J. Numer. Anal. 51 (2013) 423–444, and more recently applied by other authors to stationary or time-dependent first order Hamilton–Jacobi equations. For high order approximation schemes (where “high” stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. The work introduces a suitable local modification of these schemes by “filtering” them with a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests from mathematical finance, focussing also on the use of backward differencing formulae for constructing the high order schemes.