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

TL;DR

This paper develops and analyzes high-order filtered numerical schemes for second order Hamilton-Jacobi-Bellman equations, ensuring convergence and high accuracy for smooth solutions by combining monotone and high-order methods.

Contribution

It introduces a novel filtering technique that combines high-order schemes with monotone schemes to guarantee convergence for second order HJB equations.

Findings

Filtered schemes are proven to converge for smooth solutions.

Numerical tests demonstrate high-order accuracy in financial applications.

Backward difference formulas are effectively integrated into the schemes.

Abstract

In this paper, we present and analyse a class of "filtered" numerical schemes for second order Hamilton-Jacobi-Bellman equations. Our approach follows the ideas introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Amp\`ere partial differential equation, SIAM J. Numer. Anal., 51(1):423--444, 2013, 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 difference formulae (BDF) for constructing the high order schemes.