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.
Mots clés : Monotone schemes, high-order schemes, backward difference formulae, viscosity solutions, second order Hamilton–Jacobi–Bellman equations
@article{M2AN_2018__52_1_69_0,
author = {Bokanowski, Olivier and Picarelli, Athena and Reisinger, Christoph},
title = {High-order filtered schemes for time-dependent second order {HJB} equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {69--97},
publisher = {EDP-Sciences},
volume = {52},
number = {1},
year = {2018},
doi = {10.1051/m2an/2017039},
zbl = {1395.65012},
mrnumber = {3808153},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2017039/}
}
TY - JOUR AU - Bokanowski, Olivier AU - Picarelli, Athena AU - Reisinger, Christoph TI - High-order filtered schemes for time-dependent second order HJB equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 69 EP - 97 VL - 52 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2017039/ DO - 10.1051/m2an/2017039 LA - en ID - M2AN_2018__52_1_69_0 ER -
%0 Journal Article %A Bokanowski, Olivier %A Picarelli, Athena %A Reisinger, Christoph %T High-order filtered schemes for time-dependent second order HJB equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 69-97 %V 52 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2017039/ %R 10.1051/m2an/2017039 %G en %F M2AN_2018__52_1_69_0
Bokanowski, Olivier; Picarelli, Athena; Reisinger, Christoph. High-order filtered schemes for time-dependent second order HJB equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 69-97. doi : 10.1051/m2an/2017039. http://archive.numdam.org/articles/10.1051/m2an/2017039/
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