Issue |
ESAIM: ProcS
Volume 65, 2019
CEMRACS 2017 - Numerical methods for stochastic models: control, uncertainty quantification, mean-field
|
|
---|---|---|
Page(s) | 309 - 329 | |
DOI | https://doi.org/10.1051/proc/201965309 | |
Published online | 02 April 2019 |
Numerical approximation of general Lipschitz BSDEs with branching processes
1
Université Paris-Dauphine, PSL University, CNRS, UMR [7534], CEREMADE, 75016 PARIS, FRANCE
e-mail: bouchard@ceremade.dauphine.fr
2
Université Paris-Dauphine, PSL University, CNRS, UMR [7534], CEREMADE, 75016 PARIS, FRANCE
e-mail: tan@ceremade.dauphine.fr
3
EDF R&D & FiME, Laboratoire de Finance des Marchés de l’Energie
e-mail: warin@edf.fr
We extend the branching process based numerical algorithm of Bouchard et al. [3], that is dedicated to semilinear PDEs (or BSDEs) with Lipschitz nonlinearity, to the case where the nonlinearity involves the gradient of the solution. As in [3], this requires a localization procedure that uses a priori estimates on the true solution, so as to ensure the well-posedness of the involved Picard iteration scheme, and the global convergence of the algorithm. When, the nonlinearity depends on the gradient, the later needs to be controlled as well. This is done by using a face-lifting procedure. Convergence of our algorithm is proved without any limitation on the time horizon. We also provide numerical simulations to illustrate the performance of the algorithm.
Résumé
Nous étendons la méthode des processus de branchement de Bouchard et al. [3] dans le cas des équations semi-linéaires à non linéarités Lipschitz et dépendant du gradient de la solution. Comme dans [3], une procédure de localisation utilisant des estimations a priori sur la vraie solution permet assurer que l’algorithme de résolution par itérations de Picard est bien posé et convergent. Quand la non linéarité dépend du gradient une technique de face-lifting est utilisé. La convergence de l’algorithme donné est démontré sans limite sur la maturité du problème. Des simulations numériques illustrent les performances de l’algorithme.
© EDP Sciences, SMAI 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.