CIRM - Videos & books Library - Branching for PDEs

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Résumé : Branching methods have recently been developed to solve some PDEs. Starting from Mckean formulation, we give the initial branching method to solve the KPP equation. We then give a formulation to solve non linear equation with a non linearity polynomial in the value function u. The methodology is extended for general non linearities in the value function u. Then we develop the methodology to solve non linear equation with non linearities polynomial in u and Du with convergence results. At last we give some numerical schemes to solve the semi-linear case and even the full non linear case but currently without convergence results.

Codes MSC :
35R60 - PDEs with randomness, stochastic PDE
60H15 - Stochastic partial differential equations
60J80 - Branching processes (Galton-Watson, birth-and-death, etc.)

Ressources complémentaires :
http://smai.emath.fr/cemracs/cemracs17/Slides/warin.pdf
https://gitlab.com/stochastic-control/StOpt
https://www.fime-lab.org/

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https://videos.cirm-math.fr/2017-07-19_Warin.mp4

Informations sur la Rencontre

Nom de la rencontre : CEMRACS - Summer school: Numerical methods for stochastic models: control, uncertainty quantification, mean-field / CEMRACS - École d'été : Méthodes numériques pour équations stochastiques : contrôle, incertitude, champ moyen
Dates : 17/07/17 - 25/08/17
Année de la rencontre : 2017
URL Congrès : http://conferences.cirm-math.fr/1556.html

Données de citation

DOI : 10.24350/CIRM.V.19199103
Citer cette vidéo: (2017). Branching for PDEs. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19199103
URI : http://dx.doi.org/10.24350/CIRM.V.19199103

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Bibliographie

Bouchard, B., Tan, X., Warin, X., & Zou, Y. (2017). Numerical approximation of BSDEs using local polynomial drivers and branching processes. - https://arxiv.org/abs/1612.06790

Debrabant, K., & Jakobsen, E.R. (2013). Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Mathematics of Computation, 82(283), 433-1462 - https://doi.org/10.1090/S0025-5718-2012-02632-9

Doumbia, M., Oudjane, N., & Warin, X. (2016). Unbiased Monte Carlo estimate of stochastic differential equations expectations. - https://arxiv.org/abs/1601.03139

Fahim, A., Touzi, N., & Warin, X. (2011). A probabilistic numerical method for fully nonlinear parabolic PDEs. The Annals of Applied Probability, 21(4), 1322-1364 - http://dx.doi.org/10.1214/10-AAP723

Gevret, H., Lelong, J., & Warin, X. (2016). STochastic OPTimization library in C++. - https://hal.archives-ouvertes.fr/hal-01361291

Henry-Labordere, P., Tan, X., & Touzi, N. (2015). Unbiased simulation of stochastic differential equations. - https://arxiv.org/abs/1504.06107

Henry-Labordere, P. (2012). Counterparty risk valuation: a marked branching diffusion approach. Preprint - http://dx.doi.org/10.2139/ssrn.1995503

McKean, H.P. (1975). Application of Brownian motion to the equation of Kolmogorov-Petrovskii- Piskunov. Communications on Pure and Applied Mathematics, 28, 323-331 - http://dx.doi.org/10.1002/cpa.3160280302

Warin, X. (2017). Variations on branching methods for non linear PDEs. - https://arxiv.org/abs/1701.07660

Warin, X. (2013). Some non monotone schemes for time dependent Hamilton-Jacobi-Bellman equations in stochastic control. - https://arxiv.org/abs/1310.6121

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