Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Branching diffusion representation of semilinear PDEs and Monte Carlo approximation

Pierre Henry-Labordère, Nadia Oudjane, Xiaolu Tan, Nizar Touzi, and Xavier Warin

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Abstract

We provide a representation result of parabolic semi-linear PDEs, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod [Theory Probab. Appl. 9 (1964) 445–449], Watanabe [J. Math. Kyoto Univ. 4 (1965) 385–398] and McKean [Comm. Pure Appl. Math. 28 (1975) 323–331], by allowing for polynomial nonlinearity in the pair $(u,Du)$, where $u$ is the solution of the PDE with space gradient $Du$. Similar to the previous literature, our result requires a non-explosion condition which restrict to “small maturity” or “small nonlinearity” of the PDE. Our main ingredient is the Malliavin automatic differentiation technique as in [Ann. Appl. Probab. 27 (2017) 3305–3341], based on the Malliavin integration by parts, which allows to account for the nonlinearities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation. Indeed, this provides the first numerical method for high dimensional nonlinear PDEs with error estimate induced by the dimension-free central limit theorem. The complexity is also easily seen to be of the order of the squared dimension. The final section of this paper illustrates the efficiency of the algorithm by some high dimensional numerical experiments.

Résumé

Nous obtenons une représentation de la solution $u$ d’une EDP semi-linéaire parabolique, avec une nonlinéarité polynomiale, par le biais d’un processus de diffusion branchant. Nous étendons ainsi le résultat de représentation classique pour les équations KPP, introduit par Skorokhod [Theory Probab. Appl. 9 (1964) 445–449], Watanabe [J. Math. Kyoto Univ. 4 (1965) 385–398] et McKean [Comm. Pure Appl. Math. 28 (1975) 323–331], au cas d’une nonlinéarité polynomiale en $(u,Du)$. Bien évidemment, une telle non linéarité polynomiale requiert une condition de non explosion, qui est équivalent à une restriction de l’horizon, ou à une restriction de la taille de la perturbation nonlinéaire. L’ingrédient essentiel pour notre représentation est la technique de différentiation automatique de type Malliavin comme dans [Ann. Appl. Probab. 27 (2017) 3305–3341], qui permet de traiter la nonlinéarité en $Du$. Par conséquent, les particules de notre processus de branchement sont marquées par la nature de la nonlinéarité. Nous développons également une application importante de cette nouvelle représentation à l’approximation numérique de la solution d’une telle EDP par la méthode de Monte Carlo. Cette approximation est particulièrement intéressante en grande dimension du fait que l’estimation de l’erreur, induite par le théorème central limite, est indépendante de la dimension. La complexité de cet algorithme est de l’ordre du carré de la dimension. Dans le dernier paragraphe du papier, nous illustrons l’efficacité de cette méthode d’approximation numérique dans le cadre d’une équation de Burgers en dimension $d=20$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 184-210.

Dates
Received: 11 May 2016
Revised: 17 December 2017
Accepted: 22 December 2017
First available in Project Euclid: 18 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1547802399

Digital Object Identifier
doi:10.1214/17-AIHP880

Mathematical Reviews number (MathSciNet)
MR3901645

Zentralblatt MATH identifier
07039769

Subjects
Primary: 60J85: Applications of branching processes [See also 92Dxx] 91G60: Numerical methods (including Monte Carlo methods)

Keywords
Semilinear PDEs Branching processes Monte-Carlo methods

Citation

Henry-Labordère, Pierre; Oudjane, Nadia; Tan, Xiaolu; Touzi, Nizar; Warin, Xavier. Branching diffusion representation of semilinear PDEs and Monte Carlo approximation. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 184--210. doi:10.1214/17-AIHP880. https://projecteuclid.org/euclid.aihp/1547802399


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