Branching diffusion representation of semilinear PDEs and Monte Carlo approximation
Henry-Labordère, Pierre; Oudjane, Nadia; Tan, Xiaolu; Touzi, Nizar; Warin, Xavier (2019), Branching diffusion representation of semilinear PDEs and Monte Carlo approximation, Annales de l'Institut Henri Poincaré, Probabilités et statistiques, 55, 1, p. 184-210. 10.1214/17-AIHP880
TypeArticle accepté pour publication ou publié
Journal nameAnnales de l'Institut Henri Poincaré, Probabilités et statistiques
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Abstract (EN)We provide a representation result of parabolic semi-linear PD-Es, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod , Watanabe  and McKean , 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 automatic differentiation technique as in , based on the Malliavin integration by parts, which allows to account for the nonlin-earities 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.
Subjects / KeywordsSemilinear PDEs; branching processes; Monte-Carlo methods
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An Explicit Martingale Version of the One-dimensional Brenier's Theorem with Full Marginals Constraint Henry-Labordère, Pierre; Tan, Xiaolu; Touzi, Nizar (2016) Article accepté pour publication ou publié