TY - JOUR

T1 - Branching diffusion representation of semilinear PDEs and Monte Carlo approximation

AU - Henry-Labordère, Pierre

AU - Oudjane, Nadia

AU - Tan, Xiaolu

AU - Touzi, Nizar

AU - Warin, Xavier

N1 - Publisher Copyright:
© Association des Publications de l’Institut Henri Poincaré, 2019.

PY - 2019/2

Y1 - 2019/2

N2 - 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.

AB - 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.

KW - Branching processes

KW - Monte-Carlo methods

KW - Semilinear PDEs

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UR - http://www.scopus.com/inward/citedby.url?scp=85060914554&partnerID=8YFLogxK

U2 - 10.1214/17-AIHP880

DO - 10.1214/17-AIHP880

M3 - Article

AN - SCOPUS:85060914554

SN - 0246-0203

VL - 55

SP - 184

EP - 210

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 1

ER -