A Probabilistic numerical method for fully nonlinear parabolic pdes

Arash Fahim, Nizar Touzi, Xavier Warin

Research output: Contribution to journalArticlepeer-review


We consider the probabilistic numerical scheme for fully nonlinear partial differential equations suggested in [Comm. Pure Appl. Math. 60 (2007) 1081-1110] and show that it can be introduced naturally as a combination of Monte Carlo and finite difference schemes without appealing to the theory of backward stochastic differential equations. Our first main result proüides the conüergence of the discrete-time approximation and deriües a bound on the discretization error in terms of the time step. An explicit implementable scheme requires the approximation of the conditional expectation operators inüolüed in the discretization. This induces a further Monte Carlo error. Our second main result is to proüe the conüergence of the latter approximation scheme and to deriüe an upper bound on the approximation error. Numerical experiments are performed for the approximation of the solution of the mean curüature flow equation in dimensions two and three, and for two- and fiüedimensional (plus time) fully nonlinear Hamilton-Jacobi-Bellman equations arising in the theory of portfolio optimization in financial mathematics.

Original languageEnglish (US)
Pages (from-to)1322-1364
Number of pages43
JournalAnnals of Applied Probability
Issue number4
StatePublished - Aug 2011


  • Monotone schemes
  • Monte Carlo approximation
  • Second order backward stochastic differential equations
  • Viscosity solutions

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


Dive into the research topics of 'A Probabilistic numerical method for fully nonlinear parabolic pdes'. Together they form a unique fingerprint.

Cite this