A deterministic-control-based approach to fully nonlinear parabolic and elliptic equations

Robert V. Kohn, Sylvia Serfaty

Research output: Contribution to journalArticle

Abstract

We show that a broad class of fully nonlinear, second-order parabolic or elliptic PDEs can be realized as the Hamilton-Jacobi-Bellman equations of deterministic two-person games. More precisely: given the PDE, we identify a deterministic, discrete-time, two-person game whose value function converges in the continuous-time limit to the viscosity solution of the desired equation. Our game is, roughly speaking, a deterministic analogue of the stochastic representation recently introduced by Cheridito, Soner, Touzi, and Victoir. In the parabolic setting with no u-dependence, it amounts to a semidiscrete numerical scheme whose timestep is a min-max. Our result is interesting, because the usual control-based interpretations of second-order PDEs involve stochastic rather than deterministic control.

Original languageEnglish (US)
Pages (from-to)1298-1350
Number of pages53
JournalCommunications on Pure and Applied Mathematics
Volume63
Issue number10
DOIs
StatePublished - Oct 2010

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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