A Deterministic-Control-Based Approach to Fully Nonlinear Parabolic and Elliptic Equations

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. (C) 2010 Wiley Periodicals, Inc.