An obstacle problem for Tug-of-War games

Abstract

We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tugof-war. This is much like the case in American options where investors can exercise the option at any time up to expiry and accept a payoff equal to the intrinsic

Game theoretical methods in PDEs

“…In the case p = 1 the game is naturally related to the mean curvature flow [8] and functions of least gradient. Other extensions include the obstacle problems [15], finite difference schemes [2], equations with right hand side f = 0, mixed boundary data [1,3] and parabolic equations [9,14].…”

Kaplan's penalty operator in approximation of a diffusion-absorption problem with a one-sided constraint

“…Also u may stand for deformation in equilibrium problems in hyper-elasticity. In general, the equations of the form −∆ p u + α(u) = f (with a nonlinear function α = α(u)) arise in mathematical modelling in rheology, glaciology, radiation of heat, and plastic moulding; in description of Brownian motion and even in game theory (see mathematical tug-of-war games in [8,17,20]).…”

Discrete approximations to the double-obstacle problem and optimal stopping of tug-of-war games

“…Thus, an appropriate "mixture"of the two processes (via the parameters α and β) yields p-harmonic functions in the limit as the discrete step-size → 0. The single obstacle problem for ∆ ∞ has been studied, from this point of view, in [10]. The case p ∈ [2, ∞), still in presence of the single obstacle, has been derived in [6].…”