Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions

We study a class of stochastic target games where one player tries to find a strategy such that the state process almost-surely reaches a given target, no matter which action is chosen by the opponent. Our main result is a geometric dynamic programming principle which allows us to characterize the value function as the viscosity solution of a non-linear partial differential equation. Because abstract mea-surable selection arguments cannot be used in this context, the main obstacle is the construction of measurable almost-optimal strategies. We propose a novel approach where smooth supersolutions are used to define almost-optimal strategies of Markovian type, similarly as in ver-ification arguments for classical solutions of Hamilton--Jacobi--Bellman equations. The smooth supersolutions are constructed by an exten-sion of Krylov's method of shaken coefficients. We apply our results to a problem of option pricing under model uncertainty with different interest rates for borrowing and lending.Comment: To appear in MO

A discrete Hughes' model for pedestrian flow on graphs

In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law while the minimization principle is described by a graph eikonal equation. We show that the model is well posed and we implement some numerical examples to demonstrate the validity of the proposed model