Optimal stopping under adverse nonlinear expectation and related games

2012/12/31 by Marcel Nutz, Jianfeng Zhang · 3 citations
Mathematics · Economics, Econometrics and Finance · #math.OC #math.PR #q-fin.PR

paper · pdf · doi:10.1214/14-aap1054

published as Annals of Applied Probability 2015, Vol. 25, No. 5, 2503-2534 · Published at http://dx.doi.org/10.1214/14-AAP1054 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

arxiv created 2015/09/09 · arxiv updated 2015/09/10

Abstract

We study the existence of optimal actions in a zero-sum game inf_τsup_PE^P[X_τ] between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem inf_τE(X_τ) for a class of sublinear expectations E(⋅) such as the G-expectation. We show that the game has a value. Moreover, exploiting the theory of sublinear expectations, we define a nonlinear Snell envelope Y and prove that the first hitting time inf\t:Y_t=X_t\ is an optimal stopping time. The existence of a saddle point is shown under a compactness condition. Finally, the results are applied to the subhedging of American options under volatility uncertainty.

Cited by

• If Not Now, Then When? Model Risk in the Optimal Exercise of American Options
• Causal Second-Order States under Nondominated Martingale Laws: Restart-Stable Capacity Geometry and Dynamic Transfer
• The non-linear multiple stopping problem: between the discrete and the continuous time