Optimal stopping under adverse nonlinear expectation and related games

By Marcel Nutz, Jianfeng Zhang. 2012

71 citations recorded by the source corpus.

We study the existence of optimal actions in a zero-sum game $\inf_{\tau}\sup_PE^P[X_{\tau}]$ between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem $\inf_{\tau}\mathcal{E}(X_{\tau})$ for a class of sublinear expectations $\mathcal{E}(\cdot)$ 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.

Subfields: Control and Systems Engineering, Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/e86c0f7b5fe8e59ee7df97cdcc16fc16043a9205

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In Control and Systems Engineering

Marcel Nutz

In Economics and Econometrics

In Finance

Citations in the corpus

ROBUST FUNDAMENTAL THEOREM FOR CONTINUOUS PROCESSES (2014, background): “[5, 16, 20, 30, 36, 38, 39, 41, 44, 45], among others, or when all paths are considered possible for the stock and options are also traded; see, e.”
Measurability of semimartingale characteristics with respect to the probability law (2013, background): “Other examples where sets of semimartingale laws play a role are path-dependent PDEs [10], robust superhedging as in [16, 25] or nonlinear optimal stopping problems as in [21], to name but a few.”
Stochastic control for a class of nonlinear kernels and applications (2015, background): “2) with r = 0, and where the set of measures considered satisfy the predictable martingale representation property (that is the financial market is complete under any of the measures considered) this superhedging price has been thoroughly studied in the recent literature, see among others [2, 6, 25,”