Optimal stopping under ambiguity in continuous time

Cheng, Xue; Riedel, Frank

doi:10.1007/s11579-012-0081-6

Cited by 72 publications

(83 citation statements)

References 41 publications

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“…[26], [14], Appendix D of [20]) and has been applied to various problems stemming from mathematical finance, the most important example of which is the computation of the super hedging price of the American contingent claims [6,17,18,22]. Optimal stopping under Knightian uncertainty/nonlinear expectations/risk measures or the closely related controller-stopper-games have attracted a lot of attention in the recent years: [23,24,16,8,9,32,2,3,4,5,7,25]. In this literature, the set of probabilities is assumed to be dominated by a single probability or the controller is only allowed to influence the drift.…”

Section: {P(t ω)} (Tω)∈[0t ]×ωmentioning

confidence: 99%

On the Robust Optimal Stopping Problem

Bayraktar¹,

Yao²

2014

SIAM J. Control Optim.

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We study a robust optimal stopping problem with respect to a set P of mutually singular probabilities. This can be interpreted as a zero-sum controller-stopper game in which the stopper is trying to maximize its pay-off while an adverse player wants to minimize this payoff by choosing an evaluation criteria from P. We show that the upper Snell envelope Z of the reward process Y is a supermartingale with respect to an appropriately defined nonlinear expectation E , and Z is further an E −martingale up to the first time τ * when Z meets Y . Consequently, τ * is the optimal stopping time for the robust optimal stopping problem and the corresponding zero-sum game has a value. Although the result seems similar to the one obtained in the classical optimal stopping theory, the mutual singularity of probabilities and the game aspect of the problem give rise to major technical hurdles, which we circumvent using some new methods.

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Section: {P(t ω)} (Tω)∈[0t ]×ωmentioning

confidence: 99%

On the Robust Optimal Stopping Problem

Bayraktar¹,

Yao²

2014

SIAM J. Control Optim.

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“…Notice, Cheng and Riedel (2013) show that κ−ignorance can be applied in an infinite time-horizon. In particular they show that value functions taken under drift ambiguity in the infinite time horizon are nothing but the limits of value functions of finite time horizons T if T → ∞.…”

Section: The Modelmentioning

confidence: 98%

Ambiguity in a Real Option Game

Hellmann

Thijssen

2015

SSRN Journal

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in AbstractIn this paper we study a two-player investment game with a first mover advantage in continuous time with stochastic payoffs, driven by a geometric Brownian motion. One of the players is assumed to be ambiguous with maxmin preferences over a strongly rectangular set of priors. We develop a strategy and equilibrium concept allowing for ambiguity and show that equilibira can be preemptive (a player invests at a point where investment is Pareto dominated by waiting) or sequential (one player invests as if she were the exogenously appointed leader). Following the standard literature, the worst case prior for the ambiguous player if she is the second mover is obtained by setting the lowest possible trend in the set of priors. However, if the ambiguous player is the first mover, then the worst case prior can be given by either the lowest or the highest trend in the set of priors. This novel result shows that "worst case prior" in a setting with geometric Brownian motion and κ-ambiguity does not equate to "lowest trend".

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“…Riedel (2009) presents a unified and general theory of optimal stopping under multiple priors in discrete time and extends the theory to continuous time (da Rocha & Riedel, 2010;Cheng & Riedel, 2010). He developed a theory of optimal stopping with more than one joint distribution and with unknown distribution of the variables using and extending suitable results from the martingale theory (Williams, 1991).…”

Section: Optimal Stopping Problemmentioning

confidence: 99%

Decision Making with Dynamic Uncertain Events

Kalech

Reches

2015

jair

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When to make a decision is a key question in decision making problems characterized by uncertainty. In this paper we deal with decision making in environments where information arrives dynamically. We address the tradeoff between waiting and stopping strategies. On the one hand, waiting to obtain more information reduces uncertainty, but it comes with a cost. Stopping and making a decision based on an expected utility reduces the cost of waiting, but the decision is based on uncertain information. We propose an optimal algorithm and two approximation algorithms. We prove that one approximation is optimistic -waits at least as long as the optimal algorithm, while the other is pessimistic -stops not later than the optimal algorithm. We evaluate our algorithms theoretically and empirically and show that the quality of the decision in both approximations is near-optimal and much faster than the optimal algorithm. Also, we can conclude from the experiments that the cost function is a key factor to chose the most effective algorithm.

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Optimal stopping under ambiguity in continuous time

Cited by 72 publications

References 41 publications

On the Robust Optimal Stopping Problem

On the Robust Optimal Stopping Problem

Ambiguity in a Real Option Game

Decision Making with Dynamic Uncertain Events

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