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Abstract An unconventional approach for optimal stopping under model ambiguity is introduced. Besides ambiguity itself, we take into account howambiguity‐aversean agent is. This inclusion of ambiguity attitude, via an‐maxmin nonlinear expectation, renders the stopping problem time‐inconsistent. We look for subgame perfect equilibrium stopping policies, formulated as fixed points of an operator. For a one‐dimensional diffusion with drift and volatility uncertainty, we show that any initial stopping policy will converge to an equilibrium through a fixed‐point iteration. This allows us to capture much more diverse behavior, depending on an agent's ambiguity attitude, beyond the standard worst‐case (or best‐case) analysis. In a concrete example of real options valuation under model ambiguity, all equilibrium stopping policies, as well as thebestone among them, are fully characterized under appropriate conditions. It demonstrates explicitly the effect of ambiguity attitude on decision making: the more ambiguity‐averse, the more eager to stop—so as to withdraw from the uncertain environment. The main result hinges on a delicate analysis of continuous sample paths in the canonical space and the capacity theory. To resolve measurability issues, a generalized measurable projection theorem, new to the literature, is also established.
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General stopping behaviors of naïve and noncommitted sophisticated agents, with application to probability distortion
Abstract We consider the problem of stopping a diffusion process with a payoff functional that renders the problem time‐inconsistent. We study stopping decisions of naïve agents who reoptimize continuously in time, as well as equilibrium strategies of sophisticated agents who anticipate but lack control over their future selves' behaviors. When the state process is one dimensional and the payoff functional satisfies some regularity conditions, we prove that any equilibrium can be obtained as a fixed point of an operator. This operator represents strategic reasoning that takes the future selves' behaviors into account. We then apply the general results to the case when the agents distort probability and the diffusion process is a geometric Brownian motion. The problem is inherently time‐inconsistent as the level of distortion of a same event changes over time. We show how the strategic reasoning may turn a naïve agent into a sophisticated one. Moreover, we derive stopping strategies of the two types of agent for various parameter specifications of the problem, illustrating rich behaviors beyond the extreme ones such as “never‐stopping” or “never‐starting.”
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- Award ID(s):
- 1715439
- PAR ID:
- 10129701
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Mathematical Finance
- Volume:
- 30
- Issue:
- 1
- ISSN:
- 0960-1627
- Page Range / eLocation ID:
- p. 310-340
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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