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Abstract We consider three equilibrium concepts proposed in the literature for time‐inconsistent stopping problems, including mild equilibria (introduced in Huang and Nguyen‐Huu (2018)), weak equilibria (introduced in Christensen and Lindensjö (2018)), and strong equilibria (introduced in Bayraktar et al. (2021)). The discount function is assumed to be log subadditive and the underlying process is one‐dimensional diffusion. We first provide necessary and sufficient conditions for the characterization of weak equilibria. The smooth‐fit condition is obtained as a by‐product. Next, based on the characterization of weak equilibria, we show that an optimal mild equilibrium is also weak. Then we provide conditions under which a weak equilibrium is strong. We further show that an optimal mild equilibrium is also strong under a certain condition. Finally, we provide several examples including one showing a weak equilibrium may not be strong, and another one showing a strong equilibrium may not be optimal mild.
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Optimal equilibria for time‐inconsistent stopping problems in continuous time
Abstract For an infinite‐horizon continuous‐time optimal stopping problem under nonexponential discounting, we look for anoptimal equilibrium, which generates larger values than any other equilibrium does on theentirestate space. When the discount function is log subadditive and the state process is one‐dimensional, an optimal equilibrium is constructed in a specific form, under appropriate regularity and integrability conditions. Although there may exist other optimal equilibria, we show that they can differ from the constructed one in very limited ways. This leads to a sufficient condition for the uniqueness of optimal equilibria, up to some closedness condition. To illustrate our theoretic results, a comprehensive analysis is carried out for three specific stopping problems, concerning asset liquidation and real options valuation. For each one of them, an optimal equilibrium is characterized through an explicit formula.
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- Award ID(s):
- 1715439
- PAR ID:
- 10123563
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Mathematical Finance
- Volume:
- 30
- Issue:
- 3
- ISSN:
- 0960-1627
- Page Range / eLocation ID:
- p. 1103-1134
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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