Optimal stopping under model ambiguity: A time‐consistent equilibrium approach

Abstract

An unconventional approach for optimal stopping under model ambiguity is introduced. Besides ambiguity itself, we take into account how ambiguity‐averse an agent is. This inclusion of ambiguity attitude, via an α‐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 the best one 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.

Optimal Equilibria for Multi-dimensional Time-inconsistent Stopping Problems

“…Note that (1.1) is not restrictive in the one-dimensional case (i.e. d = 1): any regular diffusion (in the sense of [27, Definition V.45.2]) readily fulfills (1.1), as recently observed in [16,Remark 3.1].…”

“…It can be roughly categorized into two branches. The first one is the fixed-point iterative approach introduced in Huang and Nguyen-Huu [14], and further developed in [15], [19], and [16]. The second branch adapts the stochastic control formulation in [10] to the stopping context; this includes Ebert, Wei, and Zhou [9] and Christensen and Lindensjö [6,7].…”

A Time-Inconsistent Dynkin Game: from Intra-personal to Inter-personal Equilibria

“…among the player's current and future selves) in the same spirit as [22, Remark 2.1. The above fixed-point definition of an intra-personal equilibrium was introduced in [17] and followed by [18,22,20], among others. Note that there is a slightly different formulation in [21]: If we follow [21], particularly (2.5) therein, Θ T i (S) in (2.5) needs to be modified as ΘT i (S) := {x ∈ X : J i (x, 0, ρ T ) ≥ J i (x, ρ + S , ρ T )}.…”

“…One is to extend the spike variation technique from stochastic control to optimal stopping, as carried out in [12,6,7]. The other path is the iterative approach developed in [17,18,20], which circumvents spike variations via a fixed-point perspective. Let us also mention the recent work [2] which builds a connection between different concepts of equilibria in these two paths.…”

Stability of Equilibria in Time-inconsistent Stopping Problems

“…There have been a lot of papers on equilibrium strategies for time-inconsistent control problems, and we refer to [16,9,4] and the references therein. The development for theory of time-inconsistent stopping is more recent, and we refer to [12,10,11,13,15,6,5,3,19,2,1,14]. Let us also mention the work [17] which analyzes a time-inconsistent Dynkin game, and [18] which considers a time-inconsistent controller-stopper problem.…”