On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems

Christensen, Sören; Lindensjö, Kristoffer

doi:10.1137/17m1153029

Cited by 52 publications

(85 citation statements)

References 45 publications

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“…The equilibrium definitions of e.g. [1,24] are, as we have mentioned, discrete time versions of the equilibrium in the present paper and [9], but they are also discrete time versions of the equilibrium in e.g. [20], and the discrete time equilibrium definition seems unanimous.…”

Section: Motivation and Discussion Of The Definitions Of Mixed Stratementioning

confidence: 57%

“…Remark 2.7. The equilibrium condition (2.2) is in line with the one in [9] and inspired by time-inconsistent stopping problems in financial economics, see e.g. [12].…”

Section: Motivation and Discussion Of The Definitions Of Mixed Stratementioning

confidence: 82%

“…Section 4 contains further references to papers studying mean-variance and variance problems. For short surveys of time-inconsistent stopping problems we also refer to [1,9,34]. Recent papers on time-inconsistent stopping problems and the dynamic optimality and pre-commitment approaches include [31,34].…”

Section: Previous Literaturementioning

confidence: 99%

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On time-inconsistent stopping problems and mixed strategy stopping times

Christensen

Lindensjö

2020

Stochastic Processes and their Applications

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A game-theoretic framework for time-inconsistent stopping problems where the time-inconsistency is due to the consideration of a non-linear function of an expected reward is developed. A class of mixed strategy stopping times that allows the agents in the game to jointly choose the intensity function of a Cox process is introduced and motivated. A subgame perfect Nash equilibrium is defined. The equilibrium is characterized and other necessary and sufficient equilibrium conditions including a smooth fit result are proved. Existence and uniqueness are investigated. A mean-variance and a variance problem are studied. The state process is a general one-dimensional Itô diffusion.

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Section: Motivation and Discussion Of The Definitions Of Mixed Stratementioning

confidence: 57%

“…Remark 2.7. The equilibrium condition (2.2) is in line with the one in [9] and inspired by time-inconsistent stopping problems in financial economics, see e.g. [12].…”

Section: Motivation and Discussion Of The Definitions Of Mixed Stratementioning

confidence: 82%

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On time-inconsistent stopping problems and mixed strategy stopping times

Christensen

Lindensjö

2020

Stochastic Processes and their Applications

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“…Furthermore, we only consider the pure Markov stopping times commonly used in game theory; see e.g. [4] and [14]. Definition 2.1.…”

Section: Equilibrium Stopping Timesmentioning

confidence: 99%

“…Also see [13] which is a continuous-time extension of [10]. Let us also mention the recent work of [4], where the authors consider the equilibrium stopping strategies under the definition associated with first order criteria. They point out that the mean-variance problem is out of the scope of their approach.…”

Section: Introductionmentioning

confidence: 99%

Time Consistent Stopping for the Mean-Standard Deviation Problem---The Discrete Time Case

Bayraktar¹,

Zhang²,

Zhou³

2019

SIAM J. Finan. Math.

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Inspired by Strotz's consistent planning strategy, we formulate the infinite horizon mean-variance stopping problem as a subgame perfect Nash equilibrium in order to determine time consistent strategies with no regret. Equilibria among stopping times or randomized stopping times may not exist. This motivates us to consider the notion of liquidation strategies, which allows the stopping right to be divisible. We then argue that the mean-standard deviation variant of this problem makes more sense for this type of strategies in terms of time consistency. It turns out that an equilibrium liquidation strategy always exists. We then analyze whether optimal equilibrium liquidation strategies exist and whether they are unique and observe that neither may hold.Example 4.2. Consider the second example in the proof of Proposition 2.5. For mean-standard deviation problem, sup τ ∈T K p (11, τ ) = K p (11, τ ) = K p (11, τ ) = 11.1515, where τ = inf{n ≥ 0 : X 1 n ∈ {0, 18}}, τ = inf{n ≥ 0 : X 1 n ∈ {0, 17, 18}}, and sup τ ∈T K p (17, τ ) = K p (17, τ ) = 17.0022,where τ = inf{n ≥ 0 : X 1 n ∈ {0, 11, 18}}. There is an equilibrium liquidation strategy η(11) = a 3 ∈ (0, 1), η(17) = 0. We haveFor mean-variance problem, sup τ ∈T J p (11, τ ) = J p (11, τ ) = 11, sup τ ∈T J p (17, τ ) = J p (17, τ ) = 17,where τ = 0.The unique equilibrium liquidation strategy is η(11) = a ∈ (0, 1), η(17) = b ∈ (0, 1) where a ≈ 0.9312 and b ≈ 0.7629. Details on finding the equilibrium liquidation strategy can be found in Appendix A.5. We have J l (11, η) = 10.8365 < sup τ ∈T J p (11, τ ), J l (17, η) = 16.9981 < sup τ ∈T J p (17, τ ).

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