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

Abstract

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 ).

On time-inconsistent stopping problems and mixed strategy stopping times

“…To clarify the notion of pure and mixed strategy stopping times, we here formulate a simple example in discrete time, in line with the definitions of [1] (cf. the definition of time-homogeneous randomized stopping time in [1,Section 2]). Suppose X is a discrete time Markov chain living on {1, 2, 3} and consider a variance problem, i.e.…”

“…Mixed equilibria for time-inconsistent stopping are also considered in [1] in which a mean-variance problem and a mean-standard deviation problem are studied in a discrete time Markovian setting. Pure Markov stopping times are, in analogy with the present paper, defined as entry times into sets in the state space.…”

“…The development of the literature on the game-theoretic approach to time-inconsistent stopping problems is in an earlier stage. Recent papers include [1,9,11,20,21,22,24]. Section 1.2 describes references studying particular time-inconsistent stopping problems, while Section 2.1 contains a further review focusing on the choice of definition for pure and mixed strategies and equilibria.…”

“…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].…”

“…The parameter γ corresponds to risk aversion. The game-theoretic approach to stopping problems of this type is studied in Section 4.2 and [1].…”