×

Time consistent stopping for the mean-standard deviation problem – the discrete time case. (English) Zbl 1427.91251

The authors formulate the infinite horizon mean-variance stopping problem as a subgame perfect Nash equilibrium in order to determine time-consistent strategies with no regret. They study time-consistent mean-standard deviation stopping problems in discrete time with an infinite time horizon. It is proved that while a Markov equilibrium in the class of pure or randomized stopping times may not exist in general, there always exists an equilibrium liquidation strategy. In addition, it is established that an optimal equilibrium in the sense of pointwise dominance may not exist, and may not be unique if it exists. The existence of a Pareto optimal equilibrium is also established.
The main novelty of the paper is the determination of the right type of “mixed” equilibrium strategies and the appropriate modification of the criteria to make sense of the consistent planning problem. In particular, it is shown that the obvious choice of mixing, i.e., a randomized stopping strategy, does not work because when an equilibrium in this class exists it coincides with the pure equilibrium strategy. The authors propose to use the mean-standard deviation criterion and think it is more meaningful in the context of consistent planning. One reason is that the mean and standard deviation are associated with the same unit. Moreover, the scaling property of this new objective function plays along well with liquidation strategies. They compare equilibrium liquidation strategies with statically optimal ones.

MSC:

91G10 Portfolio theory
60G40 Stopping times; optimal stopping problems; gambling theory
91A55 Games of timing
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] E. Bayraktar and Z. Zhou, Arbitrage, hedging and utility maximization using semi-static trading strategies with american options, Ann. Appl. Probab., 26 (2016), pp. 3531-3558, https://doi.org/10.1214/16-AAP1184. · Zbl 1357.91046
[2] E. Bayraktar and Z. Zhou, No-arbitrage and hedging with liquid American options, Math. Oper. Res., 44 (2019), pp. 377-766, https://doi.org/10.1287/moor.2018.0932. · Zbl 1433.91169
[3] T. Björk and A. Murgoci, A theory of Markovian time-inconsistent stochastic control in discrete time, Finance Stoch., 18 (2014), pp. 545-592, https://doi.org/10.1007/s00780-014-0234-y. · Zbl 1297.49038
[4] S. Christensen and K. Lindensjö, On finding equilibrium stopping times for time-inconsistent Markovian problems, SIAM J. Control Optim., 56 (2018), pp. 4228-4255, https://doi.org/10.1137/17M1153029. · Zbl 1429.60043
[5] S. Christensen and K. Lindensjö, On Time-Inconsistent Stopping Problems and Mixed Strategy Stopping Times, https://arxiv.org/abs/1804.07018v2 (2018). · Zbl 1435.60029
[6] S. Ebert and P. Strack, Until the bitter end: On prospect theory in a dynamic context, Amer. Econom. Rev., 105 (2015), pp. 1618-33, https://doi.org/10.1257/aer.20130896.
[7] G. A. Edgar, A. Millet, and L. Sucheston, On compactness and optimality of stopping times, in Martingale Theory in Harmonic Analysis and Banach Spaces, Cleveland, Ohio, 1981, Lecture Notes in Math 939, Springer, Berlin, 1982, pp. 36-61. · Zbl 0496.60039
[8] K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Natl. Acad. Sci. USA, 38 (1952), pp. 121-126. · Zbl 0047.35103
[9] S. R. Grenadier and N. Wang, Investment under uncertainty and time-inconsistent preferences, J. Financ. Econom., 84 (2007), pp. 2-39, https://ideas.repec.org/a/eee/jfinec/v84y2007i1p2-39.html.
[10] Y.-J. Huang and Z. Zhou, The optimal equilibrium for time-inconsistent stopping problems–the discrete-time case, SIAM J. Control Optim., 57 (2019), pp. 590-609, https://doi.org/10.1137/17M1139187. · Zbl 1408.49019
[11] Y.-J. Huang and A. Nguyen-Huu, Time-consistent stopping under decreasing impatience, Finance Stoch., 22 (2018), pp. 69-95, https://doi.org/10.1007/s00780-017-0350-6. · Zbl 1391.60086
[12] Y.-J. Huang, A. Nguyen-Huu, and X. Y. Zhou, General stopping behaviors of naive and non-committed sophisticated agents, with application to probability distortion, Math. Finance, to appear. · Zbl 1508.91603
[13] Y.-J. Huang and Z. Zhou, Optimal equilibria for time-inconsistent stopping problems in continuous time, Math. Finance, to appear. · Zbl 1508.91627
[14] E. Maskin and J. Tirole, Markov perfect equilibrium: I. Observable actions, J. Econom. Theory, 100 (2001), pp. 191-219, https://doi.org/10.1006/jeth.2000.2785. · Zbl 1011.91022
[15] T. O’Donoghue and M. Rabin, Doing it now or later, Amer. Econom. Rev., 89 (1999), pp. 103-124, https://doi.org/10.1257/aer.89.1.103.
[16] J. L. Pedersen and G. Peskir, Optimal mean-variance selling strategies, Math. Finance Econ., 10 (2016), pp. 203-220, https://doi.org/10.1007/s11579-015-0156-2. · Zbl 1334.60066
[17] R. A. Pollak, Consistent planning, Rev. Econ. Stud., 35 (1968), pp. 201-208, http://www.jstor.org/stable/2296548.
[18] R. H. Strotz, Myopia and inconsistency in dynamic utility maximization, Rev. Econ. Stud., 23 (1955), pp. 165-180, https://EconPapers.repec.org/RePEc:oup:restud:v:23:y:1955:i:3:p:165-180.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.