Abstract.

This paper considers an optimal stopping problem with weighted discounting, and the state process is modeled by a general exponential Lévy process. Due to the time inconsistency, we provide a new martingale method based on a verification theorem for the equilibrium stopping strategies. As an application, we generalize an investment problem with non-exponential discounting studied by Grenadier and Wang (J. Financ. Econom., 84 (2007), pp. 2–39) and Ebert, Wei, and Zhou (J. Econom. Theory, 189 (2020), 105089) to Lévy models. Closed-form equilibrium stopping strategies are derived, which are closely related to the running maximum of the state process. The impacts of discounting preferences on the equilibrium stopping strategies are examined analytically.

Keywords

  1. weighted discounting
  2. optimal stopping
  3. time inconsistency
  4. Lévy processes

MSC codes

  1. 60G40
  2. 60G51
  3. 91B06

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Acknowledgment.

The authors would like to thank two anonymous referees for helpful comments and suggestions.

References

E. Bayraktar, J. Zhang, and Z. Zhou (2019), Time consistent stopping for the mean-standard deviation problem—the discrete time case, SIAM J. Financial Math., 10, pp. 667–697.
J. Bertoin (1998), Lévy Processes, Cambridge Tracts in Math. 121, Cambridge University Press, Cambridge.
K. Bichteler (2002), Stochastic Integration with Jumps, Encyclopedia Math. Appl., Cambridge University Press, Cambridge.
T. Björk, M. Khapko, and A. Murgoci (2017), On time-inconsistent stochastic control in continuous time, Finance Stoch., 21, pp. 331–360.
T. Bjork and A. Murgoci (2010), A General Theory of Markovian Time Inconsistent Stochastic Control Problems, preprint, SSRN 1694759.
S. Boyarchenko and S. Levendorskii (2007), Irreversible Decisions Under Uncertainty: Optimal Stopping Made Easy, Stud. Econom. Theory 27, Springer, Berlin.
M. J. Brennan and E. S. Schwartz (1985), Evaluating natural resource investments, J. Bus., 58, pp. 135–157.
S. Christensen and K. Lindensjö (2018), On finding equilibrium stopping times for time-inconsistent Markovian problems, SIAM J. Control Optim., 56, pp. 4228–4255.
R. Cont and P. Tankov (2003), Financial Modelling with Jump Processes, Chapman and Hall/CRC, Boca Raton, FL.
L. Döring and M. Savov (2011), (Non)differentiability and asymptotics for potential densities of subordinators, Electron. J. Probab., 16, pp. 470–503.
S. Ebert, W. Wei, and X. Y. Zhou (2020), Weighted discounting—on group diversity, time-inconsistency, and consequences for investment, J. Econom. Theory, 189, 105089.
S. R. Grenadier and N. Wang (2007), Investment under uncertainty and time-inconsistent preferences, J. Financ. Econom., 84, pp. 2–39.
C. M. Harvey (1986), Value functions for infinite-period planning, Manag. Sci., 32, pp. 1123–1139.
A. Hsiaw (2013), Goal-setting and self-control, J. Econom. Theory, 148, pp. 601–626.
Y.-J. Huang and A. Nguyen-Huu (2018), Time-consistent stopping under decreasing impatience, Finance Stoch., 22, pp. 69–95.
Y.-J. Huang, A. Nguyen-Huu, and X. Y. Zhou (2020), General stopping behaviors of naïve and noncommitted sophisticated agents, with application to probability distortion, Math. Finance, 30, pp. 310–340.
S. G. Kou and H. Wang (2003), First passage times of a jump diffusion process, Adv. Appl. Probab., 35, pp. 504–531.
A. Kuznetsov, A. E. Kyprianou, and J. C. Pardo (2012a), Meromorphic Lévy processes and their fluctuation identities, Ann. Appl. Probab., 22, pp. 1101–1135.
A. Kuznetsov, A. E. Kyprianou, and V. Rivero (2012b), The theory of scale functions for spectrally negative Lévy processes, in Lévy Matters II, Springer, Heidelberg, Germany, pp. 97–186.
A. E. Kyprianou (2014), Fluctuations of Lévy Processes with Applications: Introductory Lectures, Springer, Heidelberg, Germany.
G. Loewenstein and D. Prelec (1992), Anomalies in intertemporal choice: Evidence and an interpretation, Quart. J. Econom., 107, pp. 573–597.
D. M. López, J. L. Pérez, and K. Yamazaki (2021), Effects of positive jumps of assets on endogenous bankruptcy and optimal capital structure: Continuous- and periodic-observation models, SIAM J. Financial Math., 12, pp. 1112–1149.
R. McDonald and D. Siegel (1986), The value of waiting to invest, Quart. J. Econom., 101, pp. 707–727.
E. Mordecki (2002a), Optimal stopping and perpetual options for Lévy processes, Finance Stoch., 6, pp. 473–493.
E. Mordecki (2002b), The distribution of the maximum of a Lévy process with positive jumps of phase-type, Theory Stoch. Process., 8, pp. 309–316.
T. O’Donoghue and M. Rabin (1999), Doing it now or later, Amer. Econom. Rev., 89, pp. 103–124.
Z. Palmowski, J. L. Pérez, B. A. Surya, and K. Yamazaki (2020), The Leland-Toft optimal capital structure model under Poisson observations, Finance Stoch., 24, pp. 1035–1082.
Z. Palmowski, J. L. Pérez, and K. Yamazaki (2021), Double continuation regions for American options under Poisson exercise opportunities, Math. Finance, 31, pp. 722–771.
G. Peskir and A. Shiryaev (2006), Optimal Stopping and Free-Boundary Problems, Birkhäuser Basel, Basel, Germany.
P. E. Protter (2005), Stochastic Integration and Differential Equations, Springer, Berlin.
F. P. Ramsey (1928), A mathematical theory of saving, Econom. J., 38, pp. 543–559.
K. Sato (1999), Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge.
K. S. Tan, W. Wei, and X. Y. Zhou (2021), Failure of smooth pasting principle and nonexistence of equilibrium stopping rules under time-inconsistency, SIAM J. Control Optim., 59, pp. 4136–4154.
M. L. Weitzman (2001), Gamma discounting, Amer. Econom. Rev., 91, pp. 260–271.

Information & Authors

Information

Published In

cover image SIAM Journal on Financial Mathematics
SIAM Journal on Financial Mathematics
Pages: 777 - 811
ISSN (online): 1945-497X

History

Submitted: 3 August 2022
Accepted: 29 March 2023
Published online: 13 July 2023

Keywords

  1. weighted discounting
  2. optimal stopping
  3. time inconsistency
  4. Lévy processes

MSC codes

  1. 60G40
  2. 60G51
  3. 91B06

Authors

Affiliations

David Landriault
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada.
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada.
José M. Pedraza Contact the author
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada.

Funding Information

Funding: This paper is supported by the Society of Actuaries (SOA) Centers of Actuarial Excellence Grants Program. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the SOA. Support from grants from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by the first and second authors (grants 341316 and 04338, respectively). Support from the Canada Research Chair Program is gratefully acknowledged by the first author. The second author acknowledges the support from the National Natural Science Foundation of China (12271171).

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