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.


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

MSC codes

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

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


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Information & Authors


Published In

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


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


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

MSC codes

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



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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