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On optimal stopping of multidimensional diffusions

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  • Christensen, Sören
  • Crocce, Fabián
  • Mordecki, Ernesto
  • Salminen, Paavo

Abstract

This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the d-dimensional Wiener process. We first obtain some verification theorems for diffusions, based on the Green kernel representation of the value function. Specializing to the multidimensional Wiener process, we apply the Martin boundary theory to obtain a set of tractable integral equations involving only harmonic functions that characterize the stopping region of the problem in the bounded case. The approach is illustrated through the optimal stopping problem of a d-dimensional Wiener process with a positive definite quadratic form reward function.

Suggested Citation

  • Christensen, Sören & Crocce, Fabián & Mordecki, Ernesto & Salminen, Paavo, 2019. "On optimal stopping of multidimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2561-2581.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:7:p:2561-2581
    DOI: 10.1016/j.spa.2018.07.014
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    References listed on IDEAS

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    Cited by:

    1. Christensen, Sören & Irle, Albrecht, 2020. "The monotone case approach for the solution of certain multidimensional optimal stopping problems," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1972-1993.
    2. Felix Dammann & Giorgio Ferrari, 2021. "On an Irreversible Investment Problem with Two-Factor Uncertainty," Papers 2103.08258, arXiv.org, revised Jul 2021.
    3. Luis H. R. Alvarez E. & Soren Christensen, 2019. "A Class of Solvable Multidimensional Stopping Problems in the Presence of Knightian Uncertainty," Papers 1907.04046, arXiv.org.
    4. Christensen, Sören & Fischer, Simon, 2023. "A new integral equation for Brownian stopping problems with finite time horizon," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 338-360.
    5. Dammann, Felix & Ferrari, Giorgio, 2021. "On an Irreversible Investment Problem with Two-Factor Uncertainty," Center for Mathematical Economics Working Papers 646, Center for Mathematical Economics, Bielefeld University.
    6. Nunes, Cláudia & Oliveira, Carlos & Pimentel, Rita, 2021. "Quasi-analytical solution of an investment problem with decreasing investment cost due to technological innovations," Journal of Economic Dynamics and Control, Elsevier, vol. 130(C).
    7. Cheng Cai & Tiziano De Angelis, 2021. "A change of variable formula with applications to multi-dimensional optimal stopping problems," Papers 2104.05835, arXiv.org, revised Jul 2023.
    8. Lempa, Jukka & Mordecki, Ernesto & Salminen, Paavo, 2024. "Diffusion spiders: Green kernel, excessive functions and optimal stopping," Stochastic Processes and their Applications, Elsevier, vol. 167(C).
    9. Cai, Cheng & De Angelis, Tiziano, 2023. "A change of variable formula with applications to multi-dimensional optimal stopping problems," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 33-61.

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