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A new integral equation for Brownian stopping problems with finite time horizon

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  • Christensen, Sören
  • Fischer, Simon

Abstract

For classical finite time horizon stopping problems driven by a Brownian motion V(t,x)=supt≤τ≤0E(t,x)[g(τ,Wτ)],we derive a new class of Fredholm type integral equations for the stopping set. For a large class of discounted problems, we show by analytical arguments that the equation uniquely characterizes the stopping boundary of the problem. Regardless of uniqueness, we use the representation to rigorously find the limit behavior of the stopping boundary close to the terminal time. Interestingly, it turns out that the leading-order coefficient is universal for wide classes of problems. We also discuss how the representation can be used for numerical purposes.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:162:y:2023:i:c:p:338-360
    DOI: 10.1016/j.spa.2023.05.004
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    References listed on IDEAS

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    1. Stadje, Wolfgang, 1987. "An optimal stopping problem with finite horizon for sums of I.I.D. random variables," Stochastic Processes and their Applications, Elsevier, vol. 26, pages 107-121.
    2. Goran Peskir, 2005. "On The American Option Problem," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 169-181, January.
    3. Dayanik, Savas & Karatzas, Ioannis, 2003. "On the optimal stopping problem for one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 173-212, October.
    4. 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.
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    Cited by:

    1. Christensen, Sören & Fischer, Simon & Hallmann, Oskar, 2023. "Uniqueness of first passage time distributions via Fredholm integral equations," Statistics & Probability Letters, Elsevier, vol. 203(C).

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