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The shape of the value function under Poisson optimal stopping

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  • Hobson, David

Abstract

In a classical problem for the stopping of a diffusion process (Xt)t≥0, where the goal is to maximise the expected discounted value of a function of the stopped process Ex[e−βτg(Xτ)], maximisation takes place over all stopping times τ. In a Poisson optimal stopping problem, stopping is restricted to event times of an independent Poisson process. In this article we consider whether the resulting value function Vθ(x)=supτ∈T(Tθ)Ex[e−βτg(Xτ)] (where the supremum is taken over stopping times taking values in the event times of an inhomogeneous Poisson process with rate θ=(θ(Xt))t≥0) inherits monotonicity and convexity properties from g. It turns out that monotonicity (respectively convexity) of Vθ in x depends on the monotonicity (respectively convexity) of the quantity θ(x)g(x)θ(x)+β rather than g. Our main technique is stochastic coupling.

Suggested Citation

  • Hobson, David, 2021. "The shape of the value function under Poisson optimal stopping," Stochastic Processes and their Applications, Elsevier, vol. 133(C), pages 229-246.
    • Handle: RePEc:eee:spapps:v:133:y:2021:i:c:p:229-246
    DOI: 10.1016/j.spa.2020.12.001
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    References listed on IDEAS

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    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    2. Lange, Rutger-Jan & Ralph, Daniel & Støre, Kristian, 2020. "Real-Option Valuation in Multiple Dimensions Using Poisson Optional Stopping Times," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 55(2), pages 653-677, March.
    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. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
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    Cited by:

    1. Alessandro Milazzo, 2024. "On the Monotonicity of the Stopping Boundary for Time-Inhomogeneous Optimal Stopping Problems," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 336-358, October.
    2. Alvarez E., Luis H.R. & Lempa, Jukka & Saarinen, Harto & Sillanpää, Wiljami, 2024. "Solutions for Poissonian stopping problems of linear diffusions via extremal processes," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    3. Takuji Arai & Masahiko Takenaka, 2022. "Constrained optimal stopping under a regime-switching model," Papers 2204.07914, arXiv.org.
    4. David Hobson & Gechun Liang & Edward Wang, 2021. "Callable convertible bonds under liquidity constraints and hybrid priorities," Papers 2111.02554, arXiv.org, revised Oct 2024.

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