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Semimartingales on rays, Walsh diffusions, and related problems of control and stopping

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  • Karatzas, Ioannis
  • Yan, Minghan

Abstract

We introduce a class of continuous planar processes, called “semimartingales on rays”, and develop for them a change-of-variable formula involving quite general classes of test functions. Special cases of such processes are diffusions which choose, once at the origin, the rays for their subsequent voyage according to a fixed probability measure in the manner of Walsh (1978). We develop existence and uniqueness results for these “Walsh diffusions”, study their asymptotic behavior, and develop tests for explosions in finite time. We use these results to find an optimal strategy, in a problem of stochastic control with discretionary stopping involving Walsh diffusions.

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  • Karatzas, Ioannis & Yan, Minghan, 2019. "Semimartingales on rays, Walsh diffusions, and related problems of control and stopping," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 1921-1963.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:6:p:1921-1963
    DOI: 10.1016/j.spa.2018.06.012
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    References listed on IDEAS

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    1. Victor C. Pestien & William D. Sudderth, 1985. "Continuous-Time Red and Black: How to Control a Diffusion to a Goal," Mathematics of Operations Research, INFORMS, vol. 10(4), pages 599-611, November.
    2. Hajri, Hatem & Touhami, Wajdi, 2014. "Itô’s formula for Walsh’s Brownian motion and applications," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 48-53.
    3. Karatzas, Ioannis & Ocone, Daniel, 2002. "A leavable bounded-velocity stochastic control problem," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 31-51, May.
    4. 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.
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    Cited by:

    1. Bayraktar, Erhan & Zhang, Xin, 2021. "Embedding of Walsh Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 1-28.
    2. Angelos Dassios & Junyi Zhang, 2022. "First Hitting Time of Brownian Motion on Simple Graph with Skew Semiaxes," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1805-1831, September.
    3. 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).

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