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Diffusion spiders: Green kernel, excessive functions and optimal stopping

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  • Lempa, Jukka
  • Mordecki, Ernesto
  • Salminen, Paavo

Abstract

A diffusion spider is a strong Markov process with continuous paths taking values on a graph with one vertex and a finite number of edges (of infinite length). An example is Walsh’s Brownian spider where the process on each edge behaves as a Brownian motion. In this paper we calculate firstly the density of the resolvent kernel in terms of the characteristics of the underlying diffusion. Excessive functions are studied via the Martin boundary theory. A crucial result is an expression for the representing measure of a given excessive function. These results are used to solve optimal stopping problems for diffusion spiders.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:spapps:v:167:y:2024:i:c:s0304414923002016
    DOI: 10.1016/j.spa.2023.104229
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    References listed on IDEAS

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    1. 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.
    2. Vassilis G. Papanicolaou & Effie G. Papageorgiou & Dimitris C. Lepipas, 2012. "Random Motion on Simple Graphs," Methodology and Computing in Applied Probability, Springer, vol. 14(2), pages 285-297, June.
    3. Bayraktar, Erhan & Zhang, Xin, 2021. "Embedding of Walsh Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 1-28.
    4. Jacques du Toit & Goran Peskir, 2009. "Selling a stock at the ultimate maximum," Papers 0908.1014, arXiv.org.
    5. Luis H. R. Alvarez E. & Paavo Salminen, 2017. "Timing in the presence of directional predictability: optimal stopping of skew Brownian motion," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(2), pages 377-400, October.
    6. 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.
    7. Goran Peskir, 2005. "On The American Option Problem," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 169-181, January.
    8. 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.
    9. 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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