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An optimal Skorokhod embedding for diffusions

Author

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  • Cox, A. M. G.
  • Hobson, D. G.

Abstract

Given a Brownian motion (Bt)t[greater-or-equal, slanted]0 and a general target law [mu] (not necessarily centered or even in ) we show how to construct an embedding of [mu] in B. This embedding is an extension of an embedding due to Perkins, and is optimal in the sense that it simultaneously minimises the distribution of the maximum and maximises the distribution of the minimum among all embeddings of [mu]. The embedding is then applied to regular diffusions, and used to characterise the target laws for which a Hp-embedding may be found.

Suggested Citation

  • Cox, A. M. G. & Hobson, D. G., 2004. "An optimal Skorokhod embedding for diffusions," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 17-39, May.
    • Handle: RePEc:eee:spapps:v:111:y:2004:i:1:p:17-39
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    References listed on IDEAS

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    1. Vallois, P., 1992. "Quelques inégalités avec le temps local en zero du mouvement Brownien," Stochastic Processes and their Applications, Elsevier, vol. 41(1), pages 117-155, May.
    2. Jacka, S. D., 1988. "Doob's inequalities revisited: A maximal H1-embedding," Stochastic Processes and their Applications, Elsevier, vol. 29(2), pages 281-290, September.
    3. Grandits, Peter & Falkner, Neil, 2000. "Embedding in Brownian motion with drift and the Azéma-Yor construction," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 249-254, February.
    4. Pedersen, J. L. & Peskir, G., 2001. "The Azéma-Yor embedding in non-singular diffusions," Stochastic Processes and their Applications, Elsevier, vol. 96(2), pages 305-312, December.
    5. Hambly, B. M. & Kersting, G. & Kyprianou, A. E., 2003. "Law of the iterated logarithm for oscillating random walks conditioned to stay non-negative," Stochastic Processes and their Applications, Elsevier, vol. 108(2), pages 327-343, December.
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    Citations

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    Cited by:

    1. Vicky Henderson & David Hobson & Matthew Zeng, 2023. "Cautious stochastic choice, optimal stopping and deliberate randomization," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 75(3), pages 887-922, April.
    2. Cox, Alexander M.G. & Obłój, Jan, 2015. "On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3280-3300.
    3. Oblój, Jan, 2007. "An explicit solution to the Skorokhod embedding problem for functionals of excursions of Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 117(4), pages 409-431, April.

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