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Bessel Bridges Decomposition with Varying Dimension: Applications to Finance

Author

Listed:
  • Gabriel Faraud

    (Weierstrass Institute for Applied Analysis and Stochastics (WIAS))

  • Stéphane Goutte

    (Centre National de la Recherche Scientifique (CNRS), Universités Paris 7 Diderot)

Abstract

We consider a class of stochastic processes containing the classical and well-studied class of squared Bessel processes. Our model, however, allows the dimension to be a function of the time. We first give some classical results in a larger context where a time-varying drift term can be added. Then, in the non-drifted case, we extend many results already proven in the case of classical Bessel processes to our context. Our deepest result is a decomposition of the Bridge process associated with this generalized squared Bessel process, much similar to the much celebrated result of J. Pitman and M. Yor. From a more practical point of view, we give a methodology to compute the Laplace transform of additive functionals of our process and the associated bridge. In particular, this provides direct access to the joint distribution of the values at $$t$$ t of the process and its integral. We finally give some financial applications of our results.

Suggested Citation

  • Gabriel Faraud & Stéphane Goutte, 2014. "Bessel Bridges Decomposition with Varying Dimension: Applications to Finance," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1375-1403, December.
    • Handle: RePEc:spr:jotpro:v:27:y:2014:i:4:d:10.1007_s10959-013-0496-x
    DOI: 10.1007/s10959-013-0496-x
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    References listed on IDEAS

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    1. Griselda Deelstra & Freddy Delbaen, 1995. "Long-term returns in stochastic interest rate models," ULB Institutional Repository 2013/7578, ULB -- Universite Libre de Bruxelles.
    2. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    3. Deelstra, G. & Delbaen, F., 1995. "Long-term returns in stochastic interest rate models," Insurance: Mathematics and Economics, Elsevier, vol. 17(2), pages 163-169, October.
    4. Paul Glasserman & Kyoung-Kuk Kim, 2011. "Gamma expansion of the Heston stochastic volatility model," Finance and Stochastics, Springer, vol. 15(2), pages 267-296, June.
    5. Griselda Deelstra & Freddy Delbaen, 1995. "Long-term returns in stochastic interest rate models: convergence in law," ULB Institutional Repository 2013/7580, ULB -- Universite Libre de Bruxelles.
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    Cited by:

    1. David Clancy, 2021. "The Gorin–Shkolnikov Identity and Its Random Tree Generalization," Journal of Theoretical Probability, Springer, vol. 34(4), pages 2386-2420, December.

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