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First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes

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  • Andrew N. Downes

    (University of Melbourne)

  • Konstantin Borovkov

    (University of Melbourne)

Abstract

We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original boundary by a different one. In doing so we establish the existence of the first-passage time density and provide an upper bound for this function. In the case of processes with diffusion interval equal to ℝ this is extended to a lower bound, as well as bounds for the first crossing time of a lower boundary. An extension to some time-inhomogeneous diffusions is given. These results are illustrated by numerical examples.

Suggested Citation

  • Andrew N. Downes & Konstantin Borovkov, 2008. "First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes," Methodology and Computing in Applied Probability, Springer, vol. 10(4), pages 621-644, December.
    • Handle: RePEc:spr:metcap:v:10:y:2008:i:4:d:10.1007_s11009-008-9070-x
    DOI: 10.1007/s11009-008-9070-x
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    References listed on IDEAS

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    1. Liqun Wang & Klaus Pötzelberger, 2007. "Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 9(1), pages 21-40, March.
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    Cited by:

    1. Borovkov, K. & Downes, A.N., 2010. "On boundary crossing probabilities for diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 120(2), pages 105-129, February.
    2. Downes, A.N., 2009. "Bounds for the transition density of time-homogeneous diffusion processes," Statistics & Probability Letters, Elsevier, vol. 79(6), pages 835-841, March.

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