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Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries

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Abstract

We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class includes many interesting processes in real applications, e.g., Ornstein–Uhlenbeck, growth processes and geometric Brownian motion with time dependent drift. This method applies to both one-sided and two-sided general nonlinear boundaries, which may be discontinuous. Using this approach explicit formulas for boundary crossing probabilities for certain nonlinear boundaries are obtained, which are useful in evaluation and comparison of various computational algorithms. Moreover, numerical computation can be easily done by Monte Carlo integration and the approximation errors for general boundaries are automatically calculated. Some numerical examples are presented.

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Authors and Affiliations

  1. Department of Statistics, University of Manitoba, Winnipeg, R3T 2N2, Manitoba, Canada

    Liqun Wang

  2. Department of Statistics, University of Economics and Business Administration Vienna, Augasse 2-6, 1090, Vienna, Austria

    Klaus Pötzelberger

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  1. Liqun Wang
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  2. Klaus Pötzelberger
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Correspondence to Liqun Wang.

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Wang, L., Pötzelberger, K. Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries. Methodol Comput Appl Probab 9, 21–40 (2007). https://doi.org/10.1007/s11009-006-9002-6

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  • DOI: https://doi.org/10.1007/s11009-006-9002-6

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