Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries |
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Authors: | Liqun Wang Klaus Pötzelberger |
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Institution: | (1) Department of Statistics, University of Manitoba, Winnipeg, R3T 2N2, Manitoba, Canada;(2) Department of Statistics, University of Economics and Business Administration Vienna, Augasse 2-6, 1090 Vienna, Austria |
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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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Keywords: | Boundary crossing probabilities Brownian motion Diffusion process First hitting time First passage time Wiener process |
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