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Crossing probabilities for diffusion processes with piecewise continuous boundaries. (English) Zbl 1122.60070
Summary: 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.

60J60Diffusion processes
60J70Applications of Brownian motions and diffusion theory
60J25Continuous-time Markov processes on general state spaces
60G40Stopping times; optimal stopping problems; gambling theory
65C05Monte Carlo methods
Full Text: DOI
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