Methodology and Computing in Applied Probability

, Volume 9, Issue 1, pp 21-40

First online:

Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries

  • Liqun WangAffiliated withDepartment of Statistics, University of Manitoba Email author 
  • , Klaus PötzelbergerAffiliated withDepartment of Statistics, University of Economics and Business Administration Vienna

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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.


Boundary crossing probabilities Brownian motion Diffusion process First hitting time First passage time Wiener process

AMS 2000 Subject Classification

Primary 60J60 60J70 Secondary 60J25 60G40 65C05