Methodology and Computing in Applied Probability
, Volume 9, Issue 1, pp 2140
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 onetoone) 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 onesided and twosided 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.
Keywords
Boundary crossing probabilities Brownian motion Diffusion process First hitting time First passage time Wiener processAMS 2000 Subject Classification
Primary 60J60 60J70 Secondary 60J25 60G40 65C05 Title
 Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries
 Journal

Methodology and Computing in Applied Probability
Volume 9, Issue 1 , pp 2140
 Cover Date
 200703
 DOI
 10.1007/s1100900690026
 Print ISSN
 13875841
 Online ISSN
 15737713
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 Boundary crossing probabilities
 Brownian motion
 Diffusion process
 First hitting time
 First passage time
 Wiener process
 Primary 60J60
 60J70
 Secondary 60J25
 60G40
 65C05
 Industry Sectors
 Authors

 Liqun Wang ^{(1)}
 Klaus Pötzelberger ^{(2)}
 Author Affiliations

 1. Department of Statistics, University of Manitoba, Winnipeg, R3T 2N2, Manitoba, Canada
 2. Department of Statistics, University of Economics and Business Administration Vienna, Augasse 26, 1090, Vienna, Austria