A Factorisation of Diffusion Measure and Finite Sample Path Constructions

  • Alexandros Beskos
  • Omiros PapaspiliopoulosEmail author
  • Gareth O. Roberts


In this paper we introduce decompositions of diffusion measure which are used to construct an algorithm for the exact simulation of diffusion sample paths and of diffusion hitting times of smooth boundaries. We consider general classes of scalar time-inhomogeneous diffusions and certain classes of multivariate diffusions. The methodology presented in this paper is based on a novel construction of the Brownian bridge with known range for its extrema, which is of interest on its own right.


Rejection sampling Exact simulation Conditioned Brownian motion Boundary hitting times 

AMS 2000 Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. S. Asmussen, P. Glynn, and J. Pitman, “Discretization error in simulation of one-dimensional reflecting Brownian motion,” Annals of Applied Probability vol. 5 pp. 875–896, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  2. J. Bertoin, and J. Pitman, “Path transformations connecting Brownian bridge, excursion and meander,” Bulletin des Sciences Mathématiques vol. 118 pp. 147–166, 1994.zbMATHMathSciNetGoogle Scholar
  3. A. Beskos, O. Papaspiliopoulos, and G. O. Roberts, “Retrospective exact simulation of diffusion sample paths with applications,” Bernoulli vol. 12 pp. 1077–1098, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  4. A. Beskos, and G. O. Roberts, “Exact simulation of diffusions,” Annals of Applied Probability vol. 15 pp. 2422–2444, 2005.zbMATHCrossRefMathSciNetGoogle Scholar
  5. J. L. Doob, “Heuristic approach to the Kolmogorov-Smirnov theorems,” Annals of Mathematical Statistics vol. 20 pp. 393–403, 1949.zbMATHCrossRefMathSciNetGoogle Scholar
  6. J. F. C. Kingman, Poisson processes, vol. 3 of Oxford Studies in Probability, The Clarendon Press Oxford University Press, Oxford Science Publications: New York, 1993. Oxford Science Publications, 1993.Google Scholar
  7. P. E. Kloeden, and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications of Mathematics (New York). Springer-Verlag: Berlin, 1992.Google Scholar
  8. K. Pötzelberger, and L. Wang, “Boundary crossing probability for Brownian motion,” Journal of Applied Probability vol. 38 pp. 152–164, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  9. L. A. Shepp, “The joint density of the maximum and its location for a Wiener process with drift.” Journal of Applied Probability vol. 16 pp. 423–427, 1979.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Alexandros Beskos
    • 1
  • Omiros Papaspiliopoulos
    • 2
    Email author
  • Gareth O. Roberts
    • 1
  1. 1.Department of StatisticsUniversity of WarwickCoventryUK
  2. 2.Department of EconomicsUniversitat Pompeu FabraBarcelonaSpain

Personalised recommendations