Brazilian Journal of Probability and Statistics

Sample path deviations of the Wiener and the Ornstein–Uhlenbeck process from its bridges

Mátyás Barczy and Peter Kern

Full-text: Open access


We study sample path deviations of the Wiener process from three different representations of its bridge: anticipative version, integral representation and space–time transform. Although these representations of the Wiener bridge are equal in law, their sample path behavior is quite different. Our results nicely demonstrate this fact. We calculate and compare the expected absolute, quadratic and conditional quadratic path deviations of the different representations of the Wiener bridge from the original Wiener process. It is further shown that the presented qualitative behavior of sample path deviations is not restricted only to the Wiener process and its bridges. Sample path deviations of the Ornstein–Uhlenbeck process from its bridge versions are also considered and we give some quantitative answers also in this case.

Article information

Braz. J. Probab. Stat., Volume 27, Number 4 (2013), 437-466.

First available in Project Euclid: 9 September 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Sample path deviation Brownian bridge Ornstein–Uhlenbeck bridge anticipative version integral representation space–time transform


Barczy, Mátyás; Kern, Peter. Sample path deviations of the Wiener and the Ornstein–Uhlenbeck process from its bridges. Braz. J. Probab. Stat. 27 (2013), no. 4, 437--466. doi:10.1214/11-BJPS175.

Export citation


  • Balabdaoui, F. and Pitman, J. (2011). The distribution of the maximal difference between Brownian bridge and its concave majorant. Bernoulli 17(1), 466–483.
  • Barczy, M. and Kern, P. (2010a). Representations of multidimensional linear process bridges. Preprint. Available at
  • Barczy, M. and Kern, P. (2010b). Sample path deviations of the Wiener and the Ornstein–Uhlenbeck process from its bridges. Available at
  • Barczy, M. and Pap, G. (2005). Connection between deriving bridges and radial parts from multidimensional Ornstein–Uhlenbeck processes. Periodica Mathematica Hungarica 50(1–2), 47–60.
  • Bharath, K. and Dey, D. K. (2011). Test to distinguish a Brownian motion from a Brownian bridge using Polya tree process. Statistics & Probability Letters 81, 140–145.
  • Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd ed. Basel: Birkhäuser.
  • Bryc, W. and Wesołowski, J. (2009). Bridges of quadratic harnesses. Available at
  • Chaumont, L. and Uribe Bravo, G. (2011). Markovian bridges: Weak continuity and pathwise constructions. The Annals of Probability 39(2), 609–647.
  • DasGupta, A. (1996). Distinguishing a Brownian bridge from a Brownian motion with drift. Technical Report 96-8, Perdue Univ., West Lafayette.
  • Delyon, B. and Hu, Y. (2006). Simulation of conditioned diffusion and application to parameter estimation. Stochastic Processes and Their Applications 116, 1660–1675.
  • Donati-Martin, C. (1990). Le probléme de Buffon–Synge pour une corde. Advances in Applied Probability 22, 375–395.
  • Fitzsimmons, P., Pitman J. and Yor, M. (1992). Markovian bridges: Construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes (E. Çinlar et al., eds.). Progress in Probability 33, 101–134. Boston: Birkhäuser.
  • Gasbarra, D., Sottinen, T. and Valkeila, E. (2007). Gaussian bridges. In Abel Symposia 2, Stochastic Analysis and Applications, Proceedings of the Second Abel Symposium, Oslo, July 29–August 4, 2005, held in honor of Kiyosi Itô (F. E. Benth et al., eds.) 361–383. New York: Springer.
  • Horne, J. S., Garton, E. O., Krone, S. M. and Lewis, J. S. (2007). Analyzing animal movements using Brownian bridges. Ecology 88(9), 2354–2363.
  • Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. Amsterdam: North-Holland.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Berlin: Springer.
  • Øksendal, B. (2003). Stochastic Differential Equations, 6th ed. Berlin: Springer.
  • Papież, L. S. and Sandison, G. A. (1990). A diffusion model with loss of particles. Advances in Applied Probability 22, 533–547.
  • Revuz, D. and Yor, M. (2001). Continuous Martingales and Brownian Motion, corrected 2nd printing of the 3rd ed. Berlin: Springer-Verlag.
  • Shiryaev, A. N. (1996). Probability, 2nd ed. New York: Springer.