Pinned Diffusions and Markov Bridges

  • Florian HildebrandtEmail author
  • Sylvie Rœlly


In this article, we consider a family of real-valued diffusion processes on the time interval [0, 1] indexed by their prescribed initial value \(x \in \mathbb {R}\) and another point in space, \(y \in \mathbb {R}\). We first present an easy-to-check condition on their drift and diffusion coefficients ensuring that the diffusion is pinned in y at time \(t=1\). Our main result then concerns the following question: can this family of pinned diffusions be obtained as the bridges either of a Gaussian Markov process or of an Itô diffusion? We eventually illustrate our precise answer with several examples.


Pinned diffusion \(\alpha \)-Brownian bridge \(\alpha \)-Wiener bridge Gaussian Markov process Reciprocal characteristics 

Mathematics Subject Classification (2010)

60G15 60H10 60J60 



The authors thank an anonymous referee for bringing to their attention the references [7] and [8].


  1. 1.
    Barczy, M., Kern, P.: General alpha-Wiener bridges. Commun. Stoch. Anal. 5(3), 585–608 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Barczy, M., Kern, P.: Representations of multidimensional linear process bridges. Random Oper. Stoch. Equ. 21(2), 159–189 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barczy, M., Kern, P.: Gauss–Markov processes as space-time scaled stationary Ornstein–Uhlenbeck processes (2014). ArXiv:1409.7253v2
  4. 4.
    Barczy, M., Pap, G.: \(\alpha \)-Wiener bridges: singularity of induced measures and sample path properties. Stoch. Anal. Appl. 28(3), 447–466 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brennan, M.J., Schwartz, E.: Arbitrage in stock index futures. J. Bus. 63, 7–31 (1990)CrossRefGoogle Scholar
  6. 6.
    Clark, J.M.C.: A local characterization of reciprocal diffusions. In: Davis, M.H.A., Elliot, R.J. (eds.) Applied Stochastic Analysis, Stochastics Monogr, vol. 5, pp. 45–59. Gordon and Breach, New York (1991)Google Scholar
  7. 7.
    Li, X.M.: Generalised Brownian bridges: examples. Markov Process Relat. Fields 24(1), 151–163 (2018)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Li, X.M.: On the semi-classical brownian bridge measure. Electron. Commun. Probab. 22, 1–15 (2017)MathSciNetGoogle Scholar
  9. 9.
    Mansuy, R.: On a one-parameter generalization of the Brownian bridge and associated quadratic functionals. J. Theor. Probab. 17(4), 1021–1029 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Muirhead, R.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982)CrossRefGoogle Scholar
  11. 11.
    Neveu, J.: Processus aléatoires Gaussiens. Seminaire de Mathematiques Superieures. Les Presses de l’Université de Montréal, Montreal (1968)zbMATHGoogle Scholar
  12. 12.
    Thieullen, M.: Reciprocal diffusions and symmetries of parabolic PDE: the nonflat case. Potential Anal. 16(1), 1–28 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (2000)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HamburgHamburgGermany
  2. 2.Institut für Mathematik der Universität PotsdamPotsdam OT GolmGermany

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