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Pinned Diffusions and Markov Bridges

  • Florian HildebrandtEmail author
  • Sylvie Rœlly
Article
  • 12 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification (2010)

60G15 60H10 60J60 

Notes

Acknowledgements

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

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

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

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