Electronic Communications in Probability

Projections of scaled Bessel processs

Constantinos Kardaras and Johannes Ruf

Full-text: Open access


Let $X$ and $Y$ denote two independent squared Bessel processes of dimension $m$ and $n-m$, respectively, with $n\geq 2$ and $m \in [0, n)$, making $X+Y$ a squared Bessel process of dimension $n$. For appropriately chosen function $s$, the process $s (X+Y)$ is a local martingale. We study the representation and the dynamics of $s(X+Y)$, projected on the filtration generated by $X$. This projection is a strict supermartingale if, and only if, $m<2$. The finite-variation term in its Doob-Meyer decomposition only charges the support of the Markov local time of $X$ at zero.

Article information

Electron. Commun. Probab., Volume 24 (2019), paper no. 43, 11 pp.

Received: 3 January 2019
Accepted: 2 June 2019
First available in Project Euclid: 3 July 2019

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 60G44: Martingales with continuous parameter 60G48: Generalizations of martingales 60H10: Stochastic ordinary differential equations [See also 34F05] 60J55: Local time and additive functionals 60J60: Diffusion processes [See also 58J65]

Bessel process filtering local martingale local time

Creative Commons Attribution 4.0 International License.


Kardaras, Constantinos; Ruf, Johannes. Projections of scaled Bessel processs. Electron. Commun. Probab. 24 (2019), paper no. 43, 11 pp. doi:10.1214/19-ECP246. https://projecteuclid.org/euclid.ecp/1562119371

Export citation


  • [1] Sigurd Assing and Wolfgang M. Schmidt, Continuous Strong Markov Processes in Dimension One, Lecture Notes in Mathematics, vol. 1688, Springer-Verlag, Berlin, 1998, A stochastic calculus approach.
  • [2] Andrei N. Borodin and Paavo Salminen, Handbook of Brownian Motion—Facts and Formulae, second ed., Probability and its Applications, Birkhäuser Verlag, Basel, 2002.
  • [3] Cameron Bruggeman and Johannes Ruf, A one-dimensional diffusion hits points fast, Electronic Communications in Probability 21 (2016), no. 22, 1–7.
  • [4] Philippe Carmona, Frédérique Petit, and Marc Yor, Beta-gamma random variables and intertwining relations between certain Markov processes, Rev. Mat. Iberoamericana 14 (1998), no. 2, 311–367.
  • [5] Catherine Donati-Martin, Bernard Roynette, Pierre Vallois, and Marc Yor, On constants related to the choice of the local time at 0, and the corresponding Itô measure for Bessel processes with dimension $d=2(1-\alpha ),\ 0<\alpha <1$, Studia Sci. Math. Hungar. 45 (2008), no. 2, 207–221.
  • [6] Hans Föllmer and Philip Protter, Local martingales and filtration shrinkage, ESAIM: Probability and Statistics 15 (2011), 25–38.
  • [7] Mihai Gradinaru, Bernard Roynette, Pierre Vallois, and Marc Yor, Abel transform and integrals of Bessel local times, Ann. Inst. H. Poincaré Probab. Statist. 35 (1999), no. 4, 531–572.
  • [8] Constantinos Kardaras and Johannes Ruf, Projections of scaled Bessel processes, Preprint, arXiv:1805.01404, 2019.
  • [9] Shinichi Kotani, On a condition that one-dimensional diffusion processes are martingales, In Memoriam Paul-André Meyer: Séminaire de Probabilités, XXXIX, Springer, Berlin, 2006, pp. 149–156.
  • [10] Martin Larsson, Filtration shrinkage, strict local martingales and the Föllmer measure, Annals of Applied Probability 24 (2014), no. 4, 1739–1766.
  • [11] Daniel Revuz and Marc Yor, Continuous Martingales and Brownian Motion, third ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999.
  • [12] Tokuzo Shiga and Shinzo Watanabe, Bessel diffusions as a one-parameter family of diffusion processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 27 (1973), 37–46.