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Projections of scaled Bessel processs. (English) Zbl 07088984
Summary: 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.

##### MSC:
 60G44 Martingales with continuous parameter 60G48 Generalizations of martingales 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J55 Local time and additive functionals 60J60 Diffusion processes
##### Keywords:
Bessel process; filtering; local martingale; local time
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##### References:
 [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. · Zbl 0914.60008 [2] Andrei N. Borodin and Paavo Salminen, Handbook of Brownian Motion—Facts and Formulae, second ed., Probability and its Applications, Birkhäuser Verlag, Basel, 2002. · Zbl 1012.60003 [3] Cameron Bruggeman and Johannes Ruf, A one-dimensional diffusion hits points fast, Electronic Communications in Probability 21 (2016), no. 22, 1-7. · Zbl 1338.60197 [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. · Zbl 0919.60074 [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. · Zbl 1212.60070 [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. · Zbl 0937.60080 [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. · Zbl 1185.60090 [10] Martin Larsson, Filtration shrinkage, strict local martingales and the Föllmer measure, Annals of Applied Probability 24 (2014), no. 4, 1739-1766. · Zbl 1304.60050 [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. · Zbl 0917.60006 [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. · Zbl 0327.60047
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