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On Markov processes with polynomial conditional moments


Author: Paweł J. Szabłowski
Journal: Trans. Amer. Math. Soc. 367 (2015), 8487-8519
MSC (2010): Primary 60J25, 60G44; Secondary 60G99, 33C47
DOI: https://doi.org/10.1090/tran/6295
Published electronically: May 13, 2015
MathSciNet review: 3403063
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Abstract: We study properties of a subclass of Markov processes that have all moments being continuous functions of the time parameter and, more importantly, are characterized by the property that say their $ n-$th conditional moment given the past is also a polynomial of degree not exceeding $ n.$ Of course all processes with independent increments and with all moments existing, belong to this class. We give a characterization of them within the studied class. We indicate other examples of such process. Besides we indicate families of polynomials that have the property of constituting martingales. We also study conditions under which processes from the analyzed class have orthogonal polynomial martingales and further are harnesses or quadratic harnesses. Consequently we generalize some of the recent results on quadratic harnesses. We provide examples illustrating developed theory and also provide some interesting open questions. To make the paper interesting for a wider range of readers, we provide a short introduction formulated in the language of measures on the plane.


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Additional Information

Paweł J. Szabłowski
Affiliation: Department of Mathematics and Information Sciences, Warsaw University of Technology, ul Koszykowa 75, 00-662 Warsaw, Poland
Email: pawel.szablowski@gmail.com

DOI: https://doi.org/10.1090/tran/6295
Keywords: Polynomial regression, martingales, Markov processes, harnesses, quadratic harnesses, orthogonal polynomials, Cholesky decomposition, expansion of Radon-Nikodym derivative of one measure with respect to the other, Lancaster expansion
Received by editor(s): April 19, 2013
Received by editor(s) in revised form: May 30, 2013, and September 26, 2013
Published electronically: May 13, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.