The Annals of Probability

The bi-Poisson process: A quadratic harness

Włodzimierz Bryc, Wojciech Matysiak, and Jacek Wesołowski

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This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a two-parameter extension of the Al-Salam–Chihara polynomials and a relation between these polynomials for different values of parameters.

Article information

Ann. Probab., Volume 36, Number 2 (2008), 623-646.

First available in Project Euclid: 29 February 2008

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Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces

Quadratic conditional variances harnesses orthogonal martingale polynomials hypergeometric orthogonal polynomials


Bryc, Włodzimierz; Matysiak, Wojciech; Wesołowski, Jacek. The bi-Poisson process: A quadratic harness. Ann. Probab. 36 (2008), no. 2, 623--646. doi:10.1214/009117907000000268.

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