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Quadratic harnesses,  -commutations, and orthogonal martingale polynomials |
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| Authors: | Wlodzimierz Bryc Wojciech Matysiak Jacek Wesolowski |
| Institution: | Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221--0025 ; Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warszawa, Poland ; Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warszawa, Poland |
| Abstract: | We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a  -commutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest. |
| Keywords: | Quadratic conditional variances harnesses orthogonal martingale polynomials hypergeometric orthogonal polynomials |
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