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Quadratic Harnesses, q-commutations, and orthogonal martingale polynomials

Wlodzimierz Bryc·Wojciech Matysiak·Jacek Wesolowski
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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 q-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.

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Wlodzimierz Bryc, Wojciech Matysiak, Jacek Wesolowski. 2005-04-10. Quadratic Harnesses, q-commutations, and orthogonal martingale polynomials. https://arxiv.org/abs/math/0504194

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