Resumen de Quadratic harnesses, $q$-commutations, and orthogonal martingale polynomials

Wlodzimierz Bryc, Wojciech Matysiak, Jacek Weskowski

• 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.