Abstract:

Quadratic harnesses are time-inhomogeneous Markov polynomial processes with linear conditional expectations and quadratic conditional variances with respect to the past-future filtrations. Typically they are determined by five numerical constants η, θ, τ, σ and q hidden in the form of conditional variances. In this paper we derive infinitesimal generators of such processes in the case σ=0 extending previously known results. The infinitesimal generators are identified through a solution of a q-commutation equation in the algebra Q of infinite sequences of polynomials in one variable. The solution is a special element in Q whose coordinates satisfy a three-term recurrence and thus define a system of orthogonal polynomials. It turns out that the corresponding orthogonality measure υ_x,t. uniquely determines the infinitesimal generator (acting on polynomials or bounded functions with bounded continuous second derivative) as an integro-differential operator with an explicit kernel, where integration is with respect to the measure υ_x,t.

2010 AMS Mathematics Subject Classification: Primary 60J35; Secondary 46L53.

Keywords and phrases: polynomial process, quadratic harness, infinitesimal generator, orthogonal polynomials, algebra of polynomial sequences, three-step recurrence.