Time-Space Harmonic Polynomials for Continuous-Time Processes and an Extension

Abstract

A time-space harmonic polynomial for a stochastic process M=(M _t) is a polynomial P in two variables such that P(t, M _t) is a martingale. In this paper, we investigate conditions for the existence of such polynomials of each degree in the second, “space,” argument. We also describe various properties a sequence of time-space harmonic polynomials may possess and the interaction of these properties with distributional properties of the underlying process. Thus, continuous-time conterparts to the results of Goswami and Sengupta,⁽²⁾ where the analoguous problem in discrete time was considered, are derived. A few additional properties are also considered. The resulting properties of the process include independent increments, stationary independent increments and semi-stability. Finally, a generalization to a “measure” proposed by Hochberg⁽³⁾ on path space is obtained.