Time – space harmonic polynomials relative to a Lévy process

Given a stochastic process X = {Xt, t ∈ R+} with finite moments of convenient order, a time–space harmonic polynomial relative to X is a polynomial Q(x, t) such that the process Mt =Q(Xt, t) is a martingale with respect to the filtration associated with X . Major examples are the Hermite polynomials relative to a Brownian motion, the Charlier polynomials relative to a Poisson process and the Laguerre polynomials relative to a Gamma process; for these and other examples, see Feinsilver [2], Schoutens and Teugels [11], Schoutens [10] and Barrieu and Schoutens [1]. Finding a family of time–space harmonic polynomials relative to a particular process is not an easy task; see, for example, Schoutens [10]. Nevertheless, when we are dealing with a Lévy process X that has moment generating function in a neighborhood of the origin, Sengupta [13] mentions following Neveu [7] a general procedure based on the associated exponential martingale (the left-hand side of (1) below). Specifically, that martingale is an analytic function (of u) in some neighborhood of the origin and the corresponding Taylor expansion