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Time-space polynomial martingales cenerated by a discrete-time martingale
A. Goswami,
Published in Kluwer Academic Publishers-Plenum Publishers
1995
Volume: 8

Issue: 2
Pages: 417 - 432
Abstract
We investigate, for a given martingale M={Mn: n≥0}, the conditions for the existence of polynomials P(·,·) of two variables, "time" and "space," and of arbitrary degree in the latter, such that {P(n, Mn)} is a martingale for the natural filtration of M. Denoting by ℘ the vector space of all such polynomials, we ask, in particular, when such a sequence can be chosen so as to span ℘. A complete necessary and sufficient condition is obtained in the case when M has independent increments. For general M, we obtain a necessary condition which entails, under mild additional hypotheses, that M is necessarily Markovian. Considering a slightly more general class of polynomials than ℘ we obtain necessary and sufficient conditions in the case of general martingales also. It is moreover observed that in most of the cases, the set ℘ determines the law of the martingale in a certain sense. © 1995 Plenum Publishing Corporation.