Journal of Theoretical Probability

, Volume 8, Issue 2, pp 417–432 | Cite as

Time-space polynomial martingales cenerated by a discrete-time martingale

  • A. Goswami
  • Arindam Sengupta
Article

Abstract

We investigate, for a given martingaleM={Mn: n≥0}, the conditions for the existence of polynomialsP(·,·) 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 ofM. 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 whenM has independent increments. For generalM, we obtain a necessary condition which entails, under mild additional hypotheses, thatM 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.

Key Words

Discrete-parameter martingales discrete-parameter Markov processes Hermite polynomials 

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References

  1. 1.
    Billingsley, P. (1986).Probability and Measure, John Wiley & Sons.Google Scholar
  2. 2.
    Neveu, J. (1975).Discrete-Parameter Martingales, North-Holland.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • A. Goswami
  • Arindam Sengupta
    • 1
  1. 1.Stat-Math UnitIndian Statistical InstituteCalcuttaIndia

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