ISSN:
1572-9230
Keywords:
Discrete-parameter martingales
;
discrete-parameter Markov processes
;
Hermite polynomials
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We investigate, for a given martingaleM={M n: 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, M n)} 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02212886
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