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Time-Space Harmonic Polynomials for Continuous-Time Processes and an Extension (2000)

Sengupta, Arindam

Springer

Journal of theoretical probability 13 (2000), S. 951-976

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ISSN: 1572-9230

Keywords: time-space harmonic polynomials ; Lévy processes ; Semi-stable Markov processes ; Hochberg's measure

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: 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.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1023/A:1007857823002

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Time-space polynomial martingales cenerated by a discrete-time martingale (1995)

Goswami, A. ; Sengupta, Arindam

Springer

Journal of theoretical probability 8 (1995), S. 417-432

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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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