Time-Space Harmonic Polynomials for Continuous-Time Processes and an Extension |
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Authors: | Arindam Sengupta |
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Institution: | (1) Stat-Math Unit, Indian Statistical Institute, 203 B.T.Road, Calcutta, 700 035, India |
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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. |
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Keywords: | time-space harmonic polynomials Lé vy processes Semi-stable Markov processes Hochberg's measure |
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