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Time-Space Harmonic Polynomials for Continuous-Time Processes and an Extension

Author

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  • Arindam Sengupta

    (Indian Statistical Institute)

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.

Suggested Citation

  • Arindam Sengupta, 2000. "Time-Space Harmonic Polynomials for Continuous-Time Processes and an Extension," Journal of Theoretical Probability, Springer, vol. 13(4), pages 951-976, October.
    • Handle: RePEc:spr:jotpro:v:13:y:2000:i:4:d:10.1023_a:1007857823002
    DOI: 10.1023/A:1007857823002
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