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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