Finitely Polynomially Determined Lévy Processes

Arindam Sengupta (Indian Statistical Institute)
Anish Sarkar (Indian Statistical Institute (Delhi Centre))


A time-space harmonic polynomial for a continuous-time process $X=\{X_t : t \ge 0\} $ is a two-variable polynomial $ P $ such that $ \{ P(t,X_t) : t \ge 0 \} $ is a martingale for the natural filtration of $ X $. Motivated by Lévy's characterisation of Brownian motion and Watanabe's characterisation of the Poisson process, we look for classes of processes with reasonably general path properties in which a characterisation of those members whose laws are determined by a finite number of such polynomials is available. We exhibit two classes of processes, the first containing the Lévy processes, and the second a more general class of additive processes, with this property and describe the respective characterisations.

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Pages: 1-22

Publication Date: August 30, 2000

DOI: 10.1214/EJP.v6-80


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