Finitely Polynomially Determined Lévy Processes
Arindam Sengupta, Indian Statistical Institute
Anish Sarkar, Indian Statistical Institute (Delhi Centre)
Abstract
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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