## Electronic Journal of Probability

### Finitely Polynomially Determined Lévy Processes

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 6 (2001), paper no. 7, 22 pp.

Dates
Accepted: 30 August 2000
First available in Project Euclid: 19 April 2016

https://projecteuclid.org/euclid.ejp/1461097637

Digital Object Identifier
doi:10.1214/EJP.v6-80

Mathematical Reviews number (MathSciNet)
MR1831802

Zentralblatt MATH identifier
0974.60026

Subjects
Primary: 60G44: Martingales with continuous parameter
Secondary: 60J30

Rights

#### Citation

Sengupta, Arindam; Sarkar, Anish. Finitely Polynomially Determined Lévy Processes. Electron. J. Probab. 6 (2001), paper no. 7, 22 pp. doi:10.1214/EJP.v6-80. https://projecteuclid.org/euclid.ejp/1461097637

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