## Electronic Journal of Probability

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

### Finitely Polynomially Determined Lévy Processes

Arindam Sengupta and Anish Sarkar

#### 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

**Permanent link to this document**

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

**Keywords**

Lévy process additive process Lévy's characterisation Lévy measure Kolmogorov measure

**Rights**

This work is licensed under aCreative Commons Attribution 3.0 License.

#### 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