Finitely Polynomially Determined Lévy Processes

Arindam Sengupta,Anish Sarkar

doi:10.1214/ejp.v6-80

Arindam Sengupta, Anish Sarkar

Open Access

https://doi.org/10.1214/ejp.v6-80

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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 Levy'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 Levy processes, and the second a more general class of additive processes, with this property and describe the respective characterisations.

Highlights

The famous characterisation due to Paul Levy for Brownian motion contained in Doob ([D], page 384, Theorem 11.9), for instance, says that if X = (Xt)t≥0 is a martingale with continuous paths such that Xt2 − t is a martingale, (Xt) is the standard Brownian motion
The point of this paper is that for certain Levy processes, which we describe fully, finitely many time-space harmonic polynomials are enough to determine its law completely
This poses another interesting question for such processes, namely, what is the minimum number of timespace harmonic polynomials required to obtain a characterisation? We determine the number of polynomials required for such a characterisation in terms of the support of the Levy measure of the process

Summary

Introduction

This characterisation may be restated as: for a process X with continuous paths, a certain pair of polynomials in t and Xt being martingales, determines the distribution of X uniquely, and as that of a Brownian motion We call such polynomials time-space harmonic for the process X concerned; an exact definition follows. The point of this paper is that for certain Levy processes, which we describe fully, finitely many time-space harmonic polynomials are enough to determine its law completely. This poses another interesting question for such processes, namely, what is the minimum number of timespace harmonic polynomials required to obtain a characterisation? The section contains the basic definitions and the statements of the theorems, while the fourth is devoted to their proofs

Definitions and Statements of Theorems

A Key Lemma

Proofs of The Theorems

Examples