A family of martingales generated by a process with independent increments

TL;DR

This paper introduces an explicit method to construct a family of martingales from a process with independent increments, utilizing polynomials linking moments and cumulants, and Teugels martingales related to process jumps.

Contribution

It provides a novel explicit procedure for generating martingales from independent increment processes using polynomial relationships and Teugels martingales.

Findings

Explicit construction method for martingales from independent increment processes

Use of polynomials to relate moments and cumulants

Introduction of Teugels martingales associated with process jumps

Abstract

An explicit procedure to construct a family of martingales generated by a process with independent increments is presented. The main tools are the polynomials that give the relationship between the moments and cumulants, and a set of martingales related to the jumps of the process called Teugels martingales