Transformations of L\'evy Processes

TL;DR

This paper explores the structure and transformations of Lévy processes on *-bialgebras, providing a framework for their realization on Fock space and approximations between different stochastic processes.

Contribution

It introduces a general transformation framework for Lévy processes on *-bialgebras and demonstrates their realization on Bose Fock space, including convolution approximations of key martingales.

Findings

Lévy processes can be realized on Bose Fock space via infinitesimal convolution.

Transformations between Lévy processes are characterized by *-bialgebra homomorphisms.

Convolution approximations connect Lévy processes with Wiener and Azéma martingales.

Abstract

A L\'evy process on a *-bialgebra is given by its generator, a conditionally positive hermitian linear functional vanishing at the unit element. A *-algebra homomorphism k from a *-bialgebra C to a *-bialgebra B with the property that k respects the counits maps generators on B to generators on C. A tranformation between the corrresponding two L\'evy processes is given by forming infinitesimal convolution products. This general result is applied to various situations, e.g., to a *-bialgebra and its associated primitive tensor *-bialgebra (called "generator process") as well as its associated group-like *-bialgebra (called Weyl-*-bialgebra). It follows that a L\'evy process on a *-bialgebra can be realized on Bose Fock space as the infinitesimal convolution product of its generator process such that the vacuum vector is cyclic for the L\e'vy process. Moreover, we obtain convolution approximations of the Az\'ema martingale by the Wiener process and vice versa.