Holomorphic transforms with application to affine processes

TL;DR

This paper investigates holomorphic transforms of Itô-Lévy processes, establishing their relation to generator vectors, conditions for holomorphic extension, and developing series expansions, with applications to multidimensional affine processes.

Contribution

It introduces a framework for holomorphic transforms of Itô-Lévy processes, linking them to generator analytic vectors and providing series expansions for affine processes.

Findings

Holomorphic transforms relate to generator analytic vectors.

Conditions for holomorphic extension into a strip are established.

Series representations for Fourier transforms of affine processes are derived.

Abstract

In a rather general setting of It\^o-L\'evy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multidimensional affine It\^o-L\'evy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.