Holomorphic jump-diffusions

TL;DR

This paper introduces holomorphic jump-diffusions, a new class generalizing affine and polynomial processes, with a focus on their generator properties and applications to complex-valued expectations and characteristic functions.

Contribution

It defines holomorphic jump-diffusions, explores their properties, and provides conditions for their holomorphic and affine-holomorphic nature, expanding the framework of jump-diffusion processes.

Findings

Representation of power series expectations via infinite-dimensional linear ODEs

Conditions guaranteeing holomorphic and affine-holomorphic properties

Applications to various classes of jump-diffusions

Abstract

We introduce a class of jump-diffusions, called holomorphic, of which the well-known classes of affine and polynomial processes are particular instances. The defining property concerns the extended generator, which is required to map a (subset of) holomorphic functions to themselves. This leads to a representation of the expectation of power series of the process' marginals via a potentially infinite dimensional linear ODE. We apply the same procedure by considering exponentials of holomorphic functions, leading to a class of processes named affine-holomorphic for which a representation for quantities as the characteristic function of power series is provided. Relying on powerful results from complex analysis, we obtain sufficient conditions on the process' characteristics which guarantee the holomorphic and affine-holomorphic properties and provide applications to several classes of jump-diffusions.