Infinite dimensional polynomial processes

TL;DR

This paper develops a general framework for polynomial processes in infinite-dimensional Banach spaces, providing new moment representations and applications to financial models and Brownian motion signatures.

Contribution

It introduces a novel approach to infinite-dimensional polynomial processes using generator and martingale problem formulations, extending finite-dimensional results.

Findings

Derived moment formulas via ODE systems on tensor algebras.

Applied framework to polynomial forward variance curve models.

Showed the signature process of Brownian motion is polynomial and computed its expectation.

Abstract

We introduce polynomial processes taking values in an arbitrary Banach space $B$ via their infinitesimal generator $L$ and the associated martingale problem. We obtain two representations of the (conditional) moments in terms of solutions of a system of ODEs on the truncated tensor algebra of dual respectively bidual spaces. We illustrate how the well-known moment formulas for finite dimensional or probability-measure valued polynomial processes can be deduced in this general framework. As an application we consider polynomial forward variance curve models which appear in particular as Markovian lifts of (rough) Bergomi-type volatility models. Moreover, we show that the signature process of a $d$-dimensional Brownian motion is polynomial and derive its expected value via the polynomial approach.