Infinite-dimensional polynomial processes

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.