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.

Infinite dimensional polynomial processes

Christa Cuchiero;Sara Svaluto-Ferro
2021

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.
Dual processes
Infinite-dimensional Markov processes
Forward variance models Rough volatility VIX options Signature process
Forward variance models
Rough volatility
VIX options
Polynomial processes
Signature process
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11562/1051764
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