Infinite-dimensional polynomial processes

Christa Cuchiero () and Sara Svaluto-Ferro ()
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Christa Cuchiero: Vienna University
Sara Svaluto-Ferro: Vienna University

Finance and Stochastics, 2021, vol. 25, issue 2, No 7, 383-426

Abstract: Abstract We introduce polynomial processes taking values in an arbitrary Banach space B ${B}$ via their infinitesimal generator L $L$ 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 $d$ -dimensional Brownian motion is polynomial and derive its expected value via the polynomial approach.

Keywords: Polynomial processes; Infinite-dimensional Markov processes; Dual processes; Forward variance models; Rough volatility; VIX options; Signature process; 60J25; 60H15 (search for similar items in EconPapers)
JEL-codes: C60 G13 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (9)

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DOI: 10.1007/s00780-021-00450-x

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