Polynomial processes and their applications to mathematical Finance

TLDR: In this paper, the authors introduce a class of Markov processes, called $m$-polynomial, for which the calculation of (mixed) moments up to order$m$ only requires the computation of matrix exponentials.

Abstract: We introduce a class of Markov processes, called $m$-polynomial, for which the calculation of (mixed) moments up to order $m$ only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as L\'evy-driven SDEs with affine vector fields. Thus, many popular models such as exponential L\'evy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.