Polynomial processes and their applications to mathematical finance

We introduce a class of Markov stochastic processes called mpolynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains affine processes, Feller processes with quadratic squared diffusion coefficient, as well as Lévy-driven SDEs with affine vector fields. Thus, many popular models such as the classical Black-Scholes, exponential Lévy or affine models are covered by this setting. The applications range from statistical GMM estimation to option pricing. For instance, the efficient and easy computation of moments can successfully be used for variance reduction techniques in Monte Carlo simulations.