Time-inhomogeneous polynomial processes

TL;DR

This paper introduces a new class of time-inhomogeneous polynomial processes, extending existing models by allowing coefficients to depend on time, with implications for financial and energy market modeling.

Contribution

It characterizes time-inhomogeneous polynomial processes via semimartingale characteristics and explores numerical methods using Magnus series for moments calculation.

Findings

Full characterization of time-inhomogeneous polynomial processes.

Matrix exponential methods are generally not applicable for moments.

Magnus series provides a fast numerical approximation method.

Abstract

Time homogeneous polynomial processes are Markov processes whose moments can be calculated easily through matrix exponentials. In this work, we develop a notion of time inhomogeneous polynomial processes where the coeffiecients of the process may depend on time. A full characterization of this model class is given by means of their semimartingale characteristics. We show that in general, the computation of moments by matrix exponentials is no longer possible. As an alternative we explore a connection to Magnus series for fast numerical approximations. Time-inhomogeneity is important in a number of applications: in term-structure models, this allows a perfect calibration to available prices. In electricity markets, seasonality comes naturally into play and have to be captured by the used models. The model class studied in this work extends existing models, for example Sato processes and time-inhomogeneous affine processes.