Identification of Finite Dimensional L\'evy Systems in Financial Mathematics

TL;DR

This paper introduces a novel approach to identify finite dimensional Le9vy systems in financial mathematics, allowing for dependence modeling in return data by estimating system dynamics and noise characteristics using empirical characteristic functions.

Contribution

It develops new identification methods for Le9vy systems that incorporate dependence in financial return data, extending existing models that assume independence.

Findings

Proposes novel identification techniques based on empirical characteristic functions.

Provides error analysis and asymptotic covariance matrices for the estimators.

Demonstrates potential advantages over traditional prediction error methods.

Abstract

L\'evy processes are widely used in financial mathematics to model return data. Price processes are then defined as a corresponding geometric L\'evy process, implying the fact that returns are independent. In this paper we propose an alternative class of models allowing to describe dependence between return data. Technically such an alternative model class is obtained by considering finite dimensional linear stochastic SISO systems driven by a L\'evy process. In this paper we consider a discrete-time version of this model, focusing on the problem of identifying the dynamics and the noise characteristics of such a so-called L\'evy system. The special feature of this problem is that the characteristic function (c.f.) of the driving noise is explicitly known, possibly up to a few unknown parameters. We develop and analyze a variety of novel identification methods by adapting the so-called empirical characteristic function method (ECF) originally devised for estimating parameters of c.f.-s from i.i.d. samples. Precise characterization of the errors of these estimators will be given, and their asymptotic covariance matrices will be obtained. Their potential to outperform the prediction error method in estimating the system parameters will also be demonstrated.