Parameter Estimation of Multivariate Lévy Processes

TLDR: In this paper, the authors apply Levy copulas to describe the dependence structure of multivariate Levy processes and build some Levy copula-based models, which are based on maximum likelihood principles.

Abstract: In this thesis, we apply Levy copulas to describe the dependence structure of multivariate Levy processes and build some Levy copula-based models. Parameter estimation of the models is the main part of this work. The estimation procedure is based on maximum likelihood principles. For compound Poisson processes (CPP) which have finite Levy measure, we decompose the mass on the axes and outside of the axes. This decomposition for a bivariate CPP generates three independent components and shows either the jumps only in one component, or the bivariate jumps in both components. The likelihood function can be derived based on these independent parts. We also suggest a new simulation algorithm for a bivariate CPP. We apply our method to model Danish fire insurance data and estimate the parameters of the model. The extension of the method for Levy Processes with infinite Levy measure is discussed in the second part. More precisely we take a bivariate stable Levy Process and truncate all the small jumps. We base the statistical analysis on the resulting CPP. The Fisher information matrix is also calculated and the asymptotic normality of the estimators is proved as the number of jumps tends to infinity. In this model this may happen either for the observation period going to infinity, or the truncation point going to 0 for a fixed observation period. A simulation study investigates the loss of efficiency because of the truncation. Finally, a new estimation procedure is introduced in the last chapter. The main idea of this approach, which we call two-step method, is similar to IFM (inference functions for margins) for multivariate distribution functions. First, the parameters of the marginal processes are estimated. Then, given the estimates from the first step, we estimate in a second step only the dependence structure parameters. This method is applied to a bivariate α-stable Clayton subordinator with different or common marginal parameters. For the latter, the Godambe information matrix and asymptotic covariance matrix are analytically calculated. Moreover, the asymptotic normality of the estimators is proved as the time span goes to infinity or the truncation point goes to zero. A simulation study compares the quality of all three estimation methods: the two-step estimates, the MLEs of a full model and the MLEs based on joint jumps only.