Estimation of Lévy processes through stochastic programming

Abstract

Estimation of Levy processes with the use of the characteristic function has lately shifted much of its attention to nonparametric settings. However the para- metric context still offers scope for study. The nature of neighbourhoods of the minima sought for by the integrated square error estimator (ISEE), and its variants, could be meaningfully related to a number of useful properties possessed by the estimator. Furthermore the numerical problems associated with the actual computation of parameter estimates have not been given exhaustive attention. In this paper through a slight reformulation of the ISEE formula, local geometric features of the optimal solution used in ISEE are studied. This formulation is subsequently proposed within a stochastic programming framework. The latter provides a powerful, productive methodology and an alternative theoretical framework which are entertained within this study. Results are presented and discussed.