A multi-parameter regularization approach for estimating parameters in jump diffusion processes

摘要

In this paper, we consider the inverse problem of estimating simultaneously the five parameters of a jump diffusion process based on return observations of a price trajectory. We show that there occur some ill-posedness phenomena in the parameter estimation problem, because the forward operator fails to be injective and small perturbations in the data may lead to large changes in the solution. We illustrate the instability effect by a numerical case study. To obtain stable approximate solutions of the estimation problem, we use a multi-parameter regularization approach, where a least-squares fitting of empirical densities is superposed by a quadratic penalty term of fitted semi-invariants with weights. A little number of required weights is controlled by a discrepancy principle. For the realization of this control, we propose and justify a fixed point iteration, where an exponent can be chosen arbitrarily positive. A numerical case study completing the paper shows that the approach provides satisfactory results and that the amount of computation can be reduced by an appropriate choice of the free exponent.