Harnessing empirical characteristic function convergence behaviour

Parameter estimation for L´evy processes has generated much research effort lately with a strong injection of interest coming from finance. Within this context the problem can be framed as estimation using increments from an infinitely divisible distribution, for which empirical characteristic functions (ecf) are convenient tools. However convergence of ecf’s to Gaussian processes has not been exploited as fully as it might have been. In this paper we go back to strong convergence results derived from the Hungarian construction and use Brownian bridge approximations to construct new estimators. In particular we study one integrated square error estimator tailored to show deference to the variance structure of the corresponding Gaussian process. We prove some of its nice statistical properties and present simulation results obtained through its use.