Estimation and Computation of Long-Memory Continuous-Time Models

Abstract

In an article by Comte and Renault, a generalization of Stochastic Differential Equations to continuous fractional processes is presented. However, the problems in estimating such models are barely discussed there. It turns out that, at least for some of these models, the covariance structure may be simplified substantially by performing a simple integral wavelet transform, namely the Haar transform. The Haar wavelets also result in a natural sampling procedure. In this paper I analyze a new model, namely a long-memory generalization of Ornstein-Uhlenbeck type processes, which are the continuous-time analogues of long-memory autoregressions of order 1. A fractional Brownian motion with drift is a special case. These are important examples of applications in asset pricing and the term structure of interest rates. Computation is simplified in consequence of using wavelet transforms.