Lopyngnt'" ItAL Moaellng ana Lonrrol or Systems, Klagenfurt, Austria. 2001 t:L==>t:VIt:K IFAC PUBLICATIONS www.elsevier.comJlocateiifac NON-PARAMETRIC ESTIMATION OF DIFFUSION-PATHS USING WAVELET SCALING METHODS Esben P. Aarhus School of Business, Department of Information Science, Statistics Group, FlIglesangs AUt 4, DK-82 I 0 Aarhus V, Denmark. Email : eh@asb.dk Abstract: In continuous time, diffusion processes have been used to model financial dynamics for a long time. For example the Ornstein-Uhlenbeck process (the simplest mean-reverting process) has been used to model non-speculative price processes. We discuss non-parametric estimation of these processes using a wavelet filtration method, specifically the Cl trolls algorithm. Copyright © 2001lFAC Keywords: Ornstein-Uhlenbeck process, cardinal B-splines, wavelet transform, Cl trollS 1. INTRODUCTION This paper discusses the estimation and fitting of dif- fusions. In particular the subclasses of mean-reverting processes such as the Ornstein-Uhlenbeck types are interesting. The idea is to use simple Wavelet filter methods that are computationally fast. Various parametric methods to estimate diffusions have been developed in the literature. As Ornstein- Uhlenbeck type processes are the analogues in con- tinuous time of autoregressions of order 1, usually the estimation is carried out as if the process was an AR(l) . However, as this approach is unsatisfactory in some sense, other methods have been tried out, for in- stance a method based on an approximation of the true discrete-time likelihood function (Pedersen 1995), and methods based on estimating functions (Bibby and S0rensen 1995). The method in this paper, however, is quite a different approach. We use the Ito stochastic calculus together with wavelet theory to perform a continuous wavelet transform of the stochastic process in question. The purpose is to study the use of B-spline wavelets and the use of the a trollS algorithm to smooth and filter diffusion processes. 187 Wavelet transforms provide a decomposition of a stochastic process, such that temporal structures can be revealed and handled by non parametric methods. In particular the a trollS wavelet transform decom- poses the process in question into detail processes and an approximation process such that the original process can be expressed as an additive combination of details and an approximation at different resolution levels. It turns out that the coefficients produced by the a trous algorithm with B-splines obey simple differen- tial properties. 2. WAVELET DECOMPOSITION AND THE CENTRAL CARDINAL SPLINES The task is to consider the approximation of a time process at coarser and coarser resolutions. We use the a trollS (with holes) algorithm (Holschneider and Tchamitchian 1989, Shensa 1992). The central Cardinal Spline of order n ;::: 1 is defined as _ n+l(_l)J(n+l) [. n+l_ .. ]n Bn(x) - L f • X + 2 J , j=O n. J +