Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-08T07:51:44.290Z Has data issue: false hasContentIssue false
  • English
  • Français

Discrete sampling of an integrated diffusion process and parameter estimation of the diffusion coefficient

Published online by Cambridge University Press:  15 August 2002

Arnaud Gloter
Arnaud Gloter*
Affiliation:
Université de Marne-la-Vallée, Équipe d'Analyse et de Mathématiques Appliquées, 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France; e-mail: [email protected]
Get access

Abstract

Let (Xt ) be a diffusion on the interval (l,r) and Δn a sequence of positive numbers tending to zero. We define J i as the integral between iΔn and (i + 1)Δn of X s .We give an approximation of the law of (J0,...,Jn-1)by means of a Euler scheme expansion for the process (Ji ). In some special cases, an approximation by anexplicit Gaussian ARMA(1,1) process is obtained.When Δn = n-1 we deduce from this expansion estimatorsof the diffusion coefficient of X based on (Ji ). These estimatorsare shown to be asymptotically mixed normal as n tends to infinity.

Keywords

Diffusion processesdiscrete time observationhidden markov model.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 4 , 2000 , pp. 205 - 227
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

O.E. Barndorff-Nielsen and N. Shephard, Aggregation and model construction for volatility models. Working paper series No. 10. Center for Analytical Finance, University of Aarhus (1998).
Dohnal, G., On estimating the diffusion coefficient. J. Appl. Probab. 24 (1987) 105-114. CrossRef
Genon-Catalot, V. and Jacod, J., On the estimation of the diffusion coefficient for multidimensional diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 29 (1993) 119-151.
Genon-Catalot, V., Jeantheau, T. and Laredo, C., Limit theorems for discretely observed stochastic volatility models. Bernoulli 4 (1998) 283-303. CrossRef
A. Gloter, Parameter estimation for a discrete sampling of an integrated Ornstein-Uhlenbeck process. Statistics (to appear).
Hull, J. and White, A., The pricing of options on assets with stochastic volatilities. J. Finance 42 (1987) 281-300. CrossRef
J. Jacod, On continuous conditional Gaussian martingales and stable convergence in law. Séminaire de Probabilités XXXI. 1655. Springer, Berlin, Lectures Notes in Math. (1997) 232-246.
Kessler, M., Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist. 24 (1997) 211-229. CrossRef
B. Leblanc, Modélisation de la Volatilité d'un Actif Financier et Applications. Thèse, Université Paris 7 (1997).
Lefebvre, M., On the inverse of the first hitting time problem for bidimensional processes. J. Appl. Probab. 34 (1997) 610-622. CrossRef
S. Pastorello, E. Renault and N. Touzi, Statistical inference for random variance option pricing. Southern European Economics Discussion Series, D.P.136 (1994).
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Springer-Verlag, Berlin Heidelberg, second edition (1994).