Rate of convergence for parametric estimation in a stochastic volatility model

Abstract

We consider the following hidden Markov chain problem: estimate the finite-dimensional parameter θ in the equation v t=v 0+ ∫ 0 t σ(θ, v s) dW s+ drift, when we observe discrete data X i/ n at times i=0,…, n from the diffusion X t=x 0+ ∫ 0 t v s dB s+ drift . The processes ( W t ) t∈[0,1] and ( B t ) t∈[0,1] are two independent Brownian motions; asymptotics are taken as n→∞. This stochastic volatility model has been paid some attention lately, especially in financial mathematics. We prove in this note that the unusual rate n −1/4 is a lower bound for estimating θ. This rate is indeed optimal, since Gloter (CR Acad. Sci. Paris, t330, Série I, pp. 243–248), exhibited n −1/4 consistent estimators. This result shows in particular the significant difference between “high frequency data” and the ergodic framework in stochastic volatility models (compare Genon-Catalot, Jeantheau and Laredo (Bernoulli 4 (1998) 283; Bernoulli 5 (2000) 855; Bernoulli 6 (2000) 1051 and also S ørensen (Prediction-based estimating functions. Technical report, Department of Theoretical Statistics, University of Copenhagen, 1998)).