Rate of convergence for parametric estimation in a stochastic volatility model

We consider the following hidden Markov chain problem: estimate the finite-dimensional parameter θ in the equation vt=v0+∫0tσ(θ, vs) dWs+drift, when we observe discrete data Xi/n at times i=0,…,n from the diffusion Xt=x0+∫0tvs dBs+drift. The processes (Wt)t∈[0,1] and (Bt)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. ve 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 Sørensen (Prediction-based estimating functions. Technical report, Department of Theoretical Statistics, University of Copenhagen, 1998)).