Estimation of the diffusion coefficient for diffusion processes: random sampling

Abstract

We consider a diffusion process X with values in R'd, whose coefficients are smooth enough, and the diffusion coefficient is non-degenerate and depends on an unknown real parameter S. We are allowed to observe the path of X at n times only, and we study here samplings, that is sampling schemes such that the ith sampling time may depend on the previous i - 1 observations. We prove first the LAMN property as n goes to infinity, for large classes of sequences of such random sampling schemes. Second, we exhibit a sequence of random sampling schemes and associated estimators S. for S, such that nI(?n - 9) is asymptotically mixed normal, with an asymptotic conditional variance achieving the optimal (over all possible random sampling schemes) bound of the LAMN property simultaneously for all .9.

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