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Local asymptotic mixed normality property for elliptic diffusion: A Malliavin calculus approach. (English) Zbl 1003.60057
The author considers an \({\mathbb R}^d\)-valued diffusion process \[ X_t^\theta = x + \int_0^t b(\theta , s, X_s^\theta) ds + \int_0^t S(\theta , s, X^\theta_s) dB_s, \] whose law is denoted by \(P^\theta\), where \((B_t)_{t\in [0,1]}\) is a \(d\)-dimensional Brownian motion and \(\theta\) is a scalar parameter that belongs to an open interval \(\Theta\) of \({\mathbb R}\). Under regularity assumptions on \(b\) and \(S\) he proves the local asymptotic mixed normality property of the sequence \((({\mathbb R}^d)^n,{\mathcal F}_n, (P^\theta_n)_{\theta \in \Theta})_{n\geq 1}\), where \(P^\theta_n\) is the restriction of \(P^\theta\) to \({\mathcal F}_n = \sigma (X_{k/n}: 0\leq k \leq n)\). The method is to represent the log-likelihood ratio \(Z_n\) as a sum of conditional expectations of Skorokhod integrals via a Malliavin calculus integration by parts formula, and to conclude using a central limit theorem for triangular arrays of random variables. The result is new in the multidimensional case. The one-dimensional case has been considered by G. Dohnal [J. Appl. Probab. 24, 105-114 (1987; Zbl 0615.62109)] using an expansion of the transition probability \(p^\theta\) of \(X\).

MSC:
60H07 Stochastic calculus of variations and the Malliavin calculus
60J60 Diffusion processes
62F12 Asymptotic properties of parametric estimators
62M05 Markov processes: estimation; hidden Markov models
62E20 Asymptotic distribution theory in statistics
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