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Estimation for stochastic differential equations with a small diffusion coefficient. (English) Zbl 1157.62055

Summary: We consider a multidimensional diffusion \(X\) with drift coefficient \(b(X_t,\alpha )\) and diffusion coefficient \(\varepsilon a(X_t,\beta )\) where \(\alpha \) and \(\beta \) are two unknown parameters while \(\varepsilon \) is known. For a high frequency sample of observations of the diffusion at the time points \(k/n, k=1,\dots ,n\), we propose a class of contrast functions and thus obtain estimators of \((\alpha ,\beta )\). The estimators are shown to be consistent and asymptotically normal when \(n\rightarrow \infty \) and \(\varepsilon \rightarrow 0\) in such a way that \(\varepsilon ^{ - 1}n ^{- \rho} \) remains bounded for some \(\rho >0\). The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F12 Asymptotic properties of parametric estimators
65C60 Computational problems in statistics (MSC2010)
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