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Estimation for stochastic differential equations with a small diffusion coefficient

Abstract : We consider a multidimensional diffusion X with drift coefficient b(X(1), alpha) and diffusion coefficient epsilon a(X(1), beta) where alpha and beta are two unknown parameters, while epsilon is known. For a high frequency sample of observations of the diffusion at the time points k/n, k = 1, ... , 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 -> infinity and epsilon -> 0 in such a way that epsilon(-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.
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Arnaud Gloter, Michael S. Sorensen. Estimation for stochastic differential equations with a small diffusion coefficient. Stochastic Processes and their Applications, Elsevier, 2009, 119 (3), pp.679--699. ⟨10.1016/⟩. ⟨hal-00693054⟩



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