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Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-RR 2000-21
May 2000

Optimality and small $\Delta -$optimality of martingale estimating functions


Martin Jacobsen


Martingale estimating functions determined from a given collection (the base) of conditional expectations are considered for estimating the parameters of a discretely observed diffusion. It is discussed how to make the martingale estimating functions small $\Delta-$optimal, i.e. nearly efficient when the observations are close together, in particular it is shown that this is possible provided the base is large enough. It is also shown that the optimal martingale estimating function with a given base, is automatically small $\Delta-$optimal, provided only that the base is sufficiently large. In both cases the critical dimension of the base is the same and determined by the dimension of the diffusion, and on whether the squared diffusion matrix is parameter dependent or not; the critical number does not depend however on the dimension of the parameter.

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This paper has now been published in Bernoulli 8, 643-668 (2002)