MPS-RR 2002-4
March 2002

Inference for observations of integrated diffusion processes

Michael Sørensen

Susanne Ditlevsen

Abstract

Estimation of parameters in diffusion models is usually based on observations of the process at discrete time points. Here we investigate estimation when a sample of discrete observations is not available, but, instead, observations of a running integral of the process with respect to some weight function. This type of observations is, for example, obtained when a realization of the process is observed after passage through an electronic filter. Another example is provided by the ice-core data on oxygen isotopes used to investigate paleo-temperatures. Finally, such data play a role in connection with the stochastic volatility models of finance. The integrated process is no longer a Markov process which render the use of martingale estimating functions difficult. Therefore, a generalization of the martingale estimating functions, namely the prediction-based estimating functions, is applied to estimate parameters in the underlying diffusion process. The estimators are shown to be consistent and asymptotically normal. The method is applied to inference based on integrated data from Ornstein-Uhlenbeck processes and from the CIR-model for both of which an explicit estimating function can be found.

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