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Maximum contrast estimation for diffusion processes from discrete observations. (English) Zbl 0721.62082

The author considers an one-dimensional diffusion process \(X=(X_ t\), \(t\geq 0)\) defined as the solution of the stochastic differential equation \[ dX_ t=b(\theta,X_ t)dt+\epsilon dW_ t,\quad t\geq 0,\quad X_ 0=x. \] Here W is a standard Wiener process, \(\epsilon\) is a positive “small” parameter, b(\(\theta\),x) is a given nonlinear function and \(\theta\) is an unknown parameter, \(\theta \in \Theta \subset {\mathbb{R}}^ k\). The problem is to find the maximum contrast estimators (m.c.e.) of \(\theta\) on the basis of the observations of X at times \(\Delta\),2\(\Delta\),..., i.e. by using the so-called equally spaced data.
Some necessary definitions and preliminary statements are given. The relationship between the m.c.e. and the m.l.e. is studied when simultaneously \(\epsilon\) and \(\Delta\) tend to zero. It is quite interesting that the m.c.e., whose existence is established, obeys different asymptotic behavior as \(\epsilon\to 0\) and \(\Delta\to 0\), and properties like consistency and asymptotic normality primarily depend on the connection between \(\epsilon\) and \(\Delta\). All the statements are proved in detail and illustrated well by nice examples. Related topics are also discussed.
Let me note that the present paper as well as papers of other French stochasticians (to mention A. Le Breton, R. Azencott, D. Dancunha- Castelle, M. Duflo, D. Florens-Zmirou, C. Denieu, C. Laredo) are an essential contribution to the statistics of stochastic processes.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F12 Asymptotic properties of parametric estimators
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