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A class of minimum-distance estimators for diffusion processes with ergodic properties. (English) Zbl 0921.62101

Authors’ abstract: Suppose one observes a path of a stochastic process \(X=(X_t)_{t\geq 0}\) which is known to solve an equation of the form \[ d X^\theta_t= S(\theta,X^\theta_t) dt+dW_t,\quad t\geq 0,\quad \theta\in\Theta \subset\mathbb{R}^d, \] with a given functional \(S\) and given initial condition \(X_0\), where \(\Theta\) is a non-empty bounded open subset of \(\mathbb{R}^d\). In order to estimate the true but unknown parameter \(\theta_0\) the paper proposes a new class of minimum distance estimators (MDE) \(\theta_T\), which are strongly consistent and (in case \(d=1)\) asymptotically normal, provided the observed process has an ergodic property. In particular, in the generalized “signal plus noise” case \(S(\theta,x)= S_1(\theta)+ S_2(x)\), one obtains efficient estimators.

MSC:

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