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On the discrete approximation of occupation time of diffusion processes. (English) Zbl 1271.60086

Summary: Let \(X\) be a one-dimensional diffusion process. We study a simple class of estimators, which rely only on one sample data \(\{X_{\frac{i}{n}},\, 0\leq i\leq nt\}\), for the occupation time \(\int _{0}^{t}\operatorname{1}_{A}(X_{s})ds\) of process \(X\) in some set \(A\). The main concern of this paper is the rates of convergence of the estimators. First, we consider the case that \(A\) is a finite union of some intervals in \(\mathbb R\), then we show that the estimator converges at rate \(n^{-3/4}\). Second, we consider the so-called stochastic corridor in mathematical finance. More precisely, we let \(A\) be a stochastic interval, say \([X_{t_{0}},\infty)\) for some \(t_{0}\in (0,t)\), then we show that the estimator converges at rate \(n^{-1/2}\). Some discussions about the exactness of the rates are also presented.

MSC:

60J60 Diffusion processes
62M05 Markov processes: estimation; hidden Markov models
60J55 Local time and additive functionals
60F17 Functional limit theorems; invariance principles
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References:

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