Electronic Journal of Statistics

On the discrete approximation of occupation time of diffusion processes

Hoang-Long Ngo and Shigeyoshi Ogawa

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Let X be a 1-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 0tIA(Xs)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 ℝ, then we show that the estimator converges at rate n3/4. Second, we consider the so-called stochastic corridor in mathematical finance. More precisely, we let A be a stochastic interval, say [Xt0,) for some t0(0,t), then we show that the estimator converges at rate n1/2. Some discussions about the exactness of the rates are also presented.

Article information

Electron. J. Statist., Volume 5 (2011), 1374-1393.

First available in Project Euclid: 19 October 2011

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F55 60J60: Diffusion processes [See also 58J65]
Secondary: 60J55: Local time and additive functionals

Diffusion discrete approximation occupation time local time stable convergence tightness


Ngo, Hoang-Long; Ogawa, Shigeyoshi. On the discrete approximation of occupation time of diffusion processes. Electron. J. Statist. 5 (2011), 1374--1393. doi:10.1214/11-EJS645. https://projecteuclid.org/euclid.ejs/1319028572

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