On the discrete approximation of occupation time of diffusion processes

Abstract

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 ∫₀^tI_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 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_t0,∞) for some t₀∈(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.