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.


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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[1] Azaïs, J. M. (1989) Approximation des trajectoires et temps local des diffusions, Ann. Ins. H. Poincaré Probab. Statist. 25 (2) 175-194. · Zbl 0674.60032
[2] Borodin, A. N. and Salminen, P. (1996), Handbook of Brownian motion - facts and formulae. Birkhäuser. · Zbl 0859.60001
[3] Fujita, T. and Miura, R. (2002) Edokko options: A new framework of barrier options., Asia-Pacific Financial Markets 9 (2) 141-151. · Zbl 1056.91030 · doi:10.1023/A:1022294204470
[4] Fournier, N. and Printems, J. (2010) Absolute continuity for some one-dimensional processes., Bernoulli 16 (2) 343-360. · Zbl 1248.60062 · doi:10.3150/09-BEJ215
[5] Karatzas, I. and Shreve, S. E. (1991), Brownian motion and stochastic calculus. Second edition. Springer. · Zbl 0734.60060
[6] Jacod, J. (1998) Rates of convergence to the local time of a diffusion., Ann. Inst. Henri Poincaré Probab. Statist., 34 (4) 505-544. · Zbl 0911.60055 · doi:10.1016/S0246-0203(98)80026-5
[7] Jacod, J. (2008) Asymptotic properties of realized power variations and related functionals of semimartingales., Stoch. Proc. Appl., 118 (4) 517-559. · Zbl 1142.60022 · doi:10.1016/j.spa.2007.05.005
[8] Jacod, J. and Shiryaev, A. N. (2003), Limit theorems for stochastic processes. Second edition. Springer-Verlag: Berlin. · Zbl 1018.60002
[9] Labrador, B. (2009) Rates of strong uniform convergence of the, k T -occupation time density estimator. Stat. Inference Stoch. Process. 12 (3) 269-283 · Zbl 1205.62041 · doi:10.1007/s11203-009-9034-y
[10] Miura, R. (2006) Rank process, stochastic corridor and application to finance., Advances in statistical modeling and inference series in biostatistics . 3 529-542.
[11] Nualart, D., Malliavin calculus and its applications. CBMS Regional Conference Series in Mathematics, 110. · Zbl 1198.60006
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