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Smoothing and occupation measures of stochastic processes
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 1, pp. 125-156.

This is a review paper about some problems of statistical inference for one-parameter stochastic processes, mainly based upon the observation of a convolution of the path with a non-random kernel. Most of the results are known and presented without proofs. The tools are first and second order approximation theorems of the occupation measure of the path, by means of functionals defined on the smoothed paths. Various classes of stochastic processes are considered starting with the Wiener process, Gaussian processes, continuous semi-martingales and Lévy processes. Some statistical applications are also included in the text.

Cet article est une révision d’un certain nombre de problèmes statistiques concernant les processus aléatoires à un paramètre continu. En général, on suppose que l’observable est une régularisation de la trajectoire du processus, obtenue par convolution avec un noyau détérministe. La plupart des résultats ici exposés est connue et presentée sans démonstration. Les énoncés des théorèmes contiennent des approximations de la mesure d’occupation, au premier et deuxième ordre, basées sur des fonctionnelles définies sur les régularisées des trajectoires. On considère diverses classes de processus, à savoir, le processus de Wiener, les processus gaussiens, les semi-martingales continues et les processus de Lévy. Nous avons inclus les détails de certaines applications statistiques.

DOI: 10.5802/afst.1116
Mario Wschebor 1

1 Centro de Matemática, Facultad de Ciencias, Universidad de la República, Calle Iguá 4225. 11400, Montevideo (Uruguay).
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Mario Wschebor. Smoothing and occupation measures of stochastic processes. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 1, pp. 125-156. doi : 10.5802/afst.1116. https://afst.centre-mersenne.org/articles/10.5802/afst.1116/

[A-F] J.-M. Azaïs; D. Florens-Szmirou Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires, Probab. Th. Rel. Fields, Volume 76 (1987), pp. 121-132 | MR | Zbl

[A-W1] J.-M. Azaïs; M. Wschebor Almost sure oscillation of certain random processes, Bernoulli, Volume 2 (1996) no. 3, pp. 257-270 | Numdam | MR | Zbl

[A-W2] J.-M. Azaïs; M. Wschebor; J. Azéma; M. Emery; M. Yor Oscillation presque sûre de martingales continues, Séminaires de Probabilités XXXI (Lecture Notes Math.), Volume 1655, Springer-Verlag, 1997, pp. 69-76 | MR | Zbl

[A1] J.-M. Azaïs Conditions for convergence of number of crossings to the local time, Applications to stable processes with independent increments and to Gaussian processes, Probab. Math. Stat., Volume 11 (1990) no. 1, pp. 19-36 | MR | Zbl

[A2] J.-M. Azaïs Approximation des trajectoires et temps local des diffusions, Ann. Inst. H. Poincaré, B, Volume 25 (1989) no. 2, pp. 175-194 | Numdam | MR | Zbl

[B] P. Brugière Estimation de la variance d’un processus de diffusion dans le cas multidimensionel, Comptes R. Acad. Sc. Paris, Sér. I, Volume 312 (1991), pp. 999-1004 | MR | Zbl

[B-I] A. N. Borodin; I. A. Ibragimov Limit theorems for functionals of random walks, Proc. Steklov Institute Math., AMS, Providence, RI, 1995 | MR | Zbl

[B-L-O] C. Berzin; J.R. Leon; J. Ortega Level crossings and local time for regularized Gaussian processes, Probab. Math. Statist, Volume 18 (1998) no. 1, pp. 39-81 | MR | Zbl

[B-L1] C. Berzin; J.R. Leon Weak convergence of the integrated number of level crossings to the local time of the Wiener process, Comptes R. Acad. Sc. Paris, Sér. I, Volume 319 (1994), pp. 1311-1316 | MR | Zbl

[B-W] C. Berzin; M. Wschebor Approximation du temps local des surfaces gaussiennes, Probab. Th. Rel. Fields, Volume 96 (1993), pp. 1-32 | MR | Zbl

[C-R1] M. Csörgö; P. Révész Three strong approximations of the local time of a Wiener process and their applications to invariance, Limit Theorems in Probability and Statistics, Vol. I, II (Veszprém, 1982) (Coll. Math. Soc. J. Bolyai), Volume 36, North-Holland, Amsterdam, 1984, pp. 223-254 | MR | Zbl

[C-R2] M. Csörgö; P. Révész On strong invariance for local time of partial sums, Stoch. Proc. Appl., Volume 20 (1985), pp. 59-84 | MR | Zbl

[D-F] D. Dacunha-Castelle; D. Florens-Zmirou Estimation of the coefficient of a diffusion from discrete observations, Stochastics, Volume 19 (1986), pp. 263-284 | MR | Zbl

[F] D. Florens-Zmirou On estimating the diffusion coefficient from discrete observations, J. Appl. Prob., Volume 30 (1993), pp. 790-804 | MR | Zbl

[F-T] B. Fristedt; S.J. Taylor Constructions of local time for a Markov process, Z. Wahr.verw. gebiete, Volume 62 (1983), pp. 73-112 | MR | Zbl

[G-J] V. Génon-Catalot; J. Jacod On the estimation of the diffusion coefficient for multidimensional diffusion processes, Ann. Inst. H. Poincaré, Prob. Stat., Volume 29 (1993), pp. 119-151 | EuDML | Numdam | MR | Zbl

[G-J-L] V. Génon-Catalot; T. Jeantheau; C. Laredo Limit theorems for discretely observed stochastic volatility models, Bernoulli, Volume 4 (1998) no. 3, pp. 283-304 | Zbl

[G-S] I. Guikhman; A. Skorokhod Introduction à la théorie des processus aléatoires, MIR, Moscow, 1980 | Numdam | MR | Zbl

[H] M. Hoffmann L p estimation of the diffusion coefficient, Bernoulli, Volume 5 (1999) no. 3, pp. 447-481 | MR | Zbl

[I-M] K. Itô; H.P. Mc Kean Diffusion processes and their sample paths, Academic Press, 1965 | MR | Zbl

[I-W] N. Ikeda; S. Watanabe Stochastic Differential Equations and Diffusion Processes, North Holland, 1982 | MR | Zbl

[J] J. Jacod Rates of convergence to the local time of a diffusion, Ann. Inst. H. Poincaré, Prob. Stat., Volume 34 (1998), pp. 505-544 | EuDML | Numdam | MR | Zbl

[J+] J. Jacod Non-parametric kernel estimation of the diffusion coefficient of a diffusion, Scand. J. Statist., Volume 27 (2000) no. 1, pp. 83-96 | Numdam | MR | Zbl

[K-S] I. Karatzas Brownian motion and stochastic calculus, Springer-Verlag, 1998 | MR | Zbl

[L-S] R.S. Lipster; A.N. Shiryaev Statistics of Random Processes, Vol. I, II. 2d ed., Springer-Verlag, 2001 | Zbl

[M-W1] E. Mordecki; M. Wschebor Smoothing of paths and weak approximation of the occupation measure of Lévy processes (2005) (Pub. Mat. Uruguay, to appear) | Zbl

[M-W2] E. Mordecki; M. Wschebor Approximation of the occupation measure of Lévy processes, Comptes Rendus de l’Académie des Sciences, Paris, Sér. I, Volume 340 (2005), pp. 605-610 | MR | Zbl

[N-W] D. Nualart; M. Wschebor Intégration par parties dans l’espace de Wiener et approximation du temps local, Probab. Th. Rel. Fields, Volume 90 (1991), pp. 83-109 | MR | Zbl

[P-W1] G. Perera; M. Wschebor Crossings and occupation measures for a class of semimartingales, Ann. Probab., Volume 26 (1998) no. 1, pp. 253-266 | MR | Zbl

[P-W2] G. Perera; M. Wschebor Inference on the Variance and Smoothing of the Paths of Diffusions, Ann. Inst. H. Poincaré, Volume 38 (2002) no. 6, pp. 1009-1022 | EuDML | MR | Zbl

[PR] B.L.S Prakasa Rao Semimartingales and their Statistical Inference, Chapman & Hall, 1999 | Numdam | MR | Zbl

[R] P. Révész Local time and invariance, Lecture Notes in Math. (1981) no. 861, pp. 128-145 | MR | Zbl

[W1] M. Wschebor Régularisation des trajectoires et approximation du temps local, C.R. Acad. Sci. Paris, Sér. I (1984), pp. 209-212 | MR | Zbl

[W2] M. Wschebor Surfaces aléatoires. Mesure géométrique des ensembles de niveau, Lecture Notes Math., 1147, Springer-Verlag, Berlin, 1985 | MR | Zbl

[W3] M. Wschebor Crossings and local times of one-dimensional diffusions, Pub. Mat. Uruguay (1990) no. 3, pp. 69-100 | MR | Zbl

[W4] M. Wschebor Sur les accroissements du processus de Wiener, C. R. Acad. Sci. Paris, Sér. I, Volume 315 (1992), pp. 1293-1296 | Zbl

[W5] M. Wschebor Almost sure weak convergence of the increments of Lévy processes,, Stoch. Proc. Appl., Volume 55 (1995), pp. 253-270 | MR | Zbl

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