Wschebor, Mario
Smoothing and occupation measures of stochastic processes
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6 : Tome 15 (2006) no. 1 , p. 125-156
MR 2225750 | Zbl 1121.62072 | 1 citation dans Numdam
doi : 10.5802/afst.1116
URL stable : http://www.numdam.org/item?id=AFST_2006_6_15_1_125_0

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.
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.

Bibliographie

[A-F] Azaïs, J.-M.; Florens-Szmirou, D. Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires, Probab. Th. Rel. Fields, 76 (1987), p. 121-132 MR 1096937 | Zbl 0744.60091

[A-W1] Azaïs, J.-M.; Wschebor, M. Almost sure oscillation of certain random processes, Bernoulli, 2 (1996) no. 3, p. 257-270 Numdam | MR 1001025 | Zbl 0674.60032

[A-W2] Azaïs, J.-M.; Wschebor, M.; Azéma, J.; Emery, M.; Yor, M. Oscillation presque sûre de martingales continues, Séminaires de Probabilités XXXI, Springer-Verlag (Lecture Notes Math.) 1655 (1997), p. 69-76 MR 899448 | Zbl 0608.60034

[A1] Azaïs, J.-M. 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., 11 (1990) no. 1, p. 19-36 MR 1416866 | Zbl 0885.60018

[A2] Azaïs, J.-M. Approximation des trajectoires et temps local des diffusions, Ann. Inst. H. Poincaré, B, 25 (1989) no. 2, p. 175-194 Numdam | MR 1478717 | Zbl 0882.60018

[B] Brugière, P. Estimation de la variance d’un processus de diffusion dans le cas multidimensionel, Comptes R. Acad. Sc. Paris, Sér. I, 312 (1991), p. 999-1004 MR 1310678 | Zbl 0812.60069

[B-I] Borodin, A. N.; Ibragimov, I. A. Limit theorems for functionals of random walks, Proc. Steklov Institute Math., AMS, Providence, RI (1995) MR 1644037 | Zbl 0985.60035

[B-L-O] Berzin, C.; Leon, J.R.; Ortega, J. Level crossings and local time for regularized Gaussian processes, Probab. Math. Statist, 18 (1998) no. 1, p. 39-81 MR 1222362 | Zbl 0794.60030

[B-L1] Berzin, C.; Leon, J.R. Weak convergence of the integrated number of level crossings to the local time of the Wiener process, Comptes R. Acad. Sc. Paris, Sér. I, 319 (1994), p. 1311-1316 MR 1368394 | Zbl 0855.60001

[B-W] Berzin, C.; Wschebor, M. Approximation du temps local des surfaces gaussiennes, Probab. Th. Rel. Fields, 96 (1993), p. 1-32 Zbl 0794.60030 | MR 1113093 | Zbl 0751.62036

[C-R1] Csörgö, M.; Révész, P. 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), North-Holland, Amsterdam (Coll. Math. Soc. J. Bolyai) 36 (1984), p. 223-254 MR 807563 | Zbl 0567.60075

[C-R2] Csörgö, M.; Révész, P. On strong invariance for local time of partial sums, Stoch. Proc. Appl., 20 (1985), p. 59-84 MR 805116 | Zbl 0582.60073

[D-F] Dacunha-Castelle, D.; Florens-Zmirou, D. Estimation of the coefficient of a diffusion from discrete observations, Stochastics, 19 (1986), p. 263-284 MR 872464 | Zbl 0626.62085

[F] Florens-Zmirou, D. On estimating the diffusion coefficient from discrete observations, J. Appl. Prob., 30 (1993), p. 790-804 Zbl 0796.62070 | MR 684210 | Zbl 0519.60078

[F-T] Fristedt, B.; Taylor, S.J. Constructions of local time for a Markov process, Z. Wahr.verw. gebiete, 62 (1983), p. 73-112 Zbl 0519.60078 | MR 1242012 | Zbl 0796.62070

[G-J] Génon-Catalot, V.; Jacod, J. On the estimation of the diffusion coefficient for multidimensional diffusion processes, Ann. Inst. H. Poincaré, Prob. Stat., 29 (1993), p. 119-151 | Numdam | MR 1204521 | Zbl 0770.62070

[G-J-L] Génon-Catalot, V.; Jeantheau, T.; Laredo, C. Limit theorems for discretely observed stochastic volatility models, Bernoulli, 4 (1998) no. 3, p. 283-304 Zbl 0916.60075 | Zbl 0573.60003

[G-S] Guikhman, I.; Skorokhod, A. Introduction à la théorie des processus aléatoires, MIR, Moscow (1980) Zbl 0573.60003 | Numdam | MR 1204521 | Zbl 0770.62070

[H] Hoffmann, M. L p estimation of the diffusion coefficient, Bernoulli, 5 (1999) no. 3, p. 447-481 Zbl 0980.62073 | MR 1653264 | Zbl 0916.60075

[I-M] Itô, K.; Mc Kean, H.P. Diffusion processes and their sample paths, Academic Press (1965) Zbl 0127.09503 | MR 1693608 | Zbl 0980.62073

[I-W] Ikeda, N.; Watanabe, S. Stochastic Differential Equations and Diffusion Processes, North Holland (1982) MR 1011252 | Zbl 0495.60005

[J] Jacod, J. Rates of convergence to the local time of a diffusion, Ann. Inst. H. Poincaré, Prob. Stat., 34 (1998), p. 505-544 Zbl 0911.60055 | | Numdam | MR 199891

[J+] Jacod, J. Non-parametric kernel estimation of the diffusion coefficient of a diffusion, Scand. J. Statist., 27 (2000) no. 1, p. 83-96 Zbl 0938.62085 | Numdam | MR 1632849 | Zbl 0911.60055

[K-S] Karatzas, I. Brownian motion and stochastic calculus, Springer-Verlag (1998) MR 1774045 | Zbl 0938.62085

[L-S] Lipster, R.S.; Shiryaev, A.N. Statistics of Random Processes, Vol. I, II. 2d ed., Springer-Verlag (2001) Zbl 0638.60065

[M-W1] Mordecki, E.; Wschebor, M. Smoothing of paths and weak approximation of the occupation measure of Lévy processes (2005) (Pub. Mat. Uruguay, to appear) Zbl 1065.60046 | Zbl 1008.62072

[M-W2] Mordecki, E.; Wschebor, M. Approximation of the occupation measure of Lévy processes, Comptes Rendus de l’Académie des Sciences, Paris, Sér. I, 340 (2005), p. 605-610 MR 2138712 | Zbl 1065.60046

[N-W] Nualart, D.; Wschebor, M. Intégration par parties dans l’espace de Wiener et approximation du temps local, Probab. Th. Rel. Fields, 90 (1991), p. 83-109 Zbl 0727.60052 | MR 2138712 | Zbl 1065.60046

[P-W1] Perera, G.; Wschebor, M. Crossings and occupation measures for a class of semimartingales, Ann. Probab., 26 (1998) no. 1, p. 253-266 Zbl 0943.60019 | MR 1124830 | Zbl 0727.60052

[P-W2] Perera, G.; Wschebor, M. Inference on the Variance and Smoothing of the Paths of Diffusions, Ann. Inst. H. Poincaré, 38 (2002) no. 6, p. 1009-1022 Zbl 1011.62083 | | MR 1617048 | Zbl 0943.60019

[PR] Prakasa Rao, B.L.S Semimartingales and their Statistical Inference, Chapman & Hall (1999) Zbl 0960.62090 | Numdam | MR 1955349 | Zbl 1011.62083

[R] Révész, P. Local time and invariance, Lecture Notes in Math. (1981) no. 861, p. 128-145 Zbl 0456.60029 | MR 1689166 | Zbl 0960.62090

[W1] Wschebor, M. Régularisation des trajectoires et approximation du temps local, C.R. Acad. Sci. Paris, Sér. I (1984), p. 209-212 Zbl 0584.60086 | MR 655268 | Zbl 0456.60029

[W2] Wschebor, M. Surfaces aléatoires. Mesure géométrique des ensembles de niveau, Springer-Verlag, Berlin, Lecture Notes Math., 1147 (1985) Zbl 0573.60017 | MR 741097 | Zbl 0584.60086

[W3] Wschebor, M. Crossings and local times of one-dimensional diffusions, Pub. Mat. Uruguay (1990) no. 3, p. 69-100 MR 871689 | Zbl 0573.60017

[W4] Wschebor, M. Sur les accroissements du processus de Wiener, C. R. Acad. Sci. Paris, Sér. I, 315 (1992), p. 1293-1296 Zbl 0770.60075

[W5] Wschebor, M. Almost sure weak convergence of the increments of Lévy processes,, Stoch. Proc. Appl., 55 (1995), p. 253-270 Zbl 0813.60069 | MR 1194538 | Zbl 0770.60075