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
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.
@article{AFST_2006_6_15_1_125_0,
     author = {Wschebor, Mario},
     title = {Smoothing and occupation measures of stochastic processes},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 15},
     number = {1},
     year = {2006},
     pages = {125-156},
     doi = {10.5802/afst.1116},
     mrnumber = {2225750},
     zbl = {1121.62072},
     language = {en},
     url = {http://http://www.numdam.org/item/AFST_2006_6_15_1_125_0}
}
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, pp. 125-156. doi : 10.5802/afst.1116. http://www.numdam.org/item/AFST_2006_6_15_1_125_0/

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

[A-W1] Azaïs, J.-M.; Wschebor, M. Almost sure oscillation of certain random processes, Bernoulli, Tome 2 (1996) no. 3, pp. 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.) Tome 1655 (1997), pp. 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., Tome 11 (1990) no. 1, pp. 19-36 | MR 1416866 | Zbl 0885.60018

[A2] Azaïs, J.-M. Approximation des trajectoires et temps local des diffusions, Ann. Inst. H. Poincaré, B, Tome 25 (1989) no. 2, pp. 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, Tome 312 (1991), pp. 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, Tome 18 (1998) no. 1, pp. 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, Tome 319 (1994), pp. 1311-1316 | MR 1368394 | Zbl 0855.60001

[B-W] Berzin, C.; Wschebor, M. Approximation du temps local des surfaces gaussiennes, Probab. Th. Rel. Fields, Tome 96 (1993), pp. 1-32 | MR 1113093 | Zbl 0794.60030 | 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) Tome 36 (1984), pp. 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., Tome 20 (1985), pp. 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, Tome 19 (1986), pp. 263-284 | MR 872464 | Zbl 0626.62085

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

[F-T] Fristedt, B.; Taylor, S.J. Constructions of local time for a Markov process, Z. Wahr.verw. gebiete, Tome 62 (1983), pp. 73-112 | MR 1242012 | Zbl 0519.60078 | 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., Tome 29 (1993), pp. 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, Tome 4 (1998) no. 3, pp. 283-304 | Zbl 0916.60075 | Zbl 0573.60003

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

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

[I-M] Itô, K.; Mc Kean, H.P. Diffusion processes and their sample paths, Academic Press (1965) | MR 1693608 | Zbl 0127.09503 | 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., Tome 34 (1998), pp. 505-544 | Numdam | MR 199891 | Zbl 0911.60055

[J+] Jacod, J. Non-parametric kernel estimation of the diffusion coefficient of a diffusion, Scand. J. Statist., Tome 27 (2000) no. 1, pp. 83-96 | Numdam | MR 1632849 | Zbl 0938.62085 | 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, Tome 340 (2005), pp. 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, Tome 90 (1991), pp. 83-109 | MR 2138712 | Zbl 0727.60052 | Zbl 1065.60046

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

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

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

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

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

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

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

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

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