Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T14:53:40.743Z Has data issue: false hasContentIssue false
  • English
  • Français

Processus ponctuels et martingales: résultats récents sur la modélisation et le filtrage

Published online by Cambridge University Press:  01 July 2016

P. Brémaud  and
J. Jacod
P. Brémaud*
Affiliation:
CEREMADE, Université Paris IX
J. Jacod
Affiliation:
Université de Rennes
*
∗Adresse actuelle: IRIA/LABORIA, Rocquencourt, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper contains a review of the results of the martingale approach to point processes and its applications to the theory of dynamical systems (in the engineering sense), mainly estimation problems.

Some of the topics reviewed are: stochastic intensity, absolutely continuous changes of measure, martingale representations, changes of times, filtering, queues, etc.

Keywords

point processqueuing processmartingalepredictable projectionradon-nikodym derivativestieltjes stochastic integralfilteringstochastic intensity
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 2 , June 1977 , pp. 362 - 416
Copyright
Copyright © Applied Probability Trust 1977 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aalen, O. (1975) Statistical Inference for a Family of Counting Processes. Ph.D. thesis, University of California, Berkeley.Google Scholar
[2] Allain, M. F. (1975) Convergence de processus de Markov de sauts purs vers un processus de diffusion. Seminaire de Rennes 1975. Google Scholar
[3] Balakrishnan, A. V. (1972) A martingale approach to linear recursive state estimation. SIAM J. Control. 10, 754–766.CrossRefGoogle Scholar
[4] Beneš, V. (1963) General Stochastic Processes in the Theory of Queues. Addison-Wesley, Reading, Mass.Google Scholar
[5] Bismut, J. M. (1975) Contrôle des processus de sauts. C.R. Acad. Sci. Paris A 281, 767–770.Google Scholar
[6] Boel, R. (1974) Control of Jump Process. Ph.D. thesis, University of California, Berkeley, Memo ERL–M–448.Google Scholar
[7] Boel, R. and Varaiya, P. (1977) Optimal control of jump processes. SIAM J. Control. À paraître.CrossRefGoogle Scholar
[8] Boel, R., Varaiya, P. and Wong, E. (1975) Martingales on jump processes, Part I: representation results; Part II: applications. SIAM J. Control 13, 999–1061.CrossRefGoogle Scholar
[9] Brémaud, P. (1972) A Martingale Approach to Point Processes. Ph.D. thesis, University of California, Berkeley, Memo ERL–M–345.Google Scholar
[10] Brémaud, P. (1974) The martingale theory of point processes over the real half line. Control Theory, Numerical Methods and Computer System Modelling. Lecture Notes in Economics and Mathematical Systems 107, Springer-Verlag, Berlin, 519–542.Google Scholar
[11] Brémaud, P. (1975) An extension of Watanabe's theorem of characterization of Poisson processes. J. Appl. Prob. 12, 396–399.Google Scholar
[12] Brémaud, P. (1975) Estimation de l'état d'une file d'attente et du temps de panne d'une machine par la méthode de semi-martingales. Adv. Appl. Prob. 7, 845–863.CrossRefGoogle Scholar
[13] Brémaud, P. (1975) La méthode des semi-martingales en filtrage lorsque l'observation est un processus ponctuel marqué. Séminaire de Probabilité. Lecture Notes in Mathematics 511, Springer-Verlag, Berlin, 1–18.Google Scholar
[14] Brémaud, P. (1975) On the information carried by a stochastic point process. Cahiers du CETHEDEC 43, 43–70.Google Scholar
[15] Brémaud, P. (1976) Bang-bang controls of point processes. Adv. Appl. Prob. 8, 385–394.CrossRefGoogle Scholar
[16] Brémaud, P. (1976) Prédiction, filtrage et détection pour une observation mixte: méthode de la probabilité de référence. J. Appl. Math. Optimization. À paraître.Google Scholar
[17] Brémaud, P. (1977) On the output theorem of queuing theory, via filtering (preprint).Google Scholar
[18] Brémaud, P. et van Schuppen, J. (1977) Discrete time stochastic systems. Part I: Stochastic calculus and representations. Part II: Estimation theory. Report SSM 7603 a, b, Washington University, St. Louis.Google Scholar
[19] Brémaud, P. et Yor, M. (1976) Changes of filtration and of probability measures. À paraître.Google Scholar
[20] Brillinger, D. (1976) The identification of point process systems. Ann. Prob. 3, 909–929.Google Scholar
[21] Brown, M. (1971) Discrimination of point processes. Ann. Math. Statist. 42, 773–776.Google Scholar
[22] Bucy, R. et Kalman, R. (1961) New results in linear filtering and prediction theory. Trans. ASME, J. Basic. Eng. 83, 95–108.Google Scholar
[23] Cameron, H. et Martin, W. (1944) Transformation of Wiener integrals under translations. Ann. Math. 45, 386–396.CrossRefGoogle Scholar
[24] Chou, C. S. et Meyer, P. A. (1974) Sur la représentation des martingales comme intégrales stochastiques dans les processus ponctuels. C.R. Acad. Sci. Paris A 278, 1561–1563. Aussi Séminaire de Probabilité VIII. Lecture Notes in Mathematics 381, Springer-Verlag, Berlin.Google Scholar
[25] Courrege, P. (1963) Intégrale stochastique par rapport à une martingale de carré intégrable. Séminaire Brelot–Choquet–Deny. Théorie du potentiel, 7e année. Google Scholar
[26] Courrège, P. et Priouret, P. (1965) Temps d'arrêt d'une fonction aléatoire. Publ. Inst. Stat. Univ. Paris, 245–274.Google Scholar
[27] Cox, D. R. et Lewis, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London.CrossRefGoogle Scholar
[28] Davis, M. H. A. (1973) Detection of signals with point process observation. Research Report 73/8, Dept. of Computing and Control, Imperial College, London.Google Scholar
[29] Davis, M. H. A. (1973) Non-linear filtering with point process observation. Research Report 73/12, Dept. of Computing and Control, Imperial College, London.Google Scholar
[30] Davis, M. H. A. (1976) The representation of martingales of jump processes. SIAM J. Control 14, 623–638.CrossRefGoogle Scholar
[31] Davis, M. H. A. et Elliott, R. J. (1975) Optimal control of a jump process. Report, Dept. of Mathematics, University of Hull.Google Scholar
[32] Davis, M. H. A., Kailath, T. et Segall, A. (1975) Non-linear filtering with counting observations. IEEE Trans. IT–21, 143–150.Google Scholar
[33] Davis, M. H. A. et Varaiya, P. (1973) Dynamic programming conditions for partially observable stochastic systems. SIAM J. Control 11, 226–261.Google Scholar
[34] Dellacherie, C. (1972) Capacités et processus stochastiques. Springer-Verlag, Berlin.Google Scholar
[35] Dellacherie, C. (1974) Intégrales stochastiques par rapport aux processus de Wiener et de Poisson. Séminaire de Probabilité VIII, Lecture Notes in Mathematics 381, 25–26; Séminaire de Probabilité IX, Lecture Notes in Mathematics 465, 494, Springer-Verlag, Berlin.Google Scholar
[36] Doléans-Dade, C. (1970) Quelques applications de la formule de changement de variables pour les semi-martingales. Z. Wahrscheinlichkeitsth. 16, 181–194.Google Scholar
[37] Doléans-Dade, C. et Meyer, P. A. (1970) Intégrales stochastiques par rapport aux martingales locales. Séminaire de Probabilité IV, Lecture Notes in Mathematics 124, Springer-Verlag, Berlin, 77–107.Google Scholar
[38] Dolivo, F. B. (1974) Counting Processes and Integrated Conditional Rates: a Martingale Approach with Application to Detection. Ph.D. thesis, University of Michigan.Google Scholar
[39] Duncan, T. E. (1967) Probability Densities for Diffusion Processes with Application to Non-linear Filtering Theory and Detection Theory. Ph.D. thesis, Stanford University.CrossRefGoogle Scholar
[40] El-Karoui, N. et Lepeltier, J. P. (1975) Représentation des processus ponctuels multivariés à l'aide d'un processus de Poisson. Rapport, Universié du Mans.Google Scholar
[41] Elliott, R. J. (1975) The Lévy system of a point process and martingale representations. Report, Dept. of Mathematics, University of Hull.Google Scholar
[42] Elliott, R. J. (1975) Stochastic integrals for martingales of a jump process. Report, Dept. of Mathematics, University of Hull.Google Scholar
[43] Elliott, R. J. (1975) Martingales of a jump process and absolutely continuous change of measure. Symposium on Stochastic Systems, University of Kentucky.Google Scholar
[44] Fishman, P. et Snyder, D. (1975) How to track a swarm of fireflies by observing their flashes. IEEE Trans. IT–21, 692–694.Google Scholar
[45] Fishman, P. et Snyder, D. (1976) The statistical analysis of space-time point processes. IEEE Trans. IT–22, 257–274.Google Scholar
[46] Fujisaki, M., Kallianpur, G. et Kunita, H. (1972) Stochastic differential equations for the non-linear filtering problem. Osaka J. Math. 9, 19–40.Google Scholar
[47] Galtchouk, L. et Rozovskii, B. (1971) Le problème du désordre pour un processus de Poisson. Theory Prob. Appl. 16, 729–734 (en russe).Google Scholar
[48] Girsanov, I. (1960) On transforming a certain class of stochastic processes by absolutely continuous substitution of measure. Theory Prob. Appl. 5, 285–301.CrossRefGoogle Scholar
[49] Grigelionis, B. (1971) On representation of integer-valued ransom measures by means of stochastic integrals with respect to Poisson measures. Lit. Math. Sb. 11, 93–108.Google Scholar
[50] Grigelionis, B. (1972) Sur les équations stochastiques du filtrage non-linéaire des processus stochastiques. Lit. Math. Sb. 11, 37–50 (en russe).Google Scholar
[51] Grigelionis, B. (1973) On the non-linear filtering theory and absolute continuity of measures corresponding to stochastic processes. Proceedings of the Japan–USSR Symposium, Lecture Notes in Mathematics 330, Springer-Verlag, Berlin, 80–94.Google Scholar
[52] Grigelionis, B. (1974) Sur la représentation des martingales de carré intégrable comme intégrales stochastiques. Lit. Math. Sb. 14, 53–70 (en russe).Google Scholar
[53] Hawkes, A. G. et Oakes, D. (1974) A cluster process representation of a self-exciting point process. J. Appl. Prob. 11, 493–503.Google Scholar
[54] Jacod, J. (1973) On the stochastic intensity of a random point process over the half-line. Technical Report, Dept. of Statistics, Princeton University.Google Scholar
[55] Jacod, J. (1974) Transformation of measures and Radon–Nikodym derivatives for point processes. Technical Report, Dept. of Statistics, Princeton University.Google Scholar
[56] Jacod, J. (1975) Multivariate point processes: predictable projection; Radon–Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.CrossRefGoogle Scholar
[57] Jacod, J. (1976) Un théorème de représentation pour les martingales discontinues. Z. Wahrscheinlichkeitsth. 34, 225–244.Google Scholar
[58] Jacod, J. et Mémin, J. (1976) Caractéristiques locales et conditions de continuité pour les semi-martingales. Z. Wahrscheinlichkeitsth. 35, 1–37.Google Scholar
[59] Itô, K. (1944) Stochastic integral. Proc. Imp. Acad. Tokyo 20, 519–524.Google Scholar
[60] Itô, K. (1970) Poisson point processes attached to Markov processes. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 225–240.Google Scholar
[61] Kabanov, I. (1973) Representation of functionals of Wiener and Poisson processes in the form of stochastic integrals. Theory Prob. Appl. 18, 362–365 (traduction SIAM).Google Scholar
[62] Kabanov, I., Lipčer, R. et Shiryaev, A. (1975) Martingale methods on the theory of point processes. Proc. Steklov Inst. School Seminar, Part II, 269–354 (en russe).Google Scholar
[63] Kailath, T. (1970) The innovations approach to detection and estimation theory Proc. IEEE. 58, 680–695.CrossRefGoogle Scholar
[64] Kailath, T. et Segall, A. (1973) Modelling of signals in discontinuous observations. Proc. 4th Symposium on Non-linear Estimation and Applications, San Diego, Calif. Google Scholar
[65] Kailath, T. et Segall, A. (1975) Radon–Nikodym derivatives with respect to measures induced by discontinuous independent increment processes. Ann. Prob. 3, 449–464.Google Scholar
[66] Kerstan, J. (1962) Teilprozesse Poissonscher Prozesse. Trans. Third Prague Conference, 377–403.Google Scholar
[67] Kingman, J. F. C. (1967) Completely random measures. Pacific J. Math. 21, 59–78.Google Scholar
[68] Kunita, H. (1971) Asymptotic behaviour of the non-linear filtering errors of Markov processes. J. Multivariate Anal. 1, 365–393.Google Scholar
[69] Kunita, H. et Watanabe, S. (1967) On square integrable martingales. Nagoya Math. J. 30, 209–245.Google Scholar
[70] Kurtz, T. G. (1971) Limit theorem for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Prob. 8, 344–356.CrossRefGoogle Scholar
[71] Martins-Netto, , (1975) A Martingale Approach to Queuing Processes. Ph.D. Thesis, University of California, Berkeley, Memo ERL–M 475.Google Scholar
[72] Meyer, P. A. (1965) Probabilités et potentiel. Hermann, Paris.Google Scholar
[73] Meyer, P. A. (1969) Démonstration simplifiée d'un théorème de Knight. Séminaire de Probabilité V, Lecture Notes in Mathematics 191, Springer-Verlag, Berlin, 191–195.Google Scholar
[74] Meyer, P. A. (1975) Un cours sur les intégrales stochastiques. Séminaire de Probabilité X, Lecture Notes in Mathematics 511, Springer-Verlag, Berlin.Google Scholar
[75] Meyer, P. A. (1971) Processus de Poisson ponctuels, d'après K. Itô. Séminaire de Probabilité V, Lecture Notes in Mathematics 191, Springer-Verlag, Berlin, 177–190.Google Scholar
[76] Novikov, A. (1975) On discontinuous martingales. Theory Prob. Appl. 20, 11–26.Google Scholar
[77] Orey, S. (1974) Radon–Nikodym derivatives of probability measures. Report of the Dept. of the Foundations of Math. Sciences, Tokyo U. of Education.Google Scholar
[78] Papangelou, F. (1972) Integrability of expected increments of point processes and a related change of scale. Trans. Amer. Math. Soc. 165, 483–506.Google Scholar
[79] Rishel, R. (1975) A minimum principle for controlled jump processes. Control Theory, Numerical Methods and Computer Systems Modelling. Lecture Notes in Economics and Mathematical Systems 107, Springer-Verlag, Berlin, 493–508.Google Scholar
[80] Rishel, R. (1976) Controls optimal from time t onward and dynamic programming for systems of controlled jump processes. Math. Prog. Study 6, 125–153.Google Scholar
[81] Rishel, R. (1976) Optimal control of a Poisson source. Dans Proc. Joint Automatic Control Conference, 531–535.Google Scholar
[82] Ross, S. (1969) Optimal dispatching of a Poisson process. J. Appl. Prob. 6, 692–699.Google Scholar
[83] Rubin, I. (1972) Regular point processes and their detection. IEEE Trans. IT–18, 547–557.Google Scholar
[84] Rubin, I. (1974) Regular jump processes and their information processing. IEEE Trans. IT–20, 617–624.Google Scholar
[85] Rudemo, M. (1972) Doubly stochastic Poisson processes and process control. Adv. Appl. Prob. 4, 318–338.Google Scholar
[86] Segall, A. (1973) A Martingale Approach to Modelling, Estimation and Detection of Jump Processes. Ph.D. thesis, Stanford University, Technical Report 7050–21, Center for Systems Research.Google Scholar
[87] Segall, A. (1976) Dynamic file assignment in a computer network. IEEE Trans. AC–21, 161–173.Google Scholar
[88] Segall, A. et Sandell, N. (1977) Dynamic file allocation in a computer network. Part II: Random rates of demand. IEEE Trans. AC. À paraître.Google Scholar
[89] Segall, A. et Sandell, N. (1976) Recursive estimation for discrete time point processes with application to ALOHA-type computer systems. IEEE Trans. IT–22.Google Scholar
[90] Skorokhod, A. (1965) Studies in the Theory of Random Processes. Addison-Wesley, Reading, Mass.Google Scholar
[91] Snyder, D. (1972) Filtering and detection for doubly stochastic point processes. IEEE Trans. IT–18, 97–102.Google Scholar
[92] Snyder, D. (1972) Smoothing for doubly stochastic point processes. IEEE Trans. IT–18, 558–562.Google Scholar
[93] Snyder, D. (1973) Information processing for observed jump processes. Inf. and Control 22, 69–78.Google Scholar
[94] Snyder, D. (1975) Random Point Processes. Wiley, New York.Google Scholar
[95] Snyder, D. et Vaca, M. (1975) Estimation and detection for observations derived from martingales; Part I: Representations. Monograph 227. Washington University, St. Louis, Missouri.Google Scholar
[96] Stroock, D. et Varadhan, S. R. S. (1969) Diffusion processes with continuous coefficients I and II. Comm. Pure Appl. Math. 22, 345–400, 479–530.Google Scholar
[97] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
[98] Van Schuppen, J. (1973) Estimation Theory for Continuous Time Processes, a Martingale Approach. Ph.D. thesis, University of California, Berkeley. Memo ERL–M–405.Google Scholar
[99] Van Schuppen, J. (1973) Filtering for counting processes, a martingale approach. 4th Symposium on non-linear estimation and its applications, San Diego.Google Scholar
[100] Van Schuppen, J. (1975) Filtering, prediction and smoothing for counting process observations, a martingale approach. SIAM J. Appl. Math. À paraître.Google Scholar
[101] Van Schuppen, J. et Wong, E. (1974) Translation of local martingales under a change of law. Ann. Prob. 2, 879–888.Google Scholar
[102] Varaiya, P. (1975) The martingale theory of jump processes. IEEE Trans. AC–20, 34–42.Google Scholar
[103] Watanabe, S. (1964) On discontinuous additive functionals and Levy measures of a Markov process. Japanese J. Math. 34, 53–70.Google Scholar
[104] Wong, E. (1973) Recent progress in stochastic processes: a survey. IEEE Trans. IT–19, 262–275.Google Scholar
[105] Wong, E. (1972) Martingale Theory and Applications to Stochastic Problems in Dynamical Systems. Report 72/19 Dept. of Computing and Control, Imperial College London.Google Scholar
[106] Yashin, A. (1970) Filtrage des processus de saut. Automat. i. Telmeh 5, 52–58 (en russe).Google Scholar
[107] Yor, M. (1976) Représentation des martingales de carré intégrable relative aux processus de Wiener et de Poisson à n paramètres. Z. Wahrscheinlichkeitsth. 35, 121–129.Google Scholar
[108] Zakai, M. (1969) On the optimal filtering of diffusion processes. Z. Wahrscheinlichkeitsth. 11, 230–243.Google Scholar
[109] Brémaud, P. (1977) Optimal cancellation of arrivals (preprint).Google Scholar
[110] Martins-Netto, A. F. et Wong, E. (1976) A martingale approach to queues. Math. Prog. Study 6, 97–110.Google Scholar
[111] Kalman, R. E. (1960) A new approach to linear filtering and prediction problems. Trans. ASME, J. Basic Eng. 82, 35–45.Google Scholar
[112] Brémaud, P. et Pietri, J. M. (1977) Martingales and abstract dynamic programming for continuous time stochastic control. Rapport, IRIA.Google Scholar
[113] de Sam Lazaro, J. (1974) Sur les hélices du flot spécial sous une fonction. Z. Wahrscheinlichkeitsth. 30, 279–302.Google Scholar
[114] Davis, M. H. A. (1976) The structure of jump processes and related control problems. Math. Prog. Study 6, 2–14.Google Scholar
[115] Dolivo, F. B. et Beutler, F. J. (1976) Recursive integral equations for the detection of counting processes. J. Appl. Math. Optimization 3, 65–72.Google Scholar
[116] Hoversten, E. V., Rhodes, I. B. et Snyder, D. L. (1977) A separation theorem for stochastic control problems with point process observations. Automatica 13, 85–89.Google Scholar
[117] Fishman, P. M. et Snyder, D. L. (1976) The statistical analysis of space-time point processes. IEEE Trans. IT–22, 257–274.Google Scholar
[118] Snyder, D. L. (1975) Random Point Processes. Wiley, New York.Google Scholar
[119] Robin, M. (1976) Some optimal control problems for queuing systems. Math. Prog. Study 6, 154–169.Google Scholar
[120] Segall, A. (1977) Optimal control of noisy finite-state Markov processes. IEEE Trans. AC–22, 179–186.Google Scholar
[121] Segall, A. (1977) The modeling of adaptive routing in data-communication networks. IEEE Trans. COM 25, 85–95.Google Scholar
[122] Brémaud, P. (1977) Impulse-at-the-jumps control of point processes. Dans Proc. Summer Computer Simulation Conference, Chicago, 18–20 July.Google Scholar
[123] Rishel, R. (1977) Optimality for completely observed controlled jump processes. Report, Dept. of Mathematics, University of Kentucky.Google Scholar
[124] Rishel, R. (1977) State estimation for partially observed jump processes. Report, Dept. of Mathematics, University of Kentucky.Google Scholar
[125] Rebolledo, R. (1977) Remarque sur la convergence en loi des martingales vers des martingales continues. C. R. Acad. Sci. Paris A. À paraître.Google Scholar
[126] Ros-Peran, F. et Segall, A. (1977) Dynamic file assignment in a computer network; Part III: variable number of copies and computer failures. IEEE Trans. COM. À paraître.Google Scholar
[127] Segall, A. (1977) Centralized and decentralized control schemes for Gauss–Poisson processes. Report EE–302, Technion, Haifa.Google Scholar
[128] Rebolledo, R. (1977) Convergence en loi des processus ponctuels et applications à l'inférence statistique d'une famille de processus ponctuels. Rapport, Université de Reims.Google Scholar
[129] Delebecque, F. et Quadrat, J. P. (1977) Identification des caractéristiques locales dans un diffusion avec sauts. Rapport, IRIA/LABORIA.Google Scholar
[130] Bismut, J. M. (1977) Control of jump processes and applications. Bull. Soc. Math. France. A paraître.Google Scholar