Tangential existence and comparison, with applications to single and multiple integration

• Autorzy: Kallenberg O.

Identyfikatory

DOI

10.19195/0208-4147.37.1.2

• Języki publikacji: EN

Abstrakty

EN

Two semi-martingales with respect to a common filtration are said to be tangential if they have the same local characteristics. When the latter are non-random, the underlying semi-martingale is known to have independent increments. We show that every semi-martingale has a tangential process with conditionally independent increments. We also extend the Zinn-Hitchenko and related tangential comparison theorems to continuous time. Combining those results, we obtain some surprisingly general existence, convergence, and tightness criteria for broad classes of single and multiple stochastic integrals.

Słowa kluczowe

EN

tangential processes; local characteristics; conditional symmetry; marked point processes; compensators; stochastic integration; multiple stochastic integrals

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2017
• Tom: Vol. 37, Fasc. 1
• Strony: 21--52
• Opis fizyczny: Bibliogr. 27 poz.

Twórcy

autor

Kallenberg O.

• Auburn University, Auburn, Alabama 36849, USA, 221 Parker Hall

Bibliografia

• [1] D. L. Burkholder, B. J. Davis, and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, Univ. of California Press, 1972, pp. 223-240.
• [2] B. Grigelionis, Martingale characterization of stochastic processes with independent increments, Litovsk. Mat. Sb. 17 (1977), pp. 75-86.
• [3] P. Hitchenko, Comparison of moments for tangent sequences of random variables, Probab. Theory Related Fields 78 (1988), pp. 223-230.
• [4] K. Itô, Differential equations determining a Markov process, J. Pan-Japan Math. Coll. 1077 (1942).
• [5] K. Itô, Multiple Wiener integral, J. Math. Soc. Japan 3 (1951), pp. 157-169.
• [6] J. Jacod, Multivariate point processes: Predictable projection, Radon-Nikodym derivative, representation of martingales, Z. Wahrsch. Verw. Gebiete 31 (1975), pp. 235-253.
• [7] J. Jacod, Une généralisation des semimartingales: Les processus admettant un processus à accroissements indépendants tangent, in: Séminaire de Probabilités XVIII, Lecture Notes in Math., Vol. 1059, Springer, Berlin 1984, pp. 91-118.
• [8] J. Jacod and H. Sadi, Processus admettant un processus `a accroissements indépendants tangent: cas général, in: Séminaire de Probabilités XXI, Lecture Notes in Math., Vol. 1247, Springer, Berlin 1987, pp. 479-514.
• [9] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer, Berlin 1987.
• [10] A. Jakubowski, Principle of conditioning in limit theorems for sums of random variables, Ann. Probab. 14 (1986), pp. 902-915.
• [11] O. Kallenberg, Some uses of point processes in multiple stochastic integration, Math. Nachr. 151 (1991), pp. 7-31.
• [12] O. Kallenberg, Foundations of Modern Probability, second edition, Springer, New York 2002.
• [13] O. Kallenberg, Some failures, problems, and conjectures from a lifetime in probability, Lecture at the “Kallenberg Symposium,” Mittag-Leffler Institute, Djursholm, Sweden, 2013. http://www.math.uni-frankfurt.de/~ismi/Kallenberg_symposium/Presentations/Kallenberg.pdf.
• [14] O. Kallenberg, Random Measures, Theory and Applications, Springer, New York 2017.
• [15] O. Kallenberg and J. Szulga, Multiple integration with respect to Poisson and Lévy processes, Probab. Theory Related Fields 83 (1989), pp. 101-134.
• [16] W. Krakowiak and J. Szulga, Random multilinear forms, Ann. Probab. 14 (1986), pp. 955-973.
• [17] W. Krakowiak and J. Szulga, A multiple stochastic integral with respect to a strictly p-stable random measure, Ann. Probab. 16 (1988), pp. 764-777.
• [18] S. Kwapień and W. A. Woyczyński, Double stochastic integrals, random quadratic forms and random series in Orlicz spaces, Ann. Probab. 15 (1987), pp. 1072-1096.
• [19] S. Kwapień and W. A. Woyczyński, Tangent sequences of random variables: Basic inequalities and their applications, in: Proceedings of Conference on Almost Everywhere Convergence in Probability and Ergodic Theory, G. A. Edgar and L. Sucheston (Eds.), Academic Press, New York 1989, pp. 237-265.
• [20] S. Kwapień and W. A. Woyczyński, Semimartingale integrals via decoupling inequalities and tangent processes, Probab. Math. Statist. 12 (1991), pp. 165-200.
• [21] S. Kwapień and W. A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston 1992.
• [22] T. R. McConnell and M. S. Taqqu, Decoupling inequalities for multilinear forms in independent symmetric random variables, Ann. Probab. 14 (1986), pp. 943-954.
• [23] V. H. de la Peña and E. Giné, Decoupling: From Dependence to Independence, Springer, New York 1999.
• [24] V. V. Petrov, Limit Theorems of Probability Theory: Sequences of Independent Random Variables, Clarendon Press, Oxford 1995.
• [25] J. Rosiński and W. A. Woyczyński, On Itô stochastic integration with respect to p-stable motion: Inner clock, integrability of sample paths, double and multiple integrals, Ann. Probab. 14 (1986), pp. 271-286.
• [26] D. Surgailis, On the multiple stable integral, Z.Wahrsch. Verw. Gebiete 70 (1985), pp. 621-632.
• [27] J. Zinn, Comparison of martingale difference sequences, in: Probability in Banach Spaces V, Lecture Notes in Math., Vol. 1153, Springer, Berlin 1986, pp. 453-457.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).