Keywords
PII in near intervals;
additive decompositions
L2-regular martingales
continuous shadow theorem
nearby processes
variación total de un proceso PII
PII en casi intervalos
descomposiciones aditivas
martingalas L2-regulares
teorema de la sombra continua
procesos “nearby
How to Cite
Abstract
We establish a property for the total variation of a cad-lag process with independent increments wich is dichotomous in the sense that only two alternatives are possible. For this purpose we introduce the methods of nonstandard analysis with the study of PII processes in near intervals. Finally we discuss, in the case of continuos processes, an equivalent condition for one of the alternatives of the main theorem.
References
Nelson, E. (1977) “Internal set theory: a new approach to nonstandard analysis”, Bulletin of the American Mathematical Society 83(6): 1165–1198.
Nelson, E. (1987) Radically Elementary Probability Theory. Princeton University Press, Princeton.
Chuaqui, R. (1991) Truth, Possibility and Probability. North Holland, Amsterdam.
Diener, F.; Reeb, G. (1989) Analyse Non Standard. Hermann, Paris.
Lobo, J. (1998) “Descomposición de procesos de incrementos independientes en casi intervalos”, Revista de Matemática: Teoŕıa y Aplicaciones, 5(2): 163–176.
Lobo, J. (2000) “Fórmulas aproximadas del tipo Levy-Khintchine para las funciones características de procesos PII en casi intervalos”, to appear in Revista de Matemática: Teoría y Aplicaciones, 8(1).
Lobo, J. (1999) “Casi-ortogonalidad y proximidad en L2 de martingalas PII en casi intervalos”, Revista de Matemática: Teoría y Aplicaciones, 6(1): 27–33.
Stroyan, K.; Bayod, J. (1986) Foundations of Infinitesimal Stochastic Analysis. North Holland, Amsterdam.
Gihman, I.; Skorohod, A. (1980) Introduction à la Théorie des Processus Aléatoires. Mir, Moscou.
Bretagnolle, J. (1973) Processus à Accroissements Indépendants. Lecture Notes in Mathematics, Springer–Verlag, Berlin.
Millar, P.W. (1972) “Stochastic integrals and processes with stationary independent increments”, Proc. 6th Berkeley Symp. Math. Stat. Proba. 3: 307–332.
Bretagnolle, J. (1972) P-variation de fonctions aléatoires. Lecture Notes in Mathematics 258, Springer–Verlag, Berlin.
Millar, P.W. (1971) “Path behavior of processes with stationary independent increments”, Zetschrift für Wahrscheinlichkeitstheorie 17.