Abstract
For H 1-bounded sequences of martingales, we introduce a technique, related to the Kadeč-Pełczynski-decomposition for L 1 sequences, that allows us to prove compactness theorems. Roughly speaking, a bounded sequence in H 1 can be split into two sequences, one of which is weakly compact, the other forms the singular part. If the martingales are continuous then the singular part tends to zero in the semi-martingale topology. In the general case the singular parts give rise to a process of bounded variation. The technique allows to give a new proof of the Optional Decomposition Theorem in Mathematical Finance.
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References
J. P. Ansel and C. Strieker, Couverture des actifs contingents et prix maximum, Ann. Inst. Henri Poincaré, 30 (1994), 303–315.
J. Bourgain, The Komlos theorem for vector valued functions, manuscript, Vrije Univ. Brussel, (1979), 1–12.
D. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Mathematica, 124 (1970), 249–304.
D. Burkholder, Distribution function inequalities for martingales, Annals of Probability, 1 (1973), 19–42.
F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Math. Annalen, 300 (1994), 463–520.
F. Delbaen and W. Schachermayer, The fundamental theorem of asset pricing for unbounded stochastic processes, preprint, (1996).
C. Dellacherie, P. A. Meyer and M. Yor, Sur certaines propriétés des espaces de Banach H 1 et BMO, Sém. de Probabilités XII, Lecture Notes in Mathematics 649, Springer-Verlag, Berlin Heidelberg New-York, (1978), 98–113.
J. Diestel, W. Ruess and W. Schachermayer, On weak compactness in L 1 (µ, X), Proc. Am. Math. Soc., 118 (1993), 447–453.
J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., (1977).
M. Emery, Compensation de processus à variation finie non localement intégrables, Sém. de Probabilités XIV, Lecture Notes in Mathematics, 784 (1978), 152–160.
H. Föllmer and Y. M. Kabanov, On the Optional Decomposition Theorem and the Lagrange multipliers, preprint, (1995).
H. Föllmer and D. Kramkov, Optional Decomposition Theorem under Constraints, preprint, (1996).
F. R. Gantmacher, Matrizentheorie, Springer-Verlag, Berlin Heidelberg New-York, 1966.
A. M. Garsia, Martingale Inequalities. Seminar Notes on Recent Progress, Mathematics Lecture Notes Series, W. A.Benjamin, Inc, Reading, Massachusetts, 1973.
S. Guerre, La propriété de Banach-Saks ne passe pas de E à L 2 (E) d’après J. Bourgain, Sém. d’Analyse Fonctionelle, École Polytechnique, Paris, (1979/80).
J. Jacod, Calcul stochastique et problèmes de martingales, Lecture Notes in Mathematics 714, Springer-Verlag, Berlin Heidelberg New-York, (1980).
J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin Heidelberg New-York, 1987.
M. Kadeč and A. Pelczynski, Basic sequences, biorthogonal systems and norming sets in Banach and Fréchet spaces, Studia Mathematica, 25 (1965), 297–323.
N. El Karoui and M.-C. Quenez, Dynamic programming and pricing of contingent claims in an incomplete market, SIAM Journal on Control and Optimization, 33 (1) (1995), 27–66.
J. Komlos, A generalisation of a theorem of Steinhaus, Acta Math. Acad. Sci. Hungar, 18 (1967), 217–229.
D. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Probability Theory and Related Fields, 105 (1996), 459–479.
D. Kramkov, On the closure of the family of martingale measures and an optional decomposition of supermartingales, preprint, (1996).
H. Kunita and S. Watanabe, On square integrable martingales, Nagoya Mathematical Journal, 30 (1967), 209–245.
J. Marcinkiewicz and A. Zygmund, Quelques théorèmes sur les fonctions indépendantes, Studia Mathematica, 7 (1938), 104–120.
J. Memin, Espaces de semi martingales et changement de probabilités, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 52 (1980), 9–39.
P. A. Meyer, Un cours sur les intégrales stochastiques, Chap. V, Sém. de Probabilités X, Lecture Notes in Mathematics 511, Berlin Heidelberg New-York, Springer-Verlag, (1976), 245–400.
P. Monat, Remarques sur les inégalités de Burkholder-Davis-Gundy, Séminaire de Probabilités XXVIII, Lecture Notes in Mathematics 1583, Springer, Berlin Heidelberg, (1994), 92–97.
J. Neveu, Discrete Parameter Martingales, North-Holland, Amsterdam, 1975.
P. Protter, Stochastic Integration and Differential Equations, Applications of Mathematics 21, Springer, (1990).
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 2nd edition, Grundlehren der mathematischen Wissenschaften 293, Springer, (1994).
W. Schachermayer, The Banach-Saks property is not L 2 -hereditary, Israel Journal of Mathematics, 40 (1981), 340–344.
W. Schachermayer, Martingale measures for discrete time processes with infinite horizon, Mathematical Finance, 4 (1994), 25–56.
J. A. Yan, Caractérisation d’une classe d’ensembles convexes L 1 ou H 1, Séminaire de Probabilités XIV, Lecture Notes in Mathematics 784, Springer-Verlag, Berlin Heidelberg New-York, (1980), 220–222.
M. Yor, Sous-espaces denses dans L 1 ou H 1 et représentation des martingales, Séminaire de Probabilités XII, Lecture Notes in Mathematics 649, Springer-Verlag, Berlin Heidelberg New-York, 1978, 265–309.
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Delbaen, F., Schachermayer, W. (1999). A Compactness Principle for Bounded Sequences of Martingales with Applications. In: Dalang, R.C., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability, vol 45. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8681-9_10
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DOI: https://doi.org/10.1007/978-3-0348-8681-9_10
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