Towards a general Doob-Meyer decomposition theorem

• Autorzy: Jakubowski A.
• Języki publikacji: EN

Abstrakty

EN

Both the Doob-Меуег and Graversen-Rao decomposition theorems can be proved following an approach based on predictable compensators of discretizations and weak-L1 technique, which was developed by K. M. Rao. It is shown that any decomposition obtained by Rao’s method gives predictability of compensators without additional assumptions (like submartingality in the original Doob-Meyer theorem or finite energy in the Graversen-Rao theorem).

Słowa kluczowe

EN

Doob-Meyer decomposition theorem; predictable compensator; processes with finite energy; martingales

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2006
• Tom: Vol. 26, Fasc. 1
• Strony: 143--153
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Jakubowski A.

• Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12/18, 87-100 Toruń, Poland

Bibliografia

• [1] F. Coquet, A. Jakubowski, J. Mémin and L. Słomiński, Natural decomposition of processes and weak Dirichlet processes, in: Séminaire de Probabilités XXXIX, M. Émery and M. Yor (Eds.), Lecture Notes in Math. No 1874, Springer 2006, pp. 81-116.
• [2] C. Dellacherie and P. A. Meyer, Probabilités et potentiel, Vols. 1-4, Hermann, Paris 1975-1987.
• [3] S. E. Graversen and M. Rao, Quadratic variation and energy, Nagoya Math. J. 100 (1985), pp. 163-180.
• [4] Sh. He, J. Wang and J. Yan, Semimartingale Theory and Stochastic Calculus, Science Press and CRC Press, Beijing and Boca Raton 1992.
• [5] J. Jacod, Calcul stochastique et problemes de martingales, Lecture Notes in Math. No 714, Springer, 1979.
• [6] A. Jakubowski, Convergence in various topologies for stochastic integrals driven by semi- martingales, Ann. Probab. 24 (1996), pp. 2141-2153.
• [7] A. Jakubowski, A non-Skorohod topology on the Skorohod space, Electron. J. Probab. 2 (4) (1997), 21 pp.
• [8] A. Jakubowski, An almost sure approximation for the predictable process in the Doob- Meyer decomposition theorem, in: Séminaire de Probabilités XXXVIII, M. Émery, M. Ledoux and M. Yor (Eds.), Lecture Notes in Math. No 1857, Springer, 2005, pp. 158-164.
• [9] O. Kallenberg, Foundations of Modern Probability, Springer, 1997.
• [10] J. Komlós, A generalization of a problem of Steinhaus, Acta Math. Acad. Sei. Hungar. 18 (1967), pp. 217-229.
• [11] P. A. Meyer, A decomposition theorem for supermartingales, Illinois J. Math. 6 (1962), pp. 193-205.
• [12] P. A. Meyer, Decomposition of supermartingales: The uniqueness theorem, Illinois J. Math. 7 (1963), pp. 1-17.
• [13] P. Protter, Stochastic Integration and Differential Equations, Springer, 1990.
• [14] K. M. Rao, On decomposition theorems of Meyer, Math. Scand. 24 (1969), pp. 66-78.