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Convergence of stochastic integrals to a continuous local martingale with conditionally independent increments

Author: Andriy Yurachkivsky
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 90 (2014).
Journal: Theor. Probability and Math. Statist. 90 (2015), 207-221
MSC (2010): Primary 60F17; Secondary 60G44
Published electronically: August 11, 2015
MathSciNet review: 3242032
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Abstract | References | Similar Articles | Additional Information

Abstract: For each $ T>0$, let a tensor-valued stochastic process $ Y_T$ be defined by

$\displaystyle Y_T(t)=\int _0^tD Z_T(s)\otimes \vartheta _T(s), $

where $ Z_T$ is an $ \mathbf {R}^d$-valued locally square integrable martingale with respect to some filtration $ \mathbb{F}_T$ and where $ \vartheta _T$ is an $ \mathbf {R}^d$-valued $ \mathbb{F}_T$-predictable stochastic process such that $ \int _0^t\vert\vartheta _T(s)\vert^2D\operatorname {tr}\langle Z_T\rangle (s)<\infty $ for all $ t$. In this paper, conditions are found for the convergence $ (Y_T, \langle Y_T\rangle )\stackrel {{\rm law}}\longrightarrow (Y, \langle Y\rangle )$, where $ Y$ is a continuous local martingale with conditionally independent increments given $ \langle Y\rangle $.

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  • 1. R. Sh. Liptser and A. N. Shiryayev, Theory of martingales, Mathematics and its Applications (Soviet Series), vol. 49, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by K. Dzjaparidze [Kacha Dzhaparidze]. MR 1022664
  • 2. Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 1987. MR 959133
  • 3. A. Touati, Sur la convergence en loi fonctionnelle de suites de semimartingales vers un mélange de mouvements browniens, Teor. Veroyatnost. i Primenen. 36 (1991), no. 4, 744–763 (French); English transl., Theory Probab. Appl. 36 (1991), no. 4, 752–771 (1992). MR 1147174,
  • 4. Andriy Yurachkivsky, Convergence of locally square integrable martingales to a continuous local martingale, J. Probab. Stat. (2011), Art. ID 580292, 34. MR 2854289
  • 5. B. L. van der Waerden, Algebra, vol. I, Springer-Verlag, Berlin-Heidelberg-New York, 1971.
  • 6. I. M. Gel′fand, Lectures on linear algebra, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1989. With the collaboration of Z. Ya. Shapiro; Translated from the second Russian edition by A. Shenitzer; Reprint of the 1961 translation. MR 1034245
  • 7. I. I. Gikhman and A. V. Skorokhod, \cyr Stokhasticheskie differentsial′nye uravneniya i ikh prilozheniya, “Naukova Dumka”, Kiev, 1982 (Russian). MR 678374

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Additional Information

Andriy Yurachkivsky
Affiliation: Department of Operations Research, Faculty for Cybernetics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine

Keywords: Martingale, convergence, tensor
Received by editor(s): June 12, 2012
Published electronically: August 11, 2015
Article copyright: © Copyright 2015 American Mathematical Society