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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
DOI: https://doi.org/10.1090/tpms/961
Published electronically: August 11, 2015
MathSciNet review: 3242032
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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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Additional Information

Andriy Yurachkivsky
Affiliation: Department of Operations Research, Faculty for Cybernetics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email: andriy.yurachkivsky@gmail.com

DOI: https://doi.org/10.1090/tpms/961
Keywords: Martingale, convergence, tensor
Received by editor(s): June 12, 2012
Published electronically: August 11, 2015
Article copyright: © Copyright 2015 American Mathematical Society