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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Stability results for martingale representations: The general case


Authors: Antonis Papapantoleon, Dylan Possamaï and Alexandros Saplaouras
Journal: Trans. Amer. Math. Soc. 372 (2019), 5891-5946
MSC (2010): Primary 60G05, 60G07, 60G44; Secondary 60H05
DOI: https://doi.org/10.1090/tran/7880
Published electronically: July 30, 2019
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Abstract: In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales, each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense, and that the limiting martingale is quasi left continuous and admits the predictable representation property. Then we prove that each component in the martingale representation of the sequence converges to the corresponding component of the martingale representation of the limiting random variable relative to the limiting filtration, under the Skorokhod topology. This extends in several directions earlier contributions in the literature and has applications to stability results for backward stochastic differential equations with jumps, and to discretization schemes for stochastic systems.


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Additional Information

Antonis Papapantoleon
Affiliation: Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece
Email: papapan@math.ntua.gr

Dylan Possamaï
Affiliation: Department of Industrial Engineering and Operations Research, Columbia University, 500 West 120th Street, New York, New York 10027
Email: dp2917@columbia.edu

Alexandros Saplaouras
Affiliation: Department of Mathematics, University of Michigan, East Hall, 530 Church Street, Ann Arbor, Michigan 48109-1043
Email: asaplaou@umich.edu

DOI: https://doi.org/10.1090/tran/7880
Received by editor(s): July 23, 2018
Received by editor(s) in revised form: March 20, 2019
Published electronically: July 30, 2019
Additional Notes: The second author gratefully acknowledges the financial support from the ANR project PACMAN (ANR-16-CE05-0027).
The third author gratefully acknowledges the financial support from the DFG Research Training Group 1845 “Stochastic Analysis with Applications in Biology, Finance and Physics”.
The authors gratefully acknowledge the financial support from the PROCOPE project “Financial markets in transition: Mathematical models and challenges”.
Article copyright: © Copyright 2019 American Mathematical Society