\begin{document}
\setlength\parindent{0pt}

\title[Stability results for martingale representations]{Stability results for martingale representations:\\the general case}

\author[A. Papapantoleon]{Antonis Papapantoleon}
\author[D. Possama\"{i}]{Dylan Possama\"{i}}
\author[A. Saplaouras]{Alexandros Saplaouras}

\address{Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Gree-ce}
\email{papapan@math.ntua.gr}

\address{Department of Industrial Engineering and Operations Research, Columbia University, 500W 120th street, 10027, New York, NY, USA}
\email{dp2917@columbia.edu}

\address{Department of Mathematics, University of Michigan, East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, USA}
\email{asaplaou@umich.edu}

\thanks{We thank Samuel Cohen for useful discussions during the work on these topics. Alexandros Saplaouras gratefully acknowledges the financial support from the DFG Research Training Group 1845 ``Stochastic Analysis with Applications in Biology, Finance and Physics''. Dylan Possama\"i gratefully acknowledges the financial support from the ANR project {\sc Pacman} (ANR-16-CE05-0027). Moreover, all authors gratefully acknowledge the financial support from the {\sc Procope} project ``Financial markets in transition: mathematical models and challenges''.}

\keywords{}

\subjclass[2010]{}

\date{}

\begin{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 SDEs with jumps and to discretisation schemes for stochastic systems.
\end{abstract}

\maketitle \frenchspacing


\iftoggle{full}{\tableofcontents}{}

\vspace{-2em}
\section{Introduction}
\subfile{introduction}


\section{Preliminaries}\label{sec:prel}

\subsection{The stochastic basis}
\subfile{The_stochastic_basis}

\subsection{Stochastic integrals}\label{StochIntegrals}
\subfile{Stochastic_integrals}

\subsection{\texorpdfstring{The Skorokhod space $(\D,\J_1)$ and convergence in the extended sense} 
			{The Skorokhod space (D, J 1) and convergence in the extended sense}}\label{sec:skor}
\subfile{The_Skorokhod_space}


\section{Stability of martingale representations}\label{sec:RobMartRepSection}	

\subsection{Framework and statement of the main theorem}\label{subsec:RoMRFrame}
\subfile{RoMR_Framework}
\subfile{RoMR_Main_Theorem}

\subsection{Examples and applications}\label{subsec:Ex_and_Appl}
\subfile{RoMR_Examples_Applications}
 
\subsubsection{The case of processes with independent increments}
\subfile{ROMR_The_additive_case}

\subsubsection{A stronger version of the main theorem}
\subfile{RoMR_Corollary_for_BSDE_Robustness}

\subsection{Comparison with the literature}\label{sec:comp}
\subfile{RoMR_Comparison_with_the_literature}

\subsection{Outline of the proof}\label{sec:outline}
\subfile{RoMR_Outline_of_Proof}

\subfile{RoMR_StepD}
\subfile{RoMR_StepC}

\subfile{RoMR_StepB_Prep_Results}
\subfile{RoMR_StepB_Ginf_Immersed_in_F}

\subsection{Proof of the main Theorem}\label{subsec:Proof}
\subfile{RoMR_Proof_of_Main_Theorem}


\appendix

\section{Auxiliary results}\label{AppendixAux}

\subsection{\texorpdfstring{Joint convergence for the Skorokhod $\J_1$--topology}{Joint convergence for the Skorokhod J 1-topology}}\label{subsec:JointConv}
\subfile{Auxilliary_Results_Skorokhod}

\iftoggle{full}{

\subsection{Measure Theory}
\subfile{Auxilliary_Results_Measure_Theory}

} {}

\subsection{Young functions}\label{Young_functions}\label{App:Young}
\subfile{Auxilliary_Results_Young_Function}


\iftoggle{full}{

\subsection{Proof of \cref{corr:PII}}\label{Proof_Corr_main_thm}
\subfile{Auxilliary_Results_Proof_Corr_main_thm}

} {}

\bibliographystyle{abbrvnat}
\small
\bibliography{bibliographyDylan}


\end{document}
{"config": "arxiv", "file": "1806.01172/main.tex"}