\documentclass{utns}
%
\usepackage{graphicx}
\usepackage{txfonts}

\def\dj{d\kern-0.4em\char"16\kern-0.1em}
\def\DJ{\mbox{\raise0.3ex\hbox{-}\kern-0.4em D}}


\newcommand{\R}{{\bf R}}
\newcommand{\dsp}{\displaystyle}
\newcommand{\be}{\begin{equation} }
\newcommand{\ee}{\end{equation}}
\newcommand{\beq}{\begin{eqnarray} }
\newcommand{\eeq}{\end{eqnarray}}
\newcommand{\beqn}{\begin{eqnarray*} }
\newcommand{\eeqn}{\end{eqnarray*}}
\newcommand{\bi}{\begin{itemize} }
\newcommand{\ei}{\end{itemize}}
\newcommand{\bF}{{\bf F}}
\newcommand{\bH}{{\bf H}}
\newcommand{\bG}{{\bf G}}
\newcommand{\bE}{{\bf E}}
\newcommand{\kk}{\, |\!\!\! <}
\newcommand{\usu}{F_1\perp F_2|F}
\newcommand{\pr}{\hspace{0.7cm}}

\begin{document}

   \title{STATISTICAL CAUSALITY AND QUASIMARTINGALES}

   \author{VALJAREVI\' C DRAGANA
          \inst{1}\fnmsep\thanks{Corresponding author: {dragana.valjarevic@pr.ac.rs}},
          PETROVI\' C LJILJANA\inst{2},
          }

   \institute{Department of Mathematics, Faculty of Sciences and Mathematics, University of Pri\v stina, Kosovska Mitrovica, Serbia
         \and
   Department of Mathematics and Statistics, Faculty of Economics, University of Belgrade, Belgrade, Serbia
         }

  \abstract{Concept of causality is very popular and applicable nowadays,
especially when we consider the cases "what would happen if" and
"what would have happened if". Here we consider the concept of
causality based on the Granger's definition of causality,
introduced  in \cite{M1}. Many of the systems to
which it is natural to apply tests of causality take place in
continuous time, so we will consider the continuous time
processes.  Here we consider the connection between the concept of
causality and the property of being a quasimartingale.
Quasimartingales were investigated by  \cite{Fisk},
Orey and specially \cite{Rao}. Namely,  in this paper we prove an equivalence
between the given concept of causality and preservation of
quasimartingale property if the filtration is getting larger.
 We prove the same equivalence  for the stopped quasimartingale with respect to the truncated filtrations.  }

   \keywords{ Causality, filtration, martingale, quasimartingale.
   	}

   \maketitle
%
%________________________________________________________________

\section*{INTRODUCTION}

  In this paper we consider a stochastic process $X_t$ which have
a decomposition into the sum of a martingale process and a process
having almost every sample function of bounded variation on the
interval $I (I\subseteq \mathbb{R})$. Such a process is called a
quasimartingale.


After Introduction, in Section 2 we give definition of the
causality concept, based on the Granger's definition of
causa\-lity and some basic properties of that concept which will be
used later.

One of the goals of science is to find causal relations. This
cannot always be done by experiments and researchers are
restricted to observe the system they want to describe. This is
the case in, e.g., economics, demography, etc.  In the papers of
\cite{FM},  \cite{Gill}, \cite{M1}, \cite{LP3}  it is shown how the
conditional independence can serve as a basis for a general
probabilistic theory of causality for both processes and single
events.

The paper introduces a statistical concept of causality which
unifies the nonlinear Granger--causality with some related
concepts.

The linear Granger--causality was introduced by  \citealp{G}.
We shall study a nonlinear version of the concept. Like the linear
one, it defines that the process  ${\bf Y}=\{Y_t, t\in I\},
(I\subseteq \bf R)$ does not cause the process ${\bf X}=\{X_t,
t\in I\}$ if, for all $t$, the orthogonal projection of the
$L^2$-space representing $X_s, s>t$, on the space representing
$X_s$ and $Y_s, s\leq t$ is contained in the space re\-presenting
$X_s, s\leq t$. However, the spaces representing stochastic
variables are those over the $\sigma$-field generated by these
variables, while in the linear case they are the smallest closed
linear spaces containing the variables.

 We give a generalization of a
causality relationship "$\bG$ entirely causes $\bH$ within $\bF$"
which (in terms of $\sigma$-algebras) was introduced by
\cite{M1} and which is based on Granger's definition of causality
(see \cite{G})
 and discuss the relationship to nonlinear
Granger--causality.


In Section 3, we consider relations between the given causa\-lity
concept and the quasimartingale properties. More precisely, we analyze
connection between causality and the preservation of the
quasimartingale property with respect to the enlarged filtration
$\bF$ ($\bF$ is enlarged filtration of the natural filtration of
quasimartingale $\bF^X$).


The given concept of causality can be connected to the
orthogonality of martingales (see \cite{VP}) and the stable
subspaces of $H^p$ which contains the right continuous
modifications of martingales (see \cite{PV}). The preservation of
the predictable representation property, in the case when the
information $\sigma$-algebra increases, is  strongly connected to
the concept of causality (see \cite{PV1}). Also, the concept of
statistical causality can be connected to the local weak solutions
of stochastic di\-fferential equations driven with semimartingales
(see \cite{PV4}).

%__________________________________________________________________

\section*{NOTATIONS AND DEFINITIONS}

  \subsection*{CONCEPT OF CAUSALITY}

Following Granger's and Sims's pioneering papers (see
\cite{Sims}), the notion of causality in econometric is generally
defined within framework of prediction theory. This notion refers
to situations in which it is possible to reduce the size of the
information set that is taken into account for predicting a given
variable $X_1$ without affecting the precision level of the
prediction.

More precisely, a set of economic variables, denoted by $X_2$, does not cause a set of variables
$X_1$, if the information available about $X_2$ may be forgotten without any consequence
regarding the prediction of future $X_1's$. Since the content of the "available information" set is not precisely
described, the definition remains ambiguous.

% \citealp{PH} introduced the concept of instantaneous causality, which is investigating does
% the addition of some new available information set at time $t$ can improve the forecast. Granger thought that a %causal relation must be directional and can not be instantaneous because any econometric system needs a time delay
% to react to a cause.

 Modern financial econometrics is mainly devoted to the study of rapidly evolving stochastic processes.
The recent development of continuous time modelling in finance is an important motivation
for considering the concept of causality in continuous time.


In this part of the paper we give the definition of the concept of causa\-li\-ty
relationship (in continuous time) between the  flow of information (represented by
filtrations) and between the stochastic processes.

Let  $(\Omega, {\mathcal{F}}, P)$ be an arbitrary probability
space and let\ ${\bf F}=\{{\mathcal{F}}_t,\ t\in I (\subseteq \bf
R)\}$, be a family of sub-$\sigma$-algebras of ${\mathcal{F}}$.
${\mathcal{F}}_t$ can be interpreted as the set of events observed
up to  time $t$. Whether or not $\sup I=+\infty$ or $\inf
I=-\infty$ we define ${\mathcal{F}}_{\infty}$ as the smallest
$\sigma$-algebra containing all the ${\mathcal{F}}_{t}$ (even if
$\sup I<+\infty$). So, we have ${\mathcal{F}}_{\infty}=\bigvee
\limits_{t\in I}{\mathcal{F}}_t$ and
${\mathcal{F}}_{-\infty}=\bigcap \limits_{t\in I}{\mathcal{F}}_t$.

A filtration  ${\bf F}=\{{\mathcal{F}}_t,\ t\in I\}$ is a
nondecreasing family of sub-$\sigma$-algebras of ${\mathcal{F}}$,\
 i.e.  that is
$${\mathcal{ F}}_s\subseteq {\mathcal{F}}_t, s\leq t.$$

A probabilistic model for a time-dependent system is described by
$(\Omega, {\mathcal{F}}, {\mathcal{F}}_{t}, P)$, where
 $(\Omega, {\mathcal{F}}, P)$ is a probability space and  $\{
{\mathcal{F}}_{t},\ t\in I\}$ is a "framework" filtration, i.e.
$({\mathcal{F}}_{ t})$ are all events in the model up to and
including time $t$ and $({\mathcal{F}}_{ t})$ is a
sub-$\sigma$-algebra of ($\mathcal{F})$. We suppose that the
filtration $({\mathcal{F}}_t)$ satisfy the “usual conditions”,
which means that $({\mathcal{F}}_t)$ is right continuous and each
$({\mathcal{F}}_t)$ is complete.


Analogous notation will be used for filtrations\ ${\bf
H}=\{{\mathcal{H}}_t\}$ and ${\bf G}=\{{\mathcal{G}}_t\},\ t\in
I$.






It will be said that the filtrations\ $\bf G$\ and\ $\bf F$\ are
equivalent (and written as\ $\bf G=\bf F$) if\ $\bf G\subseteq\bf
F$\ and\ $\bf F\subseteq\bf G$, or equivalently, if
${\mathcal{G}}_{ t}={\mathcal{F}}_{ t}$\ for each\ $t$.



A family of $\sigma$-algebras induced by a stochastic process
${\bf X}=\{X_t, t\in I\}$ is given by ${\bf
F}^X=\{{\mathcal{F}}^X_{t}, t\in I\}$, where
$${\mathcal{F}}^X_{ t}=\sigma\{X_u, u\in I, u\leq t\},$$
being the \ smallest\  $\sigma$-algebra \ with \ respect to\ which
the \ random\ variables $X_u, u\leq t$ are measurable. The process
$X_t$ is $({\mathcal{F}}_t)$-adapted (or adapted to the filtration
$\bf F=\{{\mathcal{F}}_t\}$) if all $X_u, u\leq t$ are $\bf
F$-measurable, i.e. if $${\mathcal{F}}^X_{ t}\subseteq
{\mathcal{F}}_{ t}\ \mbox{ for each}\ t.$$ The notation $(X_t,
{\mathcal{F}}_t)$ means that $X_t$ is $({\mathcal{F}}_t)$-adapted.

A  family of $\sigma$-algebras may be induced  by several
processes, e.g. ${\bf F}^{X,Y}=\{{\mathcal{F}}^{X,Y}_{t}, t\in
I\}$, where
$${\mathcal{F}}^{X,Y}_{t}={\mathcal{F}}^X_{ t}\bigvee {\mathcal{F}}^Y_{ t}, t\in I.$$



On the probability space $(\Omega, {\mathcal{F}}, P)$ the process
${\bf Z}=\{Z_t, t\in I\}$ is a $({\mathcal{F}}_t,P)$-martingale if
$Z_t$ is $({\mathcal{F}}_t)$-adapted and
$E(Z_t|{\mathcal{F}}_s)=Z_s$ for all $t\geq s$.





The intuitively plausible notion of causality formulated in terms
of Hilbert spaces, is given in \cite{LP3}. We shall use analogue
notion of causality in terms of filtrations. Let\ $\bf F$, \ $\bf
G$\ and\ $\bf H$\ be arbitrary filtrations. We can say that "\
$\bf G$\ entirely causes\ $\bf H$\ within\ $\bf F$\ " if
\begin{equation}
{\mathcal{H}}_{\infty}\perp {\mathcal{F}}_{ t}|{\mathcal{G}}_{
t}\label{1}
\end{equation}
because the essence of (1) is that $({\mathcal{G}}_{ t})$\
contains all information from the\ $({\mathcal{F}}_{t})$\ needed
for predicting\ ${\mathcal{H}}_{\infty}$. Let us mention that the
condition\ $\bf G\subseteq\bf F$\ does not represent essential
restriction. Thus, it is natural to introduce the following
definition of causality between filtrations.



\begin{definition}\label{D21}{\rm (see \cite{LP3})} It is said that\ $\bf G$\ entirely causes\
(or just causes) \ $\bf H$\ within\ $\bf F$ relative to $P$\ (and
written as\ ${\bf H} \kk\bG; \bF; P$) if\
${\mathcal{H}}_{\infty}\subseteq {\mathcal{F}}_{\infty}$, \ $\bf
G\subseteq\bf F$\ and if\ ${\mathcal{H}}_{\infty}$ is
conditionally independent of $({\mathcal{F}}_{t})$ given
$({\mathcal{G}}_{t})$ for each $t$, i.e. \be
{\mathcal{H}}_{\infty}\perp {\mathcal{F}}_{ t}|{\mathcal{G}}_{ t}\
\mbox{for each}\ t,\ee (i.e. ${\mathcal{H}}_{u}\perp
{\mathcal{F}}_{ t}|{\mathcal{G}}_{ t}$\ holds for each\ $t$   and
each $u$), or $$(\forall A\in {\mathcal{H}}_{\infty})\,
P(A|{\mathcal{F}}_{ t})=P(A|{\mathcal{G}}_{ t}).$$
\end{definition}

If there is no doubt about $P$, we omit "relative to $P$".

The continuous time framework is fruitful, not only for the internal consistency of
economic theories but also for the statistical approach to causality analysis between stochastic processes.


Intuitively, \ $\bf H\kk\bf G;\bf F$\ means that, for arbitrary\
$t$, information about\ ${\mathcal{H}}_{\infty}$\ provided by\
$({\mathcal{F}}_{t})$\ is not "bigger" than that provided by\
$({\mathcal{G}}_{ t})$ or that it is possible to reduce available
information from $({\mathcal{F}}_{t})$ to $({\mathcal{G}}_{t})$ in
order to predict ${\mathcal{H}}_{\infty}$ .




\medskip

If\ $\bf G$\ and\ $\bf F$\ are such that\ $\bf G\kk\bf G;\bf F$,
we shall say that\ $\bf G$\ is its own cause within\ $\bf F$\
(compare with \cite{M1}). It should be mentioned that the notion
of subordination (as introduced in \cite{R}) is equivalent to the
notion of being one's own cause, as defined here. It should be
noted that "$\bf G$\ is its own cause" sometimes occurs as a
useful assumption in the theory of martingales and stochastic
integration (see \cite{BY}, \cite{RY}).

\medskip

These definitions can be applied to stochastic processes  if we
are talking about the corresponding induced filtrations. For
example, $({\mathcal{F}}_t)$-adapted stochastic process  $X_t$ is
its own cause if $({\mathcal{F}}^X_{ t})$ is its own cause within
$({\mathcal{F}}_t)$, i.e. if
$${\bf F}^{X}\kk{\bf F}^{X};{\bf F};P, \ \ \ {\mbox{holds}}.$$

\medskip


Extensions of the definitions to vector processes are usually
straightforward.



The process $X$ which is its own cause is completely described by
its behavior relative to its natural filtration ${\bf F}^X$. For
example, process $X=\{X_t, t\in I\}$ is a Markov process relative
to the filtration ${\bf F}=\{{\mathcal{F}_t}, t\in I\}$ on a
filtered probability space $(\Omega, {\mathcal{F}},
{\mathcal{F}}_t, P)$ if and only if $X$ is a Markov process
relative to ${\bf F}^X$ and it  is its own cause within ${\bf F}$
relative to $P$.

The concepts of causality in continuous time are truly relevant for economic reasons (see \cite{CR}).

In many situations we observe some system up to some random time,
 for example till the time when something happens for the first time.
  Definition 1 is extended from fixed times to stopping
times in \cite{LMJ}.

 The $\sigma$-field $({\mathcal{F}}_{T})=\{A\in {\mathcal{F}} : A\cap
\{T\leq t\}\in {\mathcal{F}}_t\}$ is usually interpreted as the
set of events that occurs before or at time $T$ (see \cite{Ell}).
For a process $X$, we set $X_{T}(\omega) = X_{T(\omega)}(\omega)$,
whenever $T(\omega) < +\infty$. We define the stopped process
$X^T=\{X_{t\wedge T}, t\in I\}$ with
$$X^T_t(\omega) = X_{t\wedge T(\omega)}(\omega)=X_t\chi_{\{t<T\}}+X_T\chi_{\{t\geq T\}}.$$
Note that if $X$ is adapted and cadlag and if $T$ is a stopping
time, then the stopped process $X^{T}$ is also adapted.

Let us mention that the truncated filtration $({\mathcal{F}}_{t
\wedge T})$ is defined as $${\mathcal{F}}_{t \wedge
T}={\mathcal{F}}_t \cap {\mathcal{F}}_{T}=
\left\{\begin{array}{ll}{\mathcal{F}}_t ,  \ t<T,
\\{\mathcal{F}}_{T}, \ t\geq T.
\end{array}\right.$$ A martingale stopped at a stopping time is still a
martingale. The natural filtration for the stopped martingale
$X_{t\wedge T}$ is $\bF^{X^T}=({\mathcal{F}}^X_{t\wedge T})$, with
respect to which the process $X_{t\wedge T}$ is completely
described. So, we have  the definition of causality which involves
the stopping times.

\begin{definition} \label{nndef2}\ {\rm(\cite{LMJ})}\ Let
$\mathbf{H}=\{{\mathcal H}_t\}$, $\mathbf{G}=\{{\mathcal G}_t\}$
and $\mathbf{E}=\{{\mathcal E}_t\}, t\in I,$ be given  filtrations
on the probability space $(\Omega, \mathcal{F}, P)$ and let $T$ be
a stopping time  with respect to filtration $\mathbf{E}$. The
filtration ${\bf G}^{T}$\ entirely causes\ ${\mathbf E}^{T}$\
within\ ${\bf H}^{T}$ relative to $P$\ (and written as\ ${\bf
E}^{T}\kk{\bf G}^{T};{\bf H}^{T};P$) if\ ${\bf E}^{T}\subseteq
{\bf H}^{T}$, ${\bf G}^{T}\subseteq{\bf H}^{T}$\ and if\
${\mathcal{E}}_{T}$ is conditionally independent of
${\mathcal{H}}_{t\wedge T}$ given ${\mathcal{G}}_{t\wedge T}$ for
each $t$, i.e.\ $ (\forall t)\,\, \ \ \ {\mathcal{E}}_{T}\perp
{\mathcal{F}}_{t\wedge \tau}\mid {\mathcal{G}}_{t\wedge \tau}, $
or \be \label{stop}(\forall t\in I)(\forall
A\in{\mathcal{E}}_{T})\,\, \ \ \ \ P(A\mid{\mathcal{H}}_{t\wedge
T})=P(A\mid{\mathcal{G}}_{ t\wedge T}).\ee
\end{definition}


The concept of causality given in Definition 2 %\ref{nndef2}
includes the stopped filtrations. Namely, the causality
relationship is defined up to a specified stopping time $T$.



\subsection*{QUASIMARTINGALES}
\indent

The term quasimartingale is for the first time used by
\cite{Fisk}. It is obvious that the sum and difference of two
quasimartingales are again quasimartingales. The difference of two
positive local martingales is necessarily a quasimartingale.  Let us
mention that there are some similarities between quasimartingales
and supermartingales. Note that every finite set of random
variables with expectations is trivially a quasimartingale. A mean
right continuous quasimartingale always has a cadlag (right continuous
with left limits)  modification. Henceforth we will assume,
unless otherwise stated, that all processes considered are cadlag
at every time point.




\begin{definition}{\rm \citep{PP} } A finite tuple of points
$\tau=(t_0,t_1,\dots,t_{n+1})$ such that $0=t_0<t_1<\dots
<t_{n+1}=\infty$ is a partition of $[0,\infty]$.
\end{definition}

\begin{definition}{\rm \citep{PP}}
 Suppose that $\tau$ is a partition of $[0,\infty]$ and that
 $X_{t_i}\in L^1,$ each $t_i\in \tau$. Define
 $$C(X,\tau)=\sum_{i=0}^n |E(X_{t_i}-X_{t_{i+1}}\mid
 {\mathcal{F}}_{t_i})|.$$ The variation of $X$ along $\tau$ is
 defined to be $$Var_{\tau}(X)=E(C(X,\tau)).$$ The variation of
 $X$ is defined to be $$Var(X)=sup_{\tau}Var_{\tau}(X),$$ where
 supremum is taken over all such partitions.
\end{definition}

\begin{definition}{\rm \citep{PP}}{\label{def-quasi}}
 An adapted, cadlag process $X$ is a quasimartingale on
 $[0,\infty]$ if $E(|X_t|)<\infty$, for each $t$, and if
 $Var(X)<\infty$.
\end{definition}

Next Theorem defines a Doob decomposition of quasimartingale.

\begin{theorem}{\rm \citep{Rao}}\label{quasi} A right continuous process $X_t$ is a
quasimartigale if and only if it has a generalised Doob
decomposition $$X_t=Y_t+M_t-B_t,$$ where $Y_t$ is a martingale,
$M_t$ is the difference of two non-negative local martingales, and
$B_t$ is the difference of two natural integrable increasing
processes. This decomposition is unique.
\end{theorem}

The definition of natural integrable increasing process is given
in \cite{Rao}.




\section*{CAUSALITY AND QUASIMARTINGALES}

The certain results, not obvious from the definition of a
quasimartingale or the fact that it is the difference of two
supermartingales, follow from the decomposition from  Theorem
6. The starting point in this section is the
decomposition \be \label{u1}X_t=M_t-B_t \ee of a quasimartingale $X_t$ into a
local martingale $M_t$ and a natural  process with finite expected total
variation $B_t$. This decomposition is unique.




 Let $({\mathcal{G}}_t)$ be a subfiltration of the
filtration $({\mathcal{F}}_t)$, i.e. $({\mathcal{G}}_t)\subseteq
({\mathcal{F}}_t)$. The next theorem  holds.

\begin{theorem}\label{Teorema1}
Every quasimartingale $X_t$ with respect to $({\mathcal{G}}_t)$ is
a quasimartingale with respect to $({\mathcal{F}}_t)$ if and only
if $\bG$ is its own cause within $\bF$, or equivalently if $$\bG
\kk \bG; \bF; P \ \ \ {\mbox{holds.}}$$
\end{theorem}

\begin{proof}
Let the process $X_t$ be a $({\mathcal{G}}_t)$ and
$({\mathcal{F}}_t)$ quasimartingale. From its unique decomposition
(\ref{u1}) it follows that process $M_t$ is a $({\mathcal{G}}_t)$
and $({\mathcal{F}}_t)$-local martingale. According to Theorem 3.3
in \cite{UT1},  the causality $\bG \kk \bG; \bF; P$ holds.

Conversely, let  $\bG \kk \bG; \bF; P$ holds and let the process
$X_t$ be a $({\mathcal{G}}_t)$-quasimartingale. Then, the  process
$X_t$ has a unique decomposition $X_t=M_t-B_t$, where $M_t$ is a
$({\mathcal{G}}_t)$-local martingale. From  $\bG \kk \bG; \bF; P$
and Theorem 3.3 in \cite{UT1} it follows that the process
$M_t$ is $({\mathcal{F}}_t)$-local martingale, too. Also, the
process $B_t$ is a natural process with finite expected total
variation with respect to filtration $({\mathcal{F}}_t)$, because
$({\mathcal{G}}_t) \subset ({\mathcal{F}}_t)$. Hence, the process
$X_t$ has a unique decomposition  $X_t=M_t-B_t$ with respect to
filtration $({\mathcal{F}}_t)$, so it is a
$({\mathcal{F}}_t)$-quasimartingale.
\end{proof}


Let $\bF^X$ be a natural filtration of the
quasimartingale $X_t$. Then the following theorem  holds.

\begin{theorem}
Process $X_t$ is a $({\mathcal{F}}_t)$-quasimartingale if and only
if it is its own cause within $({\mathcal{F}}_t)$, or equivalently
if holds
$$\bF^X \kk \bF^X; \bF; P.$$
\end{theorem}

\begin{proof}
Follows directly by Theorem 7 (we set $\bG=\bF^X$).
\end{proof}

 \begin{theorem}
 Let the process $X$ be uniformly integrable quasimartingale with respect to $\bG$, let $T$ be a $({\mathcal{G}}_t)$-stopping time and $\bG \subset \bF$. Then the
 stopped process $X^T=X_{t\wedge T}$ is quasimartingale with respect to $\bF^T=\{{\mathcal{F}}_{t\wedge T}\}$ if and only if $\bG^T$ is its own cause within $\bF^T$, i.e. if $$\bG^T \kk \bG^T; \bF^T; P \ \ \ {\mbox{holds}}.$$
 \end{theorem}

 \begin{proof}
  Let the process $X$ be uniformly integrable quasimartingale with respect to $\bG$,  $T$ be a $({\mathcal{G}}_t)$-stopping time and \be \label{procesX}\bG^T \kk \bG^T; \bF^T; P. \ee  Due to
  Lemma I.1.8.12 in \cite{Skorohod1} we have that $X_T$ is quasimartingale with respect to ${\mathcal{G}}_T$. According to the relation (\ref{procesX}), from assumption of the theorem it follows that $X^T$ is quasimartingale
  with respect to $\bG^T=\{{\mathcal{G}}_{t\wedge T}\}$. By Definition 5, %\ref{def-quasi},
  Theorem 6  %\ref{quasi}
  and assumption on the beginning of the Section it follows that the process $X^T$ can be represented as $$X^T=M^T-B^T.$$
   This decomposition is unique. Process $M^T$ is  martingale with respect to $\bG^T$. According to Theorem 6 in \cite{LMJ}, from (\ref{procesX}) it follows that the process $M^T$ is martingale with respect to $\bF^T$, too.
   Using the same technique as in the previous proof, we get  that
    $B^T$ is a process of bounded variation with respect to $\bG^T $ and $ \bF^T$, too ($\bG^T \subset \bF^T$). So, process $X^T$ can be presented as $X^T=M^T-B^T$ with respect to $\bF^T$, where $M^T$ is a local martingale and
     $B^T$ is a process of bounded variation.

   Conversely, suppose that $X^T$ is quasimartingale with respect to $\bG^T$ and $\bF^T$, where $T$ is a $({\mathcal{G}}_t)$-stopping time. Due to decomposition of the quasimartingale and its uniqueness, follows that
    $X^T=M^T-B^T$ is unique decomposition with respect to $\bG^T$ and $\bF^T$. So, $M^T$ is martingale with respect to filtrations $\bG^T$ and $\bF^T$. Due to Theorem 6 in \cite{LMJ}, it follows that  $\bG^T \kk \bG^T; \bF^T;
    P$ holds.
 \end{proof}

\bibliographystyle{aa}
\bibliography{ref}
\end{document}


%\begin{table}
%	\caption[]{Opacity sources.}
%	\label{KapSou}
%	$$
%	\begin{array}{p{0.5\linewidth}l}
%	\noalign{\smallskip}
%	Source      &  T / {[\mathrm{K}]} \\
%	\noalign{\smallskip}
%	\hline
%	\noalign{\smallskip}
%	Yorke 1979, Yorke 1980a & \leq 1700^{\mathrm{a}}     \\
%	%           Yorke 1979, Yorke 1980a & \leq 1700             \\
%	Kr\"ugel 1971           & 1700 \leq T \leq 5000 \\
%	Cox \& Stewart 1969     & 5000 \leq             \\
%	\noalign{\smallskip}
%	\hline
%	\end{array}
%	$$
%\end{table}

%_____________________________________________________________
%                 A figure as large as the width of the column
%-------------------------------------------------------------
%\begin{figure}
%	\centering
%	\includegraphics[width=\hsize]{empty.eps}
%	\caption{Vibrational stability equation of state
%		$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
%		$>0$ means vibrational stability.
%	}
%	\label{FigVibStab}
%\end{figure}
%
%_____________________________________________________________
%                                    One column rotated figure
%-------------------------------------------------------------
%\begin{figure}
%	\centering
%	\includegraphics[angle=-90,width=3cm]{empty.eps}
%	\caption{Vibrational stability equation of state
%		$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
%		$>0$ means vibrational stability.
%	}
%	\label{FigVibStab}
%\end{figure}
%
%_____________________________________________________________
%                                             Simple Table
%_____________________________________________________________
%
%\begin{table}
%	\caption{Nonlinear Model Results}             % title of Table
%	\label{table:1}      % is used to refer this table in the text
%	\centering                          % used for centering table
%	\begin{tabular}{c c c c}        % centered columns (4 columns)
%		\hline\hline                 % inserts double horizontal lines
%		HJD & $E$ & Method\#2 & Method\#3 \\    % table heading
%		\hline                        % inserts single horizontal line
%		1 & 50 & $-837$ & 970 \\      % inserting body of the table
%		2 & 47 & 877    & 230 \\
%		3 & 31 & 25     & 415 \\
%		4 & 35 & 144    & 2356 \\
%		5 & 45 & 300    & 556 \\
%		\hline                                   %inserts single line
%	\end{tabular}
%\end{table}
%
%_____________________________________________________________
%                                             Two column Table
%_____________________________________________________________
%
%\begin{table*}
%	\label{table:1}
%	\centering
%	\begin{tabular}{c c c c l l l }     % 7 columns
%		\hline\hline
%		% To combine 4 columns into a single one
%		HJD & $E$ & Method\#2 & \multicolumn{4}{c}{Method\#3}\\
%		\hline
%		1 & 50 & $-837$ & 970 & 65 & 67 & 78\\
%		2 & 47 & 877    & 230 & 567& 55 & 78\\
%		3 & 31 & 25     & 415 & 567& 55 & 78\\
%		4 & 35 & 144    & 2356& 567& 55 & 78 \\
%		5 & 45 & 300    & 556 & 567& 55 & 78\\
%		\hline
%	\end{tabular}
%\end{table*}
%
%
%-------------------------------------------------------------
%                                       Table with references
%-------------------------------------------------------------
%
%\begin{table*}[h]
%	\caption[]{\label{nearbylistaa2}List of nearby SNe used in this work.}
%	\begin{tabular}{lccc}
%		\hline \hline
%		SN name &
%		Epoch &
%		Bands &
%		References \\
%		&
%		(with respect to $B$ maximum) &
%		&
%		1981B   & 0 & {\it UBV} & 1\\
%		1986G   &  $-$3, $-$1, 0, 1, 2 & {\it BV}  & 2\\
%		1989B   & $-$5, $-$1, 0, 3, 5 & {\it UBVRI}  & 3, 4\\
%		1990N   & 2, 7 & {\it UBVRI}  & 5\\
%		1991M   & 3 & {\it VRI}  & 6\\
%		\hline
%		\noalign{\smallskip}
%		\multicolumn{4}{c}{ SNe 91bg-like} \\
%		\noalign{\smallskip}
%		\hline
%		1991bg   & 1, 2 & {\it BVRI}  & 7\\
%		1999by   & $-$5, $-$4, $-$3, 3, 4, 5 & {\it UBVRI}  & 8\\
%		\hline
%		\noalign{\smallskip}
%		\multicolumn{4}{c}{ SNe 91T-like} \\
%		\noalign{\smallskip}
%		\hline
%		1991T   & $-$3, 0 & {\it UBVRI}  &  9, 10\\
%		2000cx  & $-$3, $-$2, 0, 1, 5 & {\it UBVRI}  & 11\\ %
%		\hline
%	\end{tabular}
%	\tablebib{(1)~\citet{branch83};
%		(2) \citet{phillips87}; (3) \citet{barbon90}; (4) \citet{wells94};
%		(5) \citet{mazzali93}; (6) \citet{gomez98}; (7) \citet{kirshner93};
%		(8) \citet{patat96}; (9) \citet{salvo01}; (10) \citet{branch03};
%		(11) \citet{jha99}.
%	}
%\end{table}
%
%_____________________________________________________________
%                      A rotated Two column Table in landscape
%-------------------------------------------------------------
%\begin{sidewaystable*}
%	\caption{Summary for ISOCAM sources with mid-IR excess
%		(YSO candidates).}\label{YSOtable}
%	\centering
%	\begin{tabular}{crrlcl}
%		\hline\hline
%		ISO-L1551 & $F_{6.7}$~[mJy] & $\alpha_{6.7-14.3}$
%		& YSO type$^{d}$ & Status & Comments\\
%		\hline
%		\multicolumn{6}{c}{\it New YSO candidates}\\ % To combine 6 columns into a single one
	%	\hline
	%	1 & 1.56 $\pm$ 0.47 & --    & Class II$^{c}$ & New & Mid\\
	%	2 & 0.79:           & 0.97: & Class II ?     & New & \\
	%	3 & 4.95 $\pm$ 0.68 & 3.18  & Class II / III & New & \\
	%	5 & 1.44 $\pm$ 0.33 & 1.88  & Class II       & New & \\
	%	\hline
	%	\multicolumn{6}{c}{\it Previously known YSOs} \\
	%	\hline
	%	61 & 0.89 $\pm$ 0.58 & 1.77 & Class I & \object{HH 30} & Circumstellar disk\\
	%	96 & 38.34 $\pm$ 0.71 & 37.5& Class II& MHO 5          & Spectral type\\
	%	\hline
	%\end{tabular}
%\end{sidewaystable*}
%
%_____________________________________________________________
%                      A rotated One column Table in landscape
%-------------------------------------------------------------
%\begin{sidewaystable}
%	\caption{Summary for ISOCAM sources with mid-IR excess
%		(YSO candidates).}\label{YSOtable}
%	\centering
%	\begin{tabular}{crrlcl}
%		\hline\hline
%		ISO-L1551 & $F_{6.7}$~[mJy] & $\alpha_{6.7-14.3}$
%		& YSO type$^{d}$ & Status & Comments\\
%		\hline
%		\multicolumn{6}{c}{\it New YSO candidates}\\ % To combine 6 columns into a single one
%		\hline
%		1 & 1.56 $\pm$ 0.47 & --    & Class II$^{c}$ & New & Mid\\
%		2 & 0.79:           & 0.97: & Class II ?     & New & \\
%		3 & 4.95 $\pm$ 0.68 & 3.18  & Class II / III & New & \\
%		5 & 1.44 $\pm$ 0.33 & 1.88  & Class II       & New & \\
%		\hline
%		\multicolumn{6}{c}{\it Previously known YSOs} \\
%		\hline
%		61 & 0.89 $\pm$ 0.58 & 1.77 & Class I & \object{HH 30} & Circumstellar disk\\
%		96 & 38.34 $\pm$ 0.71 & 37.5& Class II& MHO 5          & Spectral type\\
%		\hline
%	\end{tabular}
%\end{sidewaystable}
%
%_____________________________________________________________
%                              Table longer than a single page
%-------------------------------------------------------------
% All long tables will be placed automatically at the end, after
%                                        \end{thebibliography}
%
%\begin{longtab}
%	\begin{longtable}{lllrrr}
%		\caption{\label{kstars} Sample stars with absolute magnitude}\\
%		\hline\hline
%		Catalogue& $M_{V}$ & Spectral & Distance & Mode & Count Rate \\
%		\hline
%		\endfirsthead
%		\caption{continued.}\\
%		Catalogue& $M_{V}$ & Spectral & Distance & Mode & Count Rate \\
%		\hline
%		\endhead
%		\hline
%		\endfoot
		%%
%		Gl 33    & 6.37 & K2 V & 7.46 & S & 0.043170\\
%		Gl 66AB  & 6.26 & K2 V & 8.15 & S & 0.260478\\
%		Gl 68    & 5.87 & K1 V & 7.47 & P & 0.026610\\
%		&      &      &      & H & 0.008686\\
%		Gl 86
%		\footnote{Source not included in the HRI catalog. See Sect.~5.4.2 for details.}
%		& 5.92 & K0 V & 10.91& S & 0.058230\\
%	\end{longtable}
%\end{longtab}
%