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\topmatter
\title Closing values of martingales with finite 
lifetimes\endtitle
\author Michael J. Sharpe\endauthor
\affil University of California, San Diego\endaffil
\thanks Research supported in part by NSF Grant DMS
8721347\endthanks
 \endtopmatter
\outer\def\beginsection#1\par{\vskip0pt plus .1\vsize\penalty-250
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\baselineskip16pt
\beginsection 1. Introduction

Throughout this paper, $\zz$ denotes
a  fixed stopping time that will be considered to be the time of
occurrence of some physical event which signals the end of
the lifetime of a process. We imagine
that processes with lifetime $\zz$ are observable only
strictly prior to $\zz$, so that all meaningful
constructions associated with such processes should be
based only on their values strictly prior to~$\zz$.  Of
particular interest is the case in which $\zz$ is the
lifetime of a right process $X$. In this situation, $X$ is
supposed to be sent to a dead state after its lifetime, and
no further information is generated by it after that time.

The first general study of martingales with finite lifetime
was undertaken by Maisonneuve \cite{Ma77}, who was
interested principally in continuous local martingales
with finite lifetime $\zz$, which was allowed to be an
arbitrary stopping time.  His definition of local martingale
in that case was broader than that used by later workers
such as Yan \cite{Ya82} and Zheng \cite{Zh82}.  Zheng
studied limits of martingales with finite {\it
predictable\/} lifetimes. Yan's treatment of the
foundations of the theory of local martingales
and semimartingales with finite lifetime and the associated
stochastic integral were aimed principally at at even more
general theory of local martingales on an optional set. His
ideas relied partly on the work of Jacod
\cite{Ja79, Ch\.~V}, especially the latter's notion of local
martingales up to a random time. Yan's definition of local
martingale on $\OR$ seems to be more appropriate than
Maisonneuve's in the case of discontinuous processes. See
also \cite{Sh88, \S A6 and \S53} for some further
developments along these lines.

The aim of this paper is to examine the notion of martingale on $\OR$ from the
viewpoint of processes observable only at times strictly prior to $\zeta$, so
that the question becomes one of characterizing processes on $\OR$ which have a
martingale extension, and to give a procedure for computing that extension.

\beginsection 2. Generalities concerning the lifetime interval

We follow the terminology and conventions laid down in \cite{DM75} and
\cite{DM80}. In the formulas written below, it is important to keep in mind
the convention $\inf\emptyset:=\infty$, and to recall \cite{DM75, IV.60}, that
stochastic intervals of the form $\si[U V[$, $\si]U V]$, $\GR$ are subsets of
$[0,\infty[\times\W$, and not of $[0,\infty]\times\W$. Let $\PPP$ denote the
predictable $\ss$-field on $[0,\infty[\times\W$.

For the rest of the paper, we shall avoid some trivialities by making the
blanket assumption that a\.s\., $\zz>0$. (For the general case, replace $\PR$ by
the conditional probability $\PR\{\cdot|\zz>0\}$ where necessary.)

Let
$\rho:={}^p(\IOR)$ and  $\ll:={}^p(1_{\GR})$,  the
predictable projections  of the lifetime interval and the
graph of the lifetime respectively. It was shown in
\cite{Sh88,\S A6} that 
$$\OR\subset\{\rho>0\}\subset{\dlb0,
\zz\drb}.\tag 2.1$$
 Because ${\dlb0,\zz\drb}$ is predictable,
as is $\{\rho>0\}$, the random set ${\dlb0,
\zz\drb}\setminus\{\rho>0\}$ is predictable and, by (2.1), its
sections contain at most one point. It follows then that 
${\dlb0, \zz\drb}\setminus\{\rho>0\}$ is the graph  of a
predictable stopping time that we denote by $\zz_p$.
Note too that the trivial identity 
$1_{\OR}+1_{\GR}=1_{\dlb0,\zz\drb}$ implies, after taking
predictable projections, that $\rho+\ll=1_{\dlb0,\zz\drb}$.
 As
$\rho$ and $\ll$ are positive, up to evanescence, it follows
that, up to evanescence, $0\le\rho\le1_{\dlb0,\zz\drb}$ and
$0\le\ll\le1_{\dlb0,\zz\drb}$. In particular, by (2.1),
$\{\ll=1\}\subset\GR$, and so $$\dlb
\zz_p\drb=\{\ll=1\}\subset\GR.\tag2.2$$

Meyer's well known classification of stopping times  into
totally inaccessible and accessible parts is closely related
to the properties of $\rho$ and $\ll$ cited above. Indeed,
the totally inaccessible part of~$\zz$ is the stopping time
$\zz_i$ defined by $\dlb \zz_i\drb=\GR\cap\{\ll=0\}$ and the
accessible part $\zz_a$ of~$\zz$ is the stopping time defined by
$\dlb \zz_a\drb=\GR\cap\{\ll>0\}$. 
 For, given any predictable time $T$,
$\dlb T\drb\subset\dlb \zz_i\drb$ implies, by  definition of
predictable projection, that 
$\PR\, 1_{\dlb \zz_i\drb}(T)=\PR\,
1_{\GR}1_{\{\ll=0\}}(T)=\PR\,\ll(T)1_{\{\ll=0\}}(T)=0$.
 That is, $T$ is
totally inaccessible. On the other hand,  since the
predictable projection of an optional process differs from
it only on a set with countable sections, the set
$\{\ll>0\}$ a\.s\. has countable sections, and thus may be
expressed as a countable union of graphs of predictable
times $T_n$. That is, $\dlb \zz_a\drb\subset\cup_n\dlb
T_n\drb$, so $\zz_a$ is an accessible time. The random set
$\{\ll>0\}$ is evidently the smallest (up to evanescence)
predictable set containing the graph of the accessible part
of~$\zz$.

The following elementary result is a small complement  to
the Meyer classification. 
\proclaim{(2.3) Lemma} $\zz_p$ is the
maximal predictable time with graph contained in $\GR$.
 More
precisely, given any predictable time $T$ with ${\dlb
T\drb}\subset\GR$, one has ${\dlb T\drb}\subset{\dlb
\zz_p\drb}$, up to evanescence.
\endproclaim
\demo{Proof} Given such a $T$, by definition of 
predictable projection,
 $$\PR\,
\rho_T\ITf=\PR\,\IOR\ITf=\PR\,\{T<\zz\}=0.$$  Therefore,
$\GT\subset\{\ll=1\}$, and consequently
$\GT\subset\GR\cap\{\ll=1\}= {\dlb \zz_p\drb}$. \enddemo

We call $\zz_p$ the {\it predictable part\/} of~$\zz$. 
It is an obvious consequence of the lemma and the 
preceding discussion that if we define a stopping time $\zz_s$
by $\dlb \zz_s\drb=\dlb
\zz\drb\cap\{0<\rho<1\}=\GR\cap\{0<\ll<1\}$, then $\zz_s$ is an
accessible time containing the graph of no predictable time.
Note the following obvious consequence of the definition of
$\zz_p$ and $\rho$.
$$\{\rho>0\}=\si[0 \zeta]\setminus\dlb \zz_p\drb.\tag2.4$$

 We may characterize $\zz_p$ in terms of approximation
from strictly below in the following manner.
\proclaim{(2.5) Lemma} Let $\zz$ be a stopping time with
predictable part $\zz_p$. Then there is an increasing
sequence $(T_n)$ of stopping times with (a)~$T_n\le \zz$;
(b)~$\lim_n T_n =\zz$ a\.s\.; (c)~$T_n<\zz$ a\.s\. on
$\{\zz=\zz_p\}$. Conversely, given an increasing sequence
$(T_n)$ of stopping times satisfying (a) and (b), the event
$\{T_n<\zz$ for all $n\}$ is contained in $\{\zz=\zz_p\}$.
\endproclaim
\demo{Proof} For the first assertion, choose an announcing
sequence $(T'_n)$ for $\zz_p$, and let $T_n:=T'_n\land \zz$, so that the
sequence  $(T_n)$ clearly satisfies (a), (b) and (c). For the converse, let
$\GG:=\{T_n<\zz$ for all $n\}$ and note that the stochastic intervals 
$\si ]T_n
\infty [\cap\si [0 {\zz} ]\in \PPP$ decrease to
$\dlb \zz_{\GG}\drb$ as $n\to\infty$. (Recall that
$\zz_{\GG}(\w):=\zz(\w)$ if $\w\in\GG$, $=\infty$ otherwise.)
Thus $\zz_{\GG}$ is a predictable time with graph contained
in $\GR$, and so by Lemma 2.3, $\GG\subset\{\zz=\zz_p\}$.
\enddemo

\beginsection 3. Martingales with finite lifetime

As in \cite{Sh88, A6}, we define a uniformly integrable 
martingale on $\OR$ to be a process $M_t(\w)$ defined for
$(t,\w)\in\OR$, such that there exists a (globally defined)
uniformly integrable martingale $\barm'_t$ $(0\le t\le\infty)$ whose restriction
to $\OR$ is $M$. See also the discussion in \cite{Ya82}. 
The following result
was proved in \cite{Sh88,A6}.
\proclaim{(3.1) Proposition}Let $M$ be a uniformly
integrable martingale on $\OR$. There is then a  {\it
unique\/} extension ${\barm}$ of $M$ satisfying   the two
following conditions: 
 \roster
\litem"(3.2-i)" ${\barm}$ stops at $\zz$: that is, a\.s\., $\barm_t=\barm_{\zz}$
for all $t\ge\zz$; 
\litem"(3.2-ii)" ${\barm}_\zeta\in\F_{\zz-}$.
\endroster
\medskip
\endproclaim
We call the uniquely determined $\barm$
satisfying (3.2-i) and  (3.2-ii) the {\it minimal
extension\/} of $M$. The random variable $\barm_\zeta$ will be
called the {\it closing value} of $M$. In the simplest case, in which $\zz$ is
identically $\infty$, the definition above reduces to the usual definition of a
uniformly integrable martingale, and the closing value is
$\barm_\infty:=\lim_{t\to\infty}M_t$. More generally, if $\zz$ is a predictable
time, it is easy to see that the closing value is given by
$\barm_{\zz}:=\lim_{t\ua\ua\zz}M_t$, where the $\ua\ua$ is meant to indicate a
limit taken strictly from the left. However, by considering the first jump time
$\zz$ of a Poisson process with rate 1, and $M_t=-t$ for $t<\zz$, which is the
restriction to $\OR$ of the martingale $\barm_t=-t\wedge\zz+
1_{\si[{\zz} \infty[}(t)$, it is
clear that when $\zz$ is not predictable, the closing value is not given by the
left limit.

The definition of martingale on $\OR$ given above is mathematically convenient 
 but unsatisfying in that it violates the caveat that it
should depend only on observations of $M$ strictly prior to
$\zz$. It is one of the aims of this paper to find a formulation that is
intrinsic to $\OR$.

An example of considerable importance is the case where 
$\zz$ is the lifetime of a right process $X$ with state space
$E$, $h$ a function on $E$ extended, as customary, to vanish
at the dead point $\DD$, and $M_t:=h(X_t)$. Under certain
strong conditions such as $X$ a diffusion killed at a
boundary, $M$ is a martingale on $\OR$ if and only if $h$ is
harmonic relative to the generator of $X$. Under broader
conditions on $X$, under which the lifetime may not be
predictable, the characterization of the $h$ for which $M$
is a martingale on $\OR$ is not so straight\-forward, and does not seem to
reduce to identities involving hitting operators relative to $X$.

 The proof of existence and uniqueness of $\barm$ given  in
\cite{Sh88} is correct but unnecessarily complicated,
seeming to depend on an argument involving the relative density of two dual
predictable projections. The following is a simple, direct
proof.

\proclaim{(3.3) Lemma} Fix a globally defined uniformly 
integrable martingale $\barm'$ stopping at $\zz$, let $M$
denote the restriction of $\barm'$ to $\OR$, and define
$F:=\PR\,\{\barm'_\zeta\mid\F_{\zz-}\}$. Then the process $\barm$
defined by $\barm_t:=M_t1_{\{t<\zz\}}+F1_{\{t\ge \zz\}}$ is the
(unique\!) minimal extension of $M$. \endproclaim   
\demo{Proof} For uniqueness, it suffices to prove
that if $\barm$ satisfies (3.2) and if $\barm$ vanishes on
$\OR$, then $\barm$ is evanescent. Because $\barm$ is a
martingale of integrable total variation, its compensator
$\barm\,\tilde{}$ is evanescent. By (3.2), we may write
$d\barm=Z\,d1_{\si[{\zeta} \infty [ }$ for some $Z\in\PPP$, from
which it follows that $0=d\barm\,\tilde{}=Z\,dC$, where $C$
is the compensator of $1_{\si[{\zeta} \infty [ }$. Write
$Z=Z^+-Z^-$ to see that $Z^+\,dC=Z^-\,dC$, from which we
find $|Z|\,dC=0$, hence $|Z|\,d1_{\si[{\zeta} \infty [ }=0$, so
that $\barm$ is evanescent. Turning to existence, we see
clearly that $\barm$ is
uniformly integrable and satisfies (3.2-i) and (3.2-ii), so
it suffices to prove that $\barm$ is a martingale. That is
equivalent to proving that $\barm'-\barm$ is a martingale.
But $N_t:=\barm'_t-\barm_t=(\barm'_\zeta-F)1_{\{t\ge \zz\}}$, hence
for an arbitrary $H\in\rb\Ft$,
$\PR\,N_{\infty}H=\PR(N_{\infty}H1_{\{t<\zz\}}+
N_{\infty}H1_{\{t\ge \zz\}})$, and since
$H1_{\{t<\zz\}}\in\rb\F_{\zz-}$, the first summand reduces to
$\PR\,\(\PR\,\{N_\infty\mid\F_{\zz-}\}H1_{\{t<\zz\}}\)=0$. On
the other hand, $N_\infty=N_t$ on $\{t\ge \zz\}$, and as
$N_t=0$ for $t<\zz$, we have $\PR\,\(N_\infty H1_{\{t\ge
\zz\}}\) =\PR\,\(N_t H1_{\{t\ge \zz\}}\)=\PR\,\(N_t H\)$. This
proves that $N$ is a martingale.\enddemo

 It is going to turn out that the character of a 
martingale $M$ on $\OR$ depends on the character of the
minimal extension ${\barm}$ in an essential way. This is
already evident in the work of Maisonneuve, who was
concerned principally with {\it continuous\/} martingales
and local martingales on $\OR$. He observed that the correct
definition of continuity of $M$ involves continuity of some
extension of $M$ on $\dlb0,\zz\drb$ and not just that of $M$
on $\OR$. 

The two central questions we shall focus on in this paper are:
\roster
\litem"(3.4-i)" Is there a formula by which $\barm$ may be 
computed from $M$? 
\litem"(3.4-ii)" Given a process $M$ on
$\OR$, how may we tell just by looking at $M$ and $\zz$
whether $M$ is a uniformly integrable martingale on $\OR$?
That is, how and under what conditions should we try to
construct an extension $\barm$ of $M$ that will be a global 
martingale? \endroster\medskip

It is going to turn out that the answer to the second
question includes the answer to the first. 
Note first that the extension $\barm$ of $M$ is easy to 
calculate on the predictable part of~$\zz$. The closing value
of $M$ on that predictable part is just the left limit of
$M$ at $\zz$, just as in the classical case in which
$\zz\equiv\infty$.
 \proclaim{(3.5) Proposition} Let $M$ be a
uniformly integrable martingale on $\OR$ with minimal
extension  $\barm$. Then 
$$\barm_{\zz_p}=M_{\zz_p-}
\text{\quad a\.s\. on }\{\zz_p<\infty\}.\tag3.6$$
\endproclaim
\demo{Proof} Because $\zz_p$ is predictable, $M_{\zz_p-}=\barm_{\zz_p-}=
\PR\{\barm_{\zz}\mid\F_{\zz_p-}\}$, and since $\barm_{\zz}\in\F_{\zz-}$ by
hypothesis, 
$\barm_{\zz_p}=\barm_{\zz}1_{\zz=\zz_p}\in\F_{\zz-}\subset\F_{\zz_p-}$.
 The result follows.
\enddemo 

\beginsection 4. $\PR$-measures carried by the graph of $\zz$

Let $\AAA^p$ (resp., $\AAA^m$) denote the class of
predictable (resp., measurable) processes 
$A$ with integrable total
variation. More precisely, $\AAA^p$ (resp.,
$\AAA^m$) consists of the predictable (resp.,
measurable) $A$ such that  
\roster  \litem"(4.1-i)" $A_0=0$,
$A_\infty=\lim_{t\to\infty}A_t$; 
\litem"(4.1-ii)" $t\to A_t$
is a\.s\. right continuous with left limits, and of finite
variation on finite intervals;
 \litem"(4.1-iii)"
$\PR\Int\,|dA_t|<\infty$. \endroster

Recall
\cite{DM80} that a  signed measure $\nu$  on $(\si[0
\infty[,\PPP)$ is a $\PR$-measure  provided $|\nu|(\GG)=0$
whenever $\GG\in\PPP$ is evanescent with respect to $\PR$.
It is shown there that the class of (signed)
$\PR$-measures is in 1-1 correspondence with the class
$\AAA^p$ under the correspondence given by 
$$A\leftrightarrow\nu_A\quad\iff\quad \nu_A(H)=\PR\Int
H_t\,dA_t\qquad\forall\; H\in\rb\PPP.\tag4.2$$
More generally, given $A, B\in\AAA^m$, the $\PR$-measures
$\nu_A$, $\nu_B$ are equal if and only if $A^p=B^p$,
where $A^p$ denotes the dual predictable projection,
or compensator, of $A$. Let $\AAA^p_{\loc}$ denote
the predictable processes of {\it locally} integrable variation,
and similarly for $\AAA^m_{\loc}$. Thus, for example,
$A\in\AAA^m_{\loc}$ if and only if there exists an
increasing sequence $T_n$ of stopping times with limit
$+\infty$ a\.s\. so that for every $n$, $A^{T_n}$ ($A$
stopped at $T_n$) is in $\AAA^m$. The right side of (4.2)
is then not in general defined for every $H\in\rb\PPP$, but
it is defined as a signed measure on  $\si[0 {T_n}]$
 for all~$n$.

 \proclaim{(4.3) Definition} (i) A process $A\in\AAA^m$ is
carried by $\GR$, the graph of~$\zz$, provided $\PR\,\Int
H_t\,dA_t=0$ for all $H\in\rb\PPP$ satisfying $H_\zeta=0$
a\.s\. on $\{\zz<\infty\}$. (ii) A process
$\AAA^m_{\loc}$ is carried by the graph of~$\zz$ 
if and only if there is an increasing
sequence $T_n$ of stopping times such that
$\lim_n T_n=\infty$ a\.s\., and so that for all
$n$, $A^{T_n}\in\AAA^m$ is carried by the graph
of~$\zz$.\endproclaim

Obviously, $A\in\AAA^m$ is carried by the graph of~$\zz$
if and only if the corresponding $\PR$-measure $\nu_A$ has
the property $\nu_A(H)=0$ for all $H\in\rb\PPP$ with
$H_\zeta=0$ a\.s\. on $\{\zz<\infty\}$. We say
 in these circumstances that $\nu$ is carried
by the graph of~$\zz$.

\proclaim{(4.4) Lemma} $A\in\AAA^m_{\loc}$ is carried
by the graph of $\zz$ if and only if its compensator
$A\,\tilde{}$ is carried by $\GR$. \endproclaim
\demo{Proof} Suppose $A$ is carried by $\GR$. Let $T_n$ be
stopping times such that for each $n$,
$B:=A^{T_n}\in\AAA^m$ and $B$ is carried by $\GR$. Since
the dual predictable projection has the property $(1_{\si[0
{T_n}]}A)\,\tilde{}=1_{\si[0
{T_n}]}A\,\tilde{}$, $B\,\tilde{}$ is indistinguishable from
$A\,\tilde{}$ stopped at $T_n$. However, if
$H\in\rb\PPP$ and $H_\zeta=0$, $0=\Int H_t\,dB_t=\Int
H_t\,dB_t\,\tilde{}$, from which it follows that $A\,\tilde{}$
is carried by $\GR$. To go the other direction, simply
reverse the steps in the argument.\enddemo

Note that if $\zz$ is predictable, $A\in\AAA^p$
is carried by $\zz$ if and only if $A$ has the form
$F1_{\si[{\zeta} \infty [}$ for some $F\in L^1(\F_{\zz-})$.
However, if $\zz$ is not predictable, no such simple
characterization is possible. For example, if $\zz$ is the
first jump time of a Poisson process starting at 0, it is
not hard to see that $A\in\AAA^p$ is carried by $\GR$ if and
only if $A_t(\w)1_{\OR}(t,\w)=F(t)1_{\OR}(t,\w)$ for some
deterministic function $F$, right continuous with left
limits on $\Rp$, vanishing at 0, having locally bounded
variation and satisfying $\Int e^{-t}|dF(t)|<\infty$.
Intuitively, the signed measure $dA_t$ can be spread out over the {\it least}
predictable set $\GG$ containing the graph of~$\zz$, if such
a set $\GG$ exists.
 
\proclaim{(4.5) Lemma}
 Let $A\in\AAA^m_{\loc}$, and
 let $C$ denote $\ee_\zeta\,\tilde{}$, the
compensator of (the finite part of) unit mass at $\zz$.  
Then $A$
is carried by the graph of~$\zz$
if and only if $A\,\tilde{}\abscont C$.
\endproclaim
\demo{Proof} We assume first that $A\in\AAA^m$. By (4.4),
there is no loss of generality assuming $A\in\AAA^p$, and
we shall so assume. Suppose that $A$ is carried by the graph
of~$\zz$. Considering the $\PR$-measures $\nu_A$, $\nu_C$
generated by $A$ and $C$ respectively, we may make a 
Lebesgue-Radon-Nikodym decomposition of $A$ with respect to
$C$ on the $\ss$-field $\PPP$,  say $dA=dB+Z\,dC$ with
$B\perp C$, $B\in\AAA^p$, and $Z\in\PPP$ with $\PR\Int
|Z_t|\,dC_t<\infty$. Then $Z\,dC$
is carried by the graph of~$\zz$, for if $H\in\rb\PPP$
vanishes at $\zz$, $\PR\Int H_t Z_t\,dC_t
=\PR\Int H_t Z_t\,\ee_\zeta(dt)$ by the defining property of
the compensator, and the last term clearly vanishes. It
follows now that $B$ is carried by the graph of~$\zz$. 
Let
$\si[0 \infty [=\GG\cup\GG^c$ 
be a decomposition
of $\si[0 \infty [$ such that $\GG\in\PPP$,  with  $C$  not charging
$\GG\in\PPP$, and $B$ not charging $\GG^c$. In particular,
$\PR\,1_{\GG}(\zz)1_{\{\zz<\infty\}}=\PR\Int
1_{\GG}(t)\ee_\zeta(dt)=0$, and therefore $1_{\GG}$
vanishes at $\zz$. It follows that $\PR\Int H_t
1_{\GG}(t)\,dB_t=0$ for all $H\in\rb\PPP$. As $B$ lives
on $\GG$, this proves that $B$ is evanescent. Hence
$A\abscont C$. Conversely, if $A\abscont C$, then for
$H\in\rb\PPP$ with $H_\zeta=0$ a\.s\. on $\{\zz<\infty\}$,
$0=\PR H_\zeta\IRf=\PR\Int H_t\,dC_t$ implies $\PR\Int
H_t\,dA_t =0$, hence $A$ is carried by the graph of
$\zz$. 
\enddemo

\proclaim{(4.6) Definition}
A process $M$ defined on $\OR$ is a {\it semimartingale}
on $\OR$ provided it is the restriction to $\OR$ of some globally defined
semimartingale $\barm$. It is a local semimartingale on $\OR$ provided there is
an increasing sequence $(T_n)$ of stopping times with $\lim_n T_n\ge\zz$ and
$M^{T_n}$ a semimartingale on $\OR$ for every $n$.\endproclaim

Given a process $M$ defined on $\OR$, there are two particularly simple
extensions of $M$. First, if $M_{\zz-}$ exists a\.s\. on $\{\zz<\infty\}$, we
may set $M'_t=M_t$ for $t<\zz$, $=\lim_{r\ua\ua\zz}M_r$ for $t\ge\zz$. This is
called the (unique) extension of $M$ that stops at $\zz-$. Second, if we set
$M''_t=M_t$ for $t<\zz$, $=0$ for $t\ge\zz$, then $M''$ is called the (unique)
extension of $M$ vanishing on $\si[{\zz} \infty[$. It is easy to see that $M$
is a semimartingale on $\OR$ if and only if $M''$ is a (global) semimartingale,
and in case $M_{\zz-}$ exists a\.s\. on $\{\zz<\infty\}$, an equivalent
condition is that $M'$ is a (global) semimartingale.

Suppose for the moment that $M$ is a semimartingale on $\OR$. 
Let us define $\PPP_{\zz}$ to be the trace of $\PPP$ on $\OR$. (This is what
was called the {\it natural} $\ss$-field in \cite{GS84}.)
We may then construct
stochastic integrals $H\cd M$ for $H\in\rb\PPP_{\zz}$ as follows. Let
$\bar H\in\rb\PPP$ be an arbitrary bounded extension of $H$ and let $\barm$
denote an arbitrary extension extension of $M$ as a semimartingale. 
By the localization theorem for semimartingale stochastic integrals
\cite{DM80, VIII.23}, the values of $\bar H\cd \barm$ on
$\OR$ do not depend on the particular extensions $\bar H$
and $\barm$, up to evanescence. Thus it is meaningful to let $H\cd M$ denote
the restriction of $\bar H\cd \barm$ to $\OR$.
It is  important to note 
that we don't need to know whether there is a global martingale
extension of $M$ to construct $H\cd M$.


We may now set down a criterion for a given process $M$ 
on $\OR$ to be an $\HHHH^1$
martingale on $\OR$, using only conditions involving $M$ 
strictly prior to $\zz$. Recall \cite{DM80, VII.23} or
\cite{Pr90,V.2} that a (globally defined) semimartingale
$X$ is of class $\HHHH^1$ if and only if it has at least
one decomposition $X=N+A$ with $N$ a local martingale, $A$
an optional process of locally finite variation, such that
$\PR [N,N]^{1/2}+\PR\Int|dA_t|<\infty$. In this case, $X$
is in fact a special semimartingale, and the decomposition
can be supposed to be the canonical decomposition with
$A\in\AAA^p$. See \cite{DM80, \S 98c}. 

A theorem of Yor \cite{DM80, \S104}
characterizes the class  of $\HHHH^1$ (global) semimartingales $X$
as those for which the map $H\to H\cd X$ of $\rb\PPP$ into
semimartingales has the property
$$\sup_{t>0,|H|\le 1}\|(H\cd X)_t\|_1<\infty.\tag4.7$$

The next result is our preliminary characterization of
martingales on $\OR$ and their closing values. It is preliminary because the
procedure it specifies for the closing value involves the canonical
decomposition of a global semimartingale, which is not a construction
intrinsic to $\OR$. The final version of the theorem is given in the next
section. See (5.7).

 \proclaim{(4.8)
Theorem} Let $M$ be a process defined on  $\OR$ and satisfying the conditions
\roster
\litem"(4.8-i)"  $M_{\zz-}$ exists a\.s\. on $\{\zz<\infty\}$;
\litem"(4.8-ii)" $M$ is a semimartingale on $\OR$ whose extension
$M'$ stopping at $\zz-$ is of class $\HHHH^1$. (Recall: $M'_t=M_{\zz-}$
 for $t\ge \zz$.)
\endroster
Let $\mu$ denote the functional $H\to \mu(H):= \PR(H\cd
M')_{\zz-}1_{\{\zz<\infty\}}$, $H\in\rb\PPP$. Then $\mu$ defines a signed
$\PR$-measure on $(\si[0 \infty[,\PPP)$, and $M$ is a martingale on $\OR$ if
and only if $\mu$
 is carried by the graph
of~$\zz$. 
Moreover, in this case, the
closing value $\barm_\zeta$ of $M$ is given by 
$$\barm_\zeta=M_{\zz-}-H_\zeta1_{\{\zz<\infty\}},\tag4.9$$
where $H$ is a predictable version of the relative
density $dA/dC$, and $A\in\AAA^p$ is the unique element of $\AAA^p$ for which
$N:=M'-A$ is a martingale. \endproclaim 
\demo{Proof} Under (4.8-ii),  $M'$ is necessarily a special
semimartingale by VII.98~c) of \cite{DM80}, and if $M'=N+A$ is a
decomposition of $M'$ into the sum of an $\HHHH^1$ global
martingale $N$ and $A\in\AAA^p$, then for $H\in\rb\PPP$,
$H\cd N$ is a martingale satisfying $\|H\cd
N\|_{\HHHH^1}\le \|H\|_\infty\|N\|_{\HHHH^1}<\infty$, hence 
$|\mu(H)|\le
\|H\|_\infty(\|N\|_{\HHHH^1}+\PR\Int|dA_t|)=
\|H\|_\infty\|M'\|_{\HHHH^1}$.
 Since
 $N\in\HHHH^1$, $\DD N_\zeta\in L^1(\PR)$. As 
stochastic integrals satisfy $\DD(H\cd N)=H \DD N$,
we have 
$$
\multline
\PR H\cd N_{\zz-}1_{\{\zz<\infty\}}=\PR H\cd
N_\zeta1_{\{\zz<\infty\}}-\PR \DD(H\cd N)_\zeta1_{\{\zz<\infty\}}\\
=-\PR
H_\zeta\DD N_\zeta1_{\{\zz<\infty\}}.
\endmultline\tag4.10
$$ 
Since
$A\in\AAA^p$, $\mu(H)=-\PR H_\zeta\DD N_\zeta1_{\{\zz<\infty\}}+\PR\Int
H_t\,dA_t$, from which it is clear that $\mu$ is a signed
$\PR$-measure on $(\si [0 \infty[,\PPP)$.

 Write $\mu=\mu_N+\mu_A$, where
$\mu_N(H):=\PR H\cd N_{\zz-}1_{\{\zz<\infty\}}$ and $\mu_A(H):=\PR\Int
H_t\,dA_t$. By (4.10), $\mu_N$ is carried by the graph of~$\zz$,
in the sense of~(4.3). Therefore, the condition that $\mu$ is
carried by the graph of~$\zz$ is equivalent to the condition
that $\mu_A$ is carried by the graph of~$\zz$. 
 Suppose that
$\mu$ is indeed carried by the graph of~$\zz$ so that $A\abscont C$ by~(4.5). Let
$H\in\PPP$ be a version of~$dA/dC$. 
Define then $\barm:=M'-H_\zeta1_{\si[{\zeta}
\infty [}$. Because $A$ has integrable total variation, we
know that $\infty>\PR\Int\,|dA_t| =\PR\Int\,|H_t|\,dC_t
 =\PR\Int\,|H_t|\,\ee_\zeta(dt)=\PR|H_\zeta|$.
It suffices to establish that
$\barm$ is a uniformly integrable (globally defined)
martingale. But, $\barm=N+A-H_\zeta1_{\si[{\zeta} \infty [}$, and
noting that the process $H_{\zz}1_{\si[{\zeta} \infty [}$ is the
distribution function of the random measure $H*\ee_\zeta$,
whose  compensator is $H*\ee_{\zz}\,\tilde{}=H*C=A$, it is clear
that $\barm$ is a martingale as claimed. Its uniform
integrability is a consequence of the fact that $H_\zeta\in
L^1$ (proved above) and $\PR\sup_{t<\zz}|M_t|\in L^1$ by the $\HHHH^1$
hypothesis on~$M'$. 
For the converse, suppose $M$ is a martingale on $\OR$.
Let $\barm$ denote the minimal extension of~$M$.
Because $M'$ and $\barm$ are identical on $\OR$, for
$H\in\rb\PPP$, the localization theorem
\cite{DM80,VIII.10} gives $\mu(H)=\PR(H\cd M')_{\zz-}=\PR(H\cd
\barm)_{\zz-}$, and by the same reasoning employed in the
first paragraph of the proof, the latter term is equal to
$-\PR H_\zeta\DD\barm_\zeta$, from which it is evident that $\mu$
is carried by the graph of~$\zz$.
\enddemo 

The following simple
examples illustrate the use of 
 the preceding proposition in computing the minimal extension of~$M$.

In the first case, let~$\zz$ denote the time of
first jump of a Poisson process with rate~1, 
with its natural augmented filtration.  
Let $f$ be an absolutely continuous function on $\Rp$
satisfying $\Int |f'(t)|e^{-t}\,dt<\infty$, and let
$M_t=f(t)1_{\{t<\zz\}}$. Note that $\sup_{t<\zz}|M_t|\le
\int_0^\zeta |f'(t)|\,dt$, and the expectation of the latter
term is $\Int |f'(t)|e^{-t}\,dt<\infty$. Hence $M$ is
dominated by an integrable random variable.
Observe that
$M$ is a martingale with minimal extension  $\barm_t=f(t\land
\zz)-f'(\zz)1_{\{t\ge \zz\}}$. (This is well known in case
$f(t)=t$. Let ${\tilde M}_t=t\land \zz-1_{\{t\ge \zz\}}$.
 Then $\barm=H\cd \tilde M$, where
$H_t:=f'(t)$, proving the general case.) 
When we form the measure $\mu$ of (4.8),
 because $M$ has bounded variation,
$\mu(H)$ reduces to 
$\mu(H)=\PR\,\int_{[0,\zz[}H_t\,df(t)$, and since
every predictable process $H_t$ on $\dlb0,\zz\drb$ coincides
 with some deterministic function
$h(t)$ for $t\le \zz$, 
the latter integral may be written in the
form $\PR\Int e^{-t}h(t)f'(t)\, dt$. On the other hand, for
$H_t:=h(t)1_{\si[0 {\zz}]}$, $\mu_C(H)=\PR\,H(\zz)=\Int
e^{-t}h(t)\,dt$  from which it is clear that
$(d\mu/d\mu_C)_t=f'(t)$, just as asserted by the theorem.

For a different type of example, let $X$ denote uniform motion to the right on
$\Rp$, starting at the origin and killed with probability $\a$ ($0<\a<1$) as it
passes through the state 1.  Let $R:=1$, so that
$\GR\subset\dlb R\drb$. Obviously, $R$ is predictable. 
Observe that $\F_{1-}$ is the trivial $\ss$-field. Thus, if $T$ is a stopping
time, $T$ is constant on $\{T<1\}$. It follows that if, as in section 2, we set
$\ll={}^p1_{\GR}$, then $\ll_t$ is constant for $t<1$, and
 for such $t$, $\ll_t=\PR\{t=\zz\}=0$. On the other hand,
$\ll_1=\PR\{1=\zz\}=1-\a$, so that $\ll=(1-\a)1_{\GR}$.
By (2.2), $\zz_p=\infty$ a\.s\., and in particular, $\zz$ is not
predictable. Note that in this example, $C:=\ee_{\zz}\,\tilde{}=\a\ee_R$, as
$C$ and $\ee_\zeta$ clearly generate the same potential. In order for
$A\in\AAA^p$ to satisfy $A\abscont C$, it is necessary and sufficient that
$dA=\bb\,d\ee_R$ for some scalar $\bb$. A simple argument based on (4.8) shows
then that the martingales $M$ on $\OR$ have the form 
$M=\bb_1\IOR+\bb_2 1_{\si[R \infty[}1_{\{R<\zz\}}$, for arbitrary constants
$\bb_1$, $\bb_2$.

The third example is based on a standard linear Brownian motion $B$ killed at an
independent exponential time $\zz$ with parameter 1 to give us a new process
$X$.  We seek a characterization of those functions $h$ on $\R$ for which
$h\circ X$ is a martingale on $\OR$. By results from \cite{CJPS80}, $h\circ X$
is a semimartingale over $X$ if and only if $h$ is locally a difference of
convex functions, and so $h$ has a second derivative $\nu$ (in the sense of
Schwartz) that can be identified locally with a signed measure. It is clear
that the functional $C$ in (4.8) is the additive functional $dC_t=\IOR(t)
\,dt$, and by Tanaka's formula, $2A_t=\int\nu(dx) L^x_t$, where $L^x_t$ are
local times for $X$, chosen jointly continuous in $(x,t)$. It follows then from
(4.8) that $h\circ X$ is a martingale on $\OR$ if and only if $\nu$ is
(locally) absolutely continuous with respect to Lebesgue measure $\ell$ on $\R$,
and the closing value is given by $h(X_{\zz-})-g(X_{\zz-})$, where $g$ is a
version of $d\nu/d(2\ell)$. (It is easy to verify this directly without using
(4.8).) As a particular case, we find that $X_t^2$ is a martingale on $\OR$.

The third example above is a special case of a more general problem that will
be dealt with in a future publication, in which the underlying processes involved
are (right) Markovian, and on asks for a characterization of those functions $h$
for which $h\circ X$ is a martingale on $\OR$. 

Note too that the last example above could have been discussed in the
framework of the theory of {\it grossissement de
filtration} \cite {Je80}. Indeed, it is apparent from  the statement
of (4.8) that there is a close formal connection between that theory and ours.


\beginsection 5. Natural decomposition

Given a special
semimartingale $Z$ on $\OR$, its canonical decomposition
$Z=M+A$, with $M$ a local martingale and $A$ a predictable
process of locally integrable variation, is not necessarily
intrinsic to $\OR$. The following decomposition of such a
process $Z$ has the virtue of being intrinsic. It is not
generally the same as the decomposition $Z=M+A$ above. We
need to make use of some ideas and results from \cite{GS84}.
See also \cite{Sh88}. Recall that the class $\PPP_\zeta$ of {\it
natural\/} processes (relative to the fixed time $\zz$) means
the trace of $\PPP$ on $\OR$. Define the {\it natural
projection\/} $M\to {}^nM$ of $\rp\MMM$ onto $\rp\PPP_\zeta$ by
the formula $${}^nM:={}^p(M\IOR)\IOR/\rho.\tag5.1$$
 Note that a natural process $Z$ is not in general
predictable unless $\zz$ is a predictable time.

We shall say that a global semimartingale $Z$ {\it lives
on\/} $\LL:=\{\rho>0\}$ in case $1_{\LL}\cd Z=Z-Z_0$ and
$Z_\zeta\in\F_{\zz-}$. It is easy to see that $Z$ lives on $\LL$
if and only if \roster
\litem"(5.2-i)" $Z$ stops at $\zz$ (or equivalently, $1_{\si[0
\zz ]} \cd Z=Z-Z_0$);
\litem"(5.2-ii)" $\DD Z$ vanishes off
$\GR\cap\LL$. That is, by (2.4), $\DD Z_{\zz_p}=0$, where $\zz_p$ denotes
the predictable part of~$\zz$. 
\litem"(5.2-iii)" $\DD Z_\zeta\in\F_{\zz-}$.
\endroster
Note that if a process $Z$ is the
minimal extension of a uniformly integrable martingale on
$\OR$, then $Z$ is a semimartingale that lives on $\LL$.

 Call a process
$A$ a {\it natural process of locally finite variation\/} in
case  
\roster
\litem"(5.3-i)" $A$ is right continuous with left
limits and $A_0=0$;
\litem"(5.3-ii)" $A$ is of locally finite variation;
\litem"(5.3-iii)"  $A_t=A_{\zz-}$ a\.s\. for all $t\ge \zz$;
\litem"(5.3-iv)"  the restriction of $A$ to $\OR$ is in
$\PPP_\zeta$.
\endroster
Such a process $A$ is obviously a semimartingale living on
$\LL$.

 The {\it dual natural projection\/} $A\to A^n$ of a
locally integrable process $A$ of locally integrable
variation is defined provided $A$ is carried by $\LL$ and
$A$ possesses a dual predictable projection, and is given in
this case by the formula 
$$dA^n_t:=\IOR(t)\,dA_t^p/\rho_t.\tag5.4$$
Clearly, $A^n$ satisfies the conditions (5.3),
and therefore $A^n$ lives on $\LL$.
It was proved in \cite{GS84} that, if $A$ is carried by
$\LL$, then $A^n=(A^p)^n$ and $A^p=(A^n)^p$, and
consequently $A^n-A^p$ is a local martingale. It 
is almost immediate from properties of dual predictable projections that
$A^p$ is carried by $\LL$ if $A$ is, and thus, so is $A^n-A^p$.
\proclaim{(5.5) Theorem} Let $Z$ be a
semimartingale of class $\HHHH^1$ living on $\LL$. Then
$Z$ has a unique decomposition of the form $Z=Z_0+N+B$, where $N$
is the minimal extension of an $\HHHH^1$ martingale on $\OR$, $N_0=0$, and
$B$ is a natural process of integrable variation. (This decomposition
will be called the {\bf natural} decomposition of $Z$.) \endproclaim 
\demo{Proof}
Let $Z=Z_0+M+A$ be the canonical decomposition of $Z$ into the sum of a global
$\HHHH^1$ martingale $M$ and a predictable process $A$ of integrable
total variation. Since $1_{\LL}\cd Z=Z-Z_0$,
$Z-Z_0=1_{\LL}\cd M+1_{\LL}\cd A$, and by uniqueness of the
canonical decomposition, $M=1_{\LL}\cd M$ and
$A=1_{\LL}\cd A$. As $A$ is predictable, $A_\zeta\in\F_{\zeta-}$.
Hence $A$ lives
on $\LL$. Let $B:=A^n$, the dual natural projection of $A$ described above. Then
$A-B$ is a martingale. Let $N:=M+A-A^n$, so that $N$ is a
martingale and $Z=Z_0+N+B$. Obviously, $N$ stops at $\zz$, and
$\DD Z_\zeta=\DD N_\zeta+\DD B_\zeta\in\F_{\zz-}$ gives $\DD
N_\zeta\in\F_{\zz-}$. As to uniqueness of the decomposition, it
suffices, by stopping, to verify that if $N$ is
simultaneously a uniformly integrable  martingale satisfying
$\DD N_\zeta\in\F_{\zz-}$ and also a natural process of integrable
variation, then $N$ is a\.s\. constant. Since $N$ is natural,
its dual predictable projection $N^p$ exists and $N^p-N$ is a
martingale.
It follows that $N^p=N+(N^p-N)$ is also a
martingale, and because it is predictable and of locally
bounded variation, it must be a\.s\. constant, hence evanescent.
Finally, use the fact that $N=N^n=(N^p)^n$, as discussed prior to the statement
of (5.5), to see that $N$ is evanescent.
 \enddemo
\proclaim{(5.6) Corollary} Let $Z$ be a semimartingale of class $\HHHH^1$ that
stops at $\zz-$, and let $Z=N+B$ be its natural decomposition, as in (5.5). Then
$N$ and $B$ also stop at $\zz-$.\endproclaim
\demo{Proof} That $B$ stops at $\zz-$ is clear from its construction as a dual
natural projection. The result follows at once.\enddemo

\proclaim{(5.7) Corollary} Given a process $M$ defined on
$\OR$, let $M'$ denote the extension of $M$ that stops at
$\zz-$. Assume that $M'$ is a semimartingale of class
$\HHHH^1$. Then $M'$ lives on $\LL$, and if $M'=N+B$ is the
decomposition described in (5.5), then $M$ is a martingale
on $\OR$ if and only if $B\abscont
D:=1_{\OR}\ee_\zeta\,\tilde{}$. The closing value of $M$ is
given in this case by $\barm_\zeta:=M_{\zz-}-H_\zeta$, where $H\in\PPP$
satisfies $dB=H\IOR\,dD$.
\endproclaim 
\demo{Proof} It is clear that $M'$ lives on
$\LL$. Let $\mu$ denote the signed $\PR$-measure on
$(\si [0 \infty[,\PPP)$ defined as in (4.8) by $\mu(H)
:=\PR\,(H\cd M)_{\zz-}$. Because $\DD N_\zeta=0$, we find
$\mu(H)=\PR\Int H_t\,dB_t=\PR\Int H_t\,dB_t^p$ for
$H\in\rb\PPP$. According to (4.8), $\mu$ lives on the graph of $\zz$
if and only if $B^p\abscont C:=\ee_\zeta\,\tilde{}$. The
corollary will therefore be proved once we establish the
following result.\enddemo
\proclaim{(5.8) Lemma} Let $B$ be an arbitrary
natural process of integrable total variation.
Then $B^p\abscont C:=\ee_\zeta\,\tilde{}$ with $dB^p_t=H_t\,dC_t$, $H\in\PPP$,
 if and only if $B\abscont
D:=(1_{\OR}\ee_\zeta\,\tilde{}\;)^n$, with $dB_t=H_t 1_{\OR}(t)\,dD_t$
\endproclaim
\demo{Proof} Assume first that $B^p\abscont C$ with
$dB^p_t=H_t\,dC_t$, $H\in\PPP$. We also use the notation $B^p=H*C$.
Using the definition (5.4) of dual natural projection together with the fact
that $C$ and $D$ are carried by $\LL=\{\rho>0\}$, we have
$dB_t=d(B^p)^n_t=d(H*C)^n_t=\IOR(t)H_t\,dC_t/\rho_t=H_t\,dD_t=\IOR(t)H_t\,dD_t$.
Therefore, $dB_t=\IOR(t) H_t\, dD_t$, with $\IOR H\in\rp\PPP_\zeta$.
Conversely, starting with $dB_t=\IOR(t) H_t\,dD_t=H_t\,dD_t$, with an
$H\in\rp\PPP$, essentially the same reasoning gives
$dB^p_t=d(H*D)^p_t=H_t\,dD^p_t=H_t\,d(\IOR/\rho*C)^p_t
=H_t{}^p(\IOR/\rho)_t\,dC_t=H_t\,dC_t$, the last equality because $C$ is
carried by $\LL$.

For any set $\GG\in\PPP$ with
$\PR\,\Int1_{\GG}(t)\,dC_t=0$,  $\PR\,\Int1_{\GG}(t)\,dB^p_t=0$, and therefore 
$\PR\,\Int1_{\GG}(t)\,dB_t=0$. But, by definition of $D$, 
$$\PR\,\Int1_{\GG}(t)\,dD_t=
\PR\,\Int1_{\GG}(t)\IOR(t)/\rho(t)\,dC_t=0.$$
\enddemo
\proclaim{(5.9) Lemma} The semimartingale $\HHHH^1$ norm is
equivalent to the norm
$\|M\|:=\|N\|_{\HHHH^1}+\PR\Int|dB_t|$, in which $M=N+B$ is
the natural decomposition of $M$.\endproclaim
\demo{Proof} As was proved in \cite{DM80,VII.98, remarque c)}, the $\HHHH^1$
norm, defined as the infimum of the quantities
$j(W,A):=\|W\|_{\HHHH^1}+\PR\Int|dA_t|$, in which $M=W+A$ is
some decomposition of $M$ as the sum of a martingale and a
process of locally bounded variation, is equivalent in the
case of special semimartingales to the norm $j(W,A)$ in
which $M=W+A$ is the canonical decomposition with $A$
predictable.  It follows that $\|M\|\ge
{const}\|M\|_{\HHHH^1}$. On the other hand,
$\PR\Int|dB_t|=\PR\Int|dA_t|$, since $A=B^p$ and
$dB=\IOR/\rho\,dA$ implies $|dB|=\IOR/\rho\,|dA|$, whence
$|dA|=|dB|^p$. Finally, $N=W+(A-B)$ gives
$\|N\|_{\HHHH^1}\le\|W\|_{\HHHH^1}+\PR\Int|dA_t|+\PR\Int|dB_t|
\le 2\|M\|_{\HHHH^1}$.\enddemo


\Refs 
%\input jourrefs.tex
%\input SSrefs.tex
\widestnumber\key{\bf CJPS80}

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\ref\ky DM75 \by C. Dellacherie and P.-A. Meyer 
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\ref\ky DM80 \bysame \book Probabilit\'es et
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\ref\ky
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\ref\ky Ja79 \by  J. Jacod \book  Calcul Stochastique et
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\ref\ky Je80 \by T. Jeulin \book Semi-Martingales et Grossissement d'une
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\ref\ky MS81 \by P.-A. Meyer and C. Stricker\paper 
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\bookinfo Essays dedicated to L. Schwartz, 
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\ref\ky Sh75 \by M. J. Sharpe\paper Homogeneous extension
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\ref\ky Sh80 \bysame\paper Local times and singularities of
continuous local martingales\SSXIV\pages 76--101\endref

\ref\ky Sh88 \bysame\book General Theory of Markov 
Processes
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\ref\ky Ya82 \by J.-A. Yan\paper 
Martingales locales sur un ouvert droit
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\ref\ky Zh82 \by W.-A. Zheng\paper 
Semimartingales in predictable random open
sets\SSXVI \pages 370--379\endref

\endRefs
\enddocument

