44 3 Filtrations and Martingales
3.2 Stopping Times and Associated-Fields 45 becausefT qg 2Gq Ft ifq <t. Conversely, assume thatfT < tg 2Ft
for everyt> 0. Then, for everyt0ands>t, fT tg D \
q2QC;t<q<s
fT <qg 2Fs
and it follows thatfT tg 2FtCDGt.
Then, saying thatT ^t isFt-measurable for everyt > 0is equivalent to saying that, for everys<t,fTsg 2Ft. Taking a sequence of values ofsthat increases tot, we see that the latter property implies thatfT <tg 2Ft, and so T is a stopping time of the filtration.Gt/. Conversely, ifTis a stopping time of the filtration.Gt/, we havefT sg 2GsFtwhenevers<t, and thusT^t isFt-measurable.
(ii) First, ifA2GT, we haveA\ fT tg 2Gtfor everyt0. Hence, fort> 0, A\ fT <tg D [
q2QC;q<t
A\ fTqg 2Ft
sinceA\ fT qg 2GqFt, for everyq<t.
Conversely, assume that A\ fT <tg 2Ftfor everyt> 0. Then, for every t0, ands>t,
A\ fTtg D \
q2QC;t<q<s
A\ fT <qg 2Fs:
In this way, we get thatA\ fT tg 2FtC DGtand thusA2GT.
u t Properties of stopping times and of the associated-fields
(a) For every stopping timeT, we haveFT FTC. If the filtration.Ft/is right- continuous, we haveFTC DFT.
(b) IfT Dtis a constant stopping time,FT DFtandFTC DFtC. (c) LetTbe a stopping time. ThenT isFT-measurable.
(d) LetTbe a stopping time andA2F1. Set TA.!/D
T.!/if!2A; C1 if!…A: ThenA2FTif and only ifTAis a stopping time.
(e) LetS;T be two stopping times such thatS T. ThenFS FT andFSC FTC.
(f) LetS;T be two stopping times. Then,S_TandS^T are also stopping times andFS^T DFS\FT. Furthermore,fSTg 2FS^TandfSDTg 2FS^T.
46 3 Filtrations and Martingales (g) If.Sn/is a monotone increasing sequence of stopping times, thenSDlim"Sn
is also a stopping time.
(h) If.Sn/is a monotone decreasing sequence of stopping times, thenSDlim#Sn
is a stopping time of the filtration.FtC/, and FSC D\
n
FSnC:
(i) If .Sn/is a monotone decreasing sequence of stopping times, which is also stationary (in the sense that, for every!, there exists an integerN.!/ such thatSn.!/D S.!/for everyn N.!/) thenS Dlim #Snis also a stopping time, and
FSD\
n
FSn:
( j) LetT be a stopping time. A function! 7! Y.!/defined on the setfT < 1g and taking values in the measurable set.E;E/isFT-measurable if and only if, for everyt0, the restriction ofYto the setfTtgisFt-measurable.
Remark In property ( j) we use the (obvious) notion of G-measurability for a random variable! 7! Y.!/that is defined only on aG-measurable subset of˝ (hereG is a-field on˝). This notion will be used again in Theorem3.7below.
Proof (a), (b) and (c) are almost immediate from our definitions. Let us prove the other statements.
(d) For everyt0,
fTAtg DA\ fTtg and the result follows from the definition ofFT. (e) It is enough to prove thatFSFT. IfA2FS, we have
A\ fTtg D.A\ fStg/\ fT tg 2Ft; henceA2FT.
(f) We have
fS^T tg D fStg [ fT tg 2Ft; fS_T tg D fStg \ fT tg 2Ft; so thatS^T andS_T are stopping times.
It follows from (e) thatFS^T .FS\FT/. Moreover, ifA2FS\FT, A\ fS^T tg D.A\ fStg/[.A\ fTtg/2Ft; henceA2FS^T.
3.2 Stopping Times and Associated-Fields 47
Then, for everyt0,
fSTg \ fT tg D fStg \ fTtg \ fS^tT^tg 2Ft; fSTg \ fStg D fS^tT^tg \ fStg 2Ft;
becauseS^tandT^tare bothFt-measurable by Proposition3.6(i). It follows thatfSTg 2FS\FT DFS^T. ThenfSDTg D fSTg \ fT Sg.
(g) For everyt0,
fStg D\
n
fSntg 2Ft:
(h) Similarly
fS<tg D[
n
fSn<tg 2Ft;
and we use Proposition3.6(i). Then, by (e), we haveFSC FSnCfor everyn, and conversely, ifA2T
nFSnC, A\ fS<tg D[
n
.A\ fSn<tg/2Ft;
henceA2FSC.
(i) In that case, we also have
fStg D[
n
fSntg 2Ft;
and ifA2T
nFSn,
A\ fStg D[
n
.A\ fSntg/2Ft;
so thatA2FS.
( j) First assume that, for every t 0, the restriction of Y to fT tg is Ft- measurable. Then, for every measurable subsetAofE,
fY 2Ag \ fT tg 2Ft:
Lettingt! 1, we first obtain thatfY 2Ag 2F1, and then we deduce from the previous display thatfY 2Ag 2FT.
Conversely, ifYisFT-measurable,fY2Ag 2FTand thusfY2Ag \ fT
tg 2Ft, giving the desired result. ut
48 3 Filtrations and Martingales Theorem 3.7 Let .Xt/t0 be a progressive process with values in a measurable space .E;E/, and let T be a stopping time. Then the function ! 7! XT.!/ WD XT.!/.!/, which is defined on the eventfT <1g, isFT-measurable.
Proof We use property ( j) above. Lett 0. The restriction to fT tg of the function!7!XT.!/is the composition of the two mappings
fT tg 3!7!.!;T.!/^t/
Ft Ft˝B.Œ0;t/
and
˝Œ0;t3.!;s/7!Xs.!/
Ft˝B.Œ0;t/ E
which are both measurable (the first one since T ^ t is Ft-measurable, by Proposition3.6(i), and the second one by the definition of a progressive process). It follows that the restriction tofT tgof the function!7!XT.!/isFt-measurable,
which gives the desired result by property ( j). ut
Proposition 3.8 Let T be a stopping time and let S be anFT-measurable random variable with values inŒ0;1, such that ST. Then S is also a stopping time.
In particular, if T is a stopping time, TnD
X1 kD0
kC1
2n 1fk2n<T.kC1/2ngC 1 1fTD1g; nD0; 1; 2; : : : defines a sequence of stopping times that decreases to T.
Proof For the first assertion, we write, for everyt0, fStg D fStg \ fT tg 2Ft
sincefStgisFT-measurable. The second assertion follows sinceTn T, andTn
is a function ofT, henceFT-measurable, andTn#Tasn" 1by construction. ut The following proposition will be our main tool to construct stopping times associated with random processes.
Proposition 3.9 Let.Xt/t0 be an adapted process with values in a metric space .E;d/.
(i) Assume that the sample paths of X are right-continuous, and let O be an open subset of E. Then
TODinfft0WXt2Og is a stopping time of the filtration.FtC/.
3.3 Continuous Time Martingales and Supermartingales 49