Remarks
(i) It happens frequently that instead of the assumption of the proposition we have the weaker assumption
a:s: 8t0;
Z t 0
jHs.!/j jdAs.!/j<1:
If the filtration is complete, we can still defineHAas a finite variation process under this weaker assumption. We replaceHby the processH0defined by
H0t.!/D
Ht.!/if Rn
0 jHs.!/j jdAs.!/j<1; 8n; 0 otherwise.
Thanks to the fact that the filtration is complete, the process H0 is still progressive, which allows us to defineHADH0A. We will use this extension of Proposition4.5implicitly in what follows.
(ii) Under appropriate assumptions (ifHandKare progressive andRt
0jHsj jdAsj<
1,Rt
0jHsKsj jdAsj<1for everyt0), we have the “associativity” property
K.HA/D.KH/A: (4.1)
This indeed follows from the analogous deterministic result saying informally thatk.s/ .h.s/ .ds//D.k.s/h.s// .ds/ifh.s/andk.s/h.s/are integrable with respect to the signed measureonŒ0;t.
An important special case of the proposition is the case whereAt D t. IfHis a progressive process such that
8t0; 8!2˝;
Z t 0
jHs.!/jds<1;
the processRt
0Hsdsis a finite variation process.
4.2 Continuous Local Martingales
We consider again a filtered probability space.˝;F; .Ft/;P/. IfT is a stopping time, and ifX D .Xt/t0is an adapted process with continuous sample paths, we will writeXTfor processXstopped atT, defined byXtT DXt^Tfor everyt0. It is useful to observe that, ifSis another stopping time,
.XT/S D.XS/T DXS^T:
76 4 Continuous Semimartingales Definition 4.6 An adapted process M D .Mt/t0 with continuous sample paths and such thatM0 D 0a.s. is called acontinuous local martingaleif there exists a nondecreasing sequence.Tn/n0of stopping times such thatTn" 1(i.e.Tn.!/" 1 for every!) and, for everyn, the stopped processMTn is a uniformly integrable martingale.
More generally, when we do not assume thatM0 D 0 a.s., we say that M is a continuous local martingale if the processNt D MtM0 is a continuous local martingale.
In all cases, we say that the sequence of stopping times.Tn/reduces MifTn " 1 and, for everyn, the stopped processMTnis a uniformly integrable martingale.
Remarks
(i) We do not require in the definition of a continuous local martingale that the variables Mt are in L1 (compare with the definition of martingales). In particular, the variableM0may be anyF0-measurable random variable.
(ii) Any martingale with continuous sample paths is a continuous local martingale (see property (a) below) but the converse is false, and for this reason we will sometimes speak of “true martingales” to emphasize the difference with local martingales. Let us give a few examples of continuous local martingales which are not (true) martingales. If Bis an.Ft/-Brownian motion started from 0, and Z is an F0-measurable random variable, the process Mt D Z CBt is always a continuous local martingale, but is not a martingale ifEŒjZj D 1.
If we require the propertyM0 D 0, we can also considerMt D ZBt, which is always a continuous local martingale (see Exercise4.22) but is not a martingale if EŒjZj D 1. For a less artificial example, we refer to question (8) of Exercise5.33.
(iii) One can define a notion of local martingale with càdlàg sample paths. In this course, however, we consider only continuous local martingales.
The following properties are easily established.
Properties of continuous local martingales.
(a) A martingale with continuous sample paths is a continuous local martingale, and the sequenceTnDnreducesM.
(b) In the definition of a continuous local martingale starting from 0, one can replace “uniformly integrable martingale” by “martingale” (indeed, one can then observe thatMTn^nis uniformly integrable, and we still haveTn^n" 1).
(c) IfMis a continuous local martingale, then, for every stopping timeT,MTis a continuous local martingale (this follows from Corollary3.24).
(d) If.Tn/reducesMand if.Sn/is a sequence of stopping times such thatSn" 1, then the sequence.Tn^Sn/also reducesM(use Corollary3.24again).
4.2 Continuous Local Martingales 77 (e) The space of all continuous local martingales is a vector space (to check stability under addition, note that ifMandM0are two continuous local martingales such thatM0 D 0andM00 D 0, if the sequence.Tn/reducesMand if the sequence .Tn0/reducesM0, property (d) shows that the sequenceTn^Tn0reducesMCM0).
The next proposition gives three other useful properties of local martingales.
Proposition 4.7
(i) A nonnegative continuous local martingale M such that M0 2 L1 is a supermartingale.
(ii) A continuous local martingale M such that there exists a random variable Z2 L1 withjMtj Z for every t 0 (in particular a bounded continuous local martingale) is a uniformly integrable martingale.
(iii) If M is a continuous local martingale and M0 D 0(or more generally M0 2 L1), the sequence of stopping times
Tn Dinfft0W jMtj ng reduces M.
Proof
(i) WriteMt D M0CNt. By definition, there exists a sequence.Tn/of stopping times that reducesN. Then, ifst, we have for everyn,
Ns^Tn DEŒNt^Tn jFs:
We can add on both sides the random variableM0 (which is F0-measurable and inL1by assumption), and we get
Ms^Tn DEŒMt^Tn jFs:
SinceMtakes nonnegative values, we can now letntend to1and apply the version of Fatou’s lemma for conditional expectations, which gives
MsEŒMtjFs:
TakingsD 0, we getEŒMt EŒM0 < 1, henceMt 2 L1for everyt 0.
The previous inequality now shows thatMis a supermartingale.
(ii) By the same argument as in (i), we get for0st,
Ms^Tn DEŒMt^Tn jFs: (4.2)
Since jMt^Tnj Z, we can use dominated convergence to obtain that the sequenceMt^Tn converges toMt inL1. We can thus pass to the limitn ! 1 in (4.2), and get thatMsDEŒMtjFs.
78 4 Continuous Semimartingales (iii) Suppose thatM0 D 0. The random timesTn are stopping times by Proposi- tion3.9. The desired result is an immediate consequence of (ii) sinceMTn is a continuous local martingale andjMTnj n. If we only assume thatM0 2 L1, we observe thatMTnis dominated bynC jM0j. ut Remark Considering property (ii) of the proposition, one might expect that a continuous local martingale M such that the collection .Mt/t0 is uniformly integrable (or even a continuous local martingale satisfying the stronger property of being bounded in Lp for some p > 1) is automatically a martingale. This is incorrect!! For instance, ifBis a three-dimensional Brownian motion started from x6D0, the processMtD1=jBtjis a continuous local martingale bounded inL2, but is not a martingale: see Exercise5.33.
Theorem 4.8 Let M be a continuous local martingale. Assume that M is also a finite variation process (in particular M0D0). Then Mt D0for every t0, a.s.
Proof Set
n Dinfft0W Z t
0
jdMsj ng
for every integern0. By Proposition3.9,nis a stopping time (recall thatRt 0jdMsj is an increasing process ifMis a finite variation process).
Fixn0and setNDMn. Note that, for everyt0, jNtj D jMt^nj
Z t^n 0
jdMsj n:
By Proposition4.7,Nis a (bounded) martingale. Lett > 0and let0 Dt0 < t1 <
<tpDtbe any subdivision ofŒ0;t. Then, from Proposition3.14, we have EŒNt2D
Xp iD1
EŒ.Nti Nti1/2
E h
1supipjNtiNti1jXp
iD1
jNtiNti1ji n E
h sup
1ip
jNtiNti1ji
noting thatRt
0jdNsj nby the definition ofn, and using Proposition4.2.
We now apply the preceding bound to a sequence0Dtk0<tk1< <tkpk Dtof subdivisions ofŒ0;twhose mesh tends to0. Using the continuity of sample paths,