116 7 A Primer in Stochastic Integration
Example 7.2.9.IfLis a local martingale, it is tempting to use the following
sequence of stopping times as a “localising” sequence
τn=inf{t||Lt|≥n}It is rather obvious thatτnindeed defines a localising sequence in the case
ofcontinuouslocal martingalesL.ButifLfails to be continuous this is not
a good choice in general. In fact there is a martingale (Mn,Fn)∞n=1(indexed
bythetimesetT=Nfor convenience) such that forT=inf{t|Mt=0},
the stopped processMT is not uniformly integrable. To construct such an
example we start with a sequence of Bernoulli variables, this is a sequence of
independent and identically distributed (i.i.d.) variablesrnsuch thatP[rn=
1] =P[rn=−1] =^12 .WealsoneedavariableXdefined on Ω, independent of
the sequencernand such that also forXwe haveP[X=1]=P[X=−1] =^12.
We define the random timeT asT =inf{n|rn=+1}. Hence we have
P[T=n]=2−n.
We now define a process (Mn)∞n=0, indeed byN.Forn<T, we putMn=0
and at timeTwe putMT=X 2 n.AftertimeTtheprocessdoesnotmove
anymore, meaningMn=MTforn≥T. The filtration (Fn)∞n=0is defined
asFn=σ(M 1 ...Mn)sothatTis a stopping time for (Fn)∞n=0. Clearly the
processMis a martingale, hence a local martingale. ButT=inf{n|Mn=0}
and we have thatMT is not integrable! Therefore the stopped processMT
cannot be a uniformly integrable martingale.
For positive local martingales one can do better as Remark 7.2.11 below
shows. We first need a preparatory result.
Proposition 7.2.10.IfL=(Lt)t≥ 0 is a local martingale such that
sup{E[|LT|]|Tfinite stopping time}<∞ (7.2)then
Tn=inf{t||Lt|≥n} (7.3)
is a localising sequence. More precisely
(i) P[Tn=∞]↗ 1 , i.e.,Tnincreases in a stationary way to∞.
(ii)LTn∈H^1 , i.e.E
[
L∗Tn]
=E
[
sup 0 ≤t≤Tn|Lt|]
<∞,foreachn∈N.Proof.(i): Denote byKthe sup appearing in (7.2). By Fatou’s lemma we have
E[|LTn| (^1) {Tn<∞}]≤K. HenceP[Tn<∞]≤Kn which gives (i).
(ii): Clearly|L∗Tn|≤max(n,|LTn|)∈L^1 and henceLTn∈H^1.
Remark 7.2.11.(useful but often forgotten!) An R-valued local martingale
which is uniformly bounded from below certainly satisfies the hypothesis
of Proposition 7.2.10. Indeed, for a stopping timeτwe have by the super-
martingale property ofL(Proposition 7.2.7) thatE[Lτ]≤E[L 0 ].
The seemingly unimportant fact thatP[Tn=∞]↗1 for the sequence
of stopping times (Tn)∞n=1defined by (7.3), will be used when we deal with
boundedness properties in the spaceL^0 (Ω,F,P).