7.2 Introductory on Stochastic Processes 115SequencesTn↗∞, such that eachXTnsatisfies (P), are called localising
sequences. In particular we have the following definition (see [D 72]).
Definition 7.2.6.(i) AprocessS:R+×Ω→Rdis locally bounded if there
is an increasing sequence of stopping times(Tn)∞n=1tending to∞a.s. and
asequence(Kn)∞n=1inR+such that|S (^1) [[ 0,Tn]]|≤Kn.
(ii)AprocessX:R+×Ω→Ris a local martingale if there is an increasing
sequence of stopping times(Tn)∞n=1tending to∞a.s. so that, for eachn,
the processXTnis a uniformly integrable martingale.
One can show thatXis a local martingale if and only if there is an increas-
ing sequences of stopping timesTn↗∞such that eachXTnis a martingale
or, equivalently, is a martingale bounded inH^1 (P) i.e. sup 0 ≤t≤Tn|Xt|∈L^1 (P)
(compare Proposition 14.2.6 below).
Recall that, for a martingaleM,theHp(Ω,F,P)-norm is defined by
‖M‖Hp=
(
E
[(
sup
t|Mt|)p])^1 p
, for 1≤p<∞. (7.1)The following proposition is almost obvious, but it has important conse-
quences in mathematical finance.
Proposition 7.2.7.IfL:R+×Ω→Ris a local martingale such thatL≥
− 1 ,thenLis a super-martingale.
Proof.LetTn↗∞be a localising sequence forLand letU≤V be fi-
nite stopping times. For eachnthe processLTn is a uniformly integrable
martingale with respect to the filtration (Ft)t≥ 0 .ForeachA ∈FU and
eachn≤mwe therefore have
∫
A∩{U≤Tn}LU∧Tn=∫
∫ A∩{U≤Tn}LV∧Tm. Hence
A∩{U≤Tn}LU=
∫
A∩{U≤Tn}LV∧Tm.Ifweletm→∞,observethatLV∩Tm≥
−1 and use Fatou’s lemma to obtain that
∫
A∩{U≤Tn}LU≥∫
∫ A∩{U≤Tn}LV=
A∩{U≤Tn}E[LV|FU]. Hence on{U≤Tn}we haveLU≥E[LV|FU]. We
now letntend to∞to conclude.
Example 7.2.8.The archetype example of a local martingale which fails to be
a martingale is the inverse Bessel (3) process. If we takeX:R+×Ω→R^3 to
be a three dimensional Brownian motion, starting atX 0 =(1, 0 ,0), then with
respect to the natural filtration ofX,L=|X^1 |is a strictly positive local mar-
tingale that is not a martingale. The family (Lt)t≥ 0 is uniformly integrable,
but the family{LT |Tfinite stopping time}isnot uniformly integrable!
One can show that the natural filtration ofLis the filtration generated by
a one-dimensional Brownian motion. Bessel processes are thoroughly studied
by Pitman and Yor [PY 82]. See also [DS 95c] for applications of this theory
to finance.