The Mathematics of Arbitrage

124 7 A Primer in Stochastic Integration

E[|Xt|]=

∫t

0

∣

∣

∣

∣

B

u

∣

∣

∣

∣dP[T=u]=

∫t

0

1

u

2 e−^2 udu=∞.

HenceX is not a martingale asE[Xt] does not make sense. In fact, also
stopping does not help to remedy the integrability problem! It is not hard to
check that, for every stopping timeτw.r. to the filtration (Ft)t≥ 0 such that
P[τ>0]>0, we still have (see [E 80] for the details)

E[|Xτ|]=∞.

HenceX even fails to be a local martingale. In particular,H·M is not
defined as a stochastic integral in the sense of integration with respect to a
local martingale as developed above, asin this theory the integral of a local
martingale is necessarily a local martingale.
In Sect. 8.3 below we shall define the notion of a sigma-martingale which
will yield the proper framework enabling us to also interpret processes such
asX=H·Mabove still as a “fair game”.

This example ofEmery is very simple, but it shows that one has to be ́
careful when dealing with stochastic integrals. For a martingaleMthe integral
H·Mmight exist as a semi-martingale but not as a local martingale! We
conclude that the local martingales do not form a closed subspace (w.r. to the
semi-martingale topology) of the space of semi-martingales and by the same
example we see that the space of special semi-martingales is not closed in the
space of semi-martingales. SeeEmery and M ́ ́ emin for details [E 79], [M 80].

The proof of Theorem 7.3.3 will be based on two results, the first stating
that under the assumptions of Theorem 7.3.3 the stochastic integralH·A
necessarily exists as a Lebesgue-Stieltjes integral. The second is a necessary
and sufficient condition for a stochastic integral of a local martingale to be a
local martingale.

Lemma 7.3.5.IfS is a special Rd-valued semi-martingale with canonical
decompositionS=M+A,ifH is anRd-valued predictable process, if the
stochastic integralH·Sis special, then the processH·Aexists as a Lebesgue-
Stieltjes integral.

Proof.We start by localising which allows us to assume that the special semi-
martingalesSandH·Sare of the formS=M+Aand (H·S)=N+B
such thatMandNare martingales bounded inH^1 (P)andAandBare
predictable processes of integrable variation. We also may represent theRd-
valued processAasdA=βd|A|,where(|A|t)t≥ 0 is the total variation process
of (At)t≥ 0 andβis anRd-valued predictable process taking values in the unit
sphere ofRd.
We now define the{− 1 , 0 , 1 }-valued predictable process

Kt=sign(Ht,βt).