128 7 A Primer in Stochastic Integration
Corollary 7.3.8.IfMis a local martingale, ifHisM-integrable and if(H·
M)−is locally integrable, i.e., there is a sequence(τn)∞n=1of stopping times
increasing to infinity such thatE[−inf 0 ≤t≤Tn(H·M)t]<∞;thenH·Mis a
local martingale.
Proof.We only have to verify the hypothesis of theorem 7.3.7. For eachn
letRnbe defined asRn=inf{t|(H·M)t≥n}.Letalsoτnbe chosen in
such a way thatH·Mτn≥θn,whereeachθnis integrable andτn↗∞.
LetTn=min(Rn,τn). ClearlyTn↗∞and the jumpsH∆MTn≥−n+θn.
Theorem 7.3.7 gives the desired result.
Proof of Theorem 7.3.3.LetHbeS-integrable. IfH·Sis special then by
Lemma 7.3.5 the processH·Aexists as an ordinary Lebesgue-Stieltjes integral
which yields (ii). AsH·Sis special it is locally integrable, i.e., there is
a sequence of integrable functionsθn∈L^1 and an increasing sequence of
stopping timesTn↗∞such that (H·S)Tn≥θn.TakeRn↗∞such that
(H·A)Rnhas integrable variation. This is possible since the integralH·Ais an
ordinary integral and the result therefore is a predictable process. For eachn
we find (H·M)Rn∧Tn≥θn−
∫Rn
0 |HudAu|. We now apply the Ansel-Stricker
theorem to show that (i) holds true.
Conversely, if (i) and (ii) hold true thenH·Sis the sum of the local mar-
tingaleH·Mand the predictable finite variation processH·Aand therefore
special.