The Mathematics of Arbitrage

7.3 Strategies, Semi-martingales and Stochastic Integration 125

Clearly the Lebesgue-Stieltjes integrability ofHwith respect toAis tan-
tamount to that ofKH. The latter is easier to check asKH·Ais an increasing
process. Note thatKH·S=K·N+K·Bis still the sum of anH^1 (P)-bounded
martingale and a process of integrable variation so that, for eacht<∞,

E

[

sup

0 ≤u≤t

(KH·S)u

]

<∞.

Fixt<∞.LetHn=H (^1) {|H|≤n},forn∈N, and define the stopping times
τnby
τn=inf{u≤t||(KH·S)u−(KHn·S)u|> 1 }∧t.
AsKHisS-integrablewehavethat(τn)∞n=1increases stationary tot.We
may estimate
|(KHn·S)τn|≤|(KH·S)τn|+1+|Hτn∆Sτn|
≤3sup
0 ≤u≤t
|((KH)·S)u|+1.
AsKHnis bounded andMis bounded inH^1 (P), the process (KHn)·M
is anH^1 (P)-bounded martingale too (starting at ((KHn)·M) 0 =0)sothat
E[((KHn)·A)τn]=E[((KHn)·S)τn]−E[((KHn)·M)τn]
≤ 3 E

[

sup

0 ≤u≤t

((KH)·S)u

]

+1<∞.

Lettingn→∞we obtain by the monotone convergence theorem that

E[((KH)·A)t]<∞,

which proves the lemma.

Example 7.3.6.LetSbe a special semi-martingale with canonical decomposi-
tionS=M+A. For anS-integrable predictable processH, the integralH·S
may exist, butH·Amay not. Such an example can be made up as follows.
Take a random variableTwhich is exponentially distributed with parame-

ter 1 and letSt= (^1) {t≥T}with its natural filtration. Clearly the canonical
decomposition ofSis given byS=M+Awhere
Au=
∫u∧T
0
ds=(u∧T)
is the compensator of the processS.
LetHbe defined byHt= 1 −^1 t,for0≤t<1, andHt=0fort≥1. We
find that
(H·S)t=

⎧

⎨

⎩

0 , fort<T,

1

1 −T, forT≤tandT<^1 ,

0 , otherwise.