The Mathematics of Arbitrage

9.2 Definitions and Preliminary Results 157

The theorem and more precisely its Corollary 9.2.4, will be used in
Sect. 9.4. It allows to control the jumps of the martingale part in the canonical
decomposition of a special semi-martingale.

Theorem 9.2.3.IfXis a semi-martingale satisfying‖(∆X)∗‖p<∞,where
1 <p≤∞,then

(a)Xis special and has a canonical decompositionX=M+A
(b)Asatisfies‖(∆A)∗‖p≤p−p 1 ‖(∆X)∗‖p;

(c)Msatisfies‖(∆M)∗‖p≤^2 pp−− 11 ‖(∆X)∗‖p.

Proof.SinceX is locallyp-integrable it is certainly locally integrable and
hence is special. (a) is therefore proved. LetX=M+Abe the canonical
decomposition whereAis the predictable process of finite variation andM
is the local martingale part. LetYbe the cadlag martingale defined asYt=
E[(∆X)∗|Ft].
SinceAis predictable the set{∆A=0}is the union of a sequence of sets
of the form [[Tn]] w h e r eTnare predictable stopping times. For each predictable
stopping timeTwe have that ∆AT=E[∆XT|FT−] and hence

|∆AT|≤E[|∆XT||FT−]≤E[(∆X)∗|FT−]=YT−≤Y∗.

This implies that (∆A)∗ ≤ Y∗. From Doob’s maximal inequality, see
[DM 80], it now follows that

‖Y∗‖p≤

p

p− 1

‖(∆X)∗‖p and therefore

‖(∆A)∗‖p≤

p

p− 1

‖(∆X)∗‖p and ‖(∆M)∗‖p≤

2 p− 1

p− 1

‖(∆X)∗‖p. 

Corollary 9.2.4.IfTis a stopping time then:

‖(∆A)T‖p≤

p

p− 1

‖(∆X)∗‖p;

‖(∆M)T‖p≤

2 p− 1

p− 1

‖(∆X)∗‖p.

Corollary 9.2.5.If 1 <p≤∞and the semi-martingale X satisfies
sup{‖(∆X)T‖p |T stopping time}=N<∞, then forp′ <pthere is
constantk(p, p′)depending only anpandp′such that

‖(∆A)∗‖p′≤k(p, p′)N.

Proof.Let the stopping timeT be defined asT =inf{t||(∆X)t|≥c}.
Fromthehypothesiswededucethat

∫

|∆XT|≤Npand this implies, by the
Markov-Tchebycheff inequality, thatcpP[(∆X)∗>c]≤Np. The rest follows
easily.