The Mathematics of Arbitrage

9.7 Simple Integrands 193

where 0≤T 0 ≤...≤TNn+1<∞are stopping times and the functions

fknareFTkn-measurable functions, bounded by 1.

For eachnwe putζtnthe process defined as

ζtn=

∑

(STkn+1−STkn)^2 ,

where the summation is done over the set of indicesk=0,...,Nnsuch
thatTkn+1≤t.
Since by the preceding lemmaGis bounded inL^0 ,thereisc>0such
thatP[ζn∞≥c]≤ ε 2 .Letforeachnthe stopping timeT

′
nbe defined as
T
′
n=inf{t|ζ
n
t ≥c}. This definition implies thatT

′
ntakesvaluesinthe
set{T 0 ,...,Tn+1,∞}and is a stopping time with respect to the discrete
time filtration (FTkn)k=0,...,Nn+1. The boundζTn′
n
≤c+4 (since|S|≤1) and

P[T

′

n<∞]≤

ε

2 are straightforward. Take nowK

n=Hn 1
[[ 0,Tn′]] and observe
thatP[(Kn·S)∞≥n]≥ε 2.
Each discrete time, stopped, process

(

Smin(Tkn,Tn′)

)

k=0,...,Nn+1is now de-
composed according to the discrete time Doob decomposition:

AnTkn+1−AnTkn=E

[

Smin(Tkn+1,Tn′)−Smin(Tkn,Tn′)

∣

∣

∣FTkn

]

MTnkn+1−MTnkn=

(

Smin(Tkn+1,Tn′)−Smin(Tkn,Tn′)

)

−E

[

Smin(Tkn+1,Tn′)−Smin(Tkn,Tn′)

∣

∣

∣FTkn

]

.

(

MTnkn

)

k=0,...,Nn+1is now a martingale bounded inL

(^2). Indeed

E

[(

MTnNn

n+1

) 2 ]

=

∑Nn

k=0

E

[(

MTnkn+1−MTnkn

) 2 ]

+E

[(

MTn 0 n

) 2 ]

≤

∑Nn

k=0

E

[(

Smin(Tkn+1,Tn′)−Smin(Tkn,Tn′)

) 2 ]

+E

[(

Smin(T 0 n,Tn′)

) 2 ]

≤

[

(ζTnn′)^2

]

+1≤c+5.

For eachtwe putMtn=E

[

MTnNn

n+1

∣

∣

∣Ft

]

and we take a c`adl`ag version

of this martingale. Because of the optional sampling theorem this definition
coincides with the previously given construction ofMtnfor timest=TNnk.In

the definition ofHnwe now replace eachfknbyf ̃kn=fknsign

(

AnTn

k+1

−AnTn

k

)

.

The functionsf ̃knare still measurable with respect to theσ-algebraFTkn.The

resulting process is denoted byK ̃ni.e.

K ̃n=

∑Nn

k=0

f ̃kn (^1) ]]Tn
k,Tkn+1]].