The Mathematics of Arbitrage

9.4 Proof of the Main Theorem 169

Proof.Suppose to the contrary that there isα>0, sequences (nk,mk)k≥ 1
tending to∞and for eachk:P

[

supt≥ 0

(

(Hnk·S)t−(Hmk·S)t

)

>α

]

≥α.

Define the stopping timesTkas

Tk=inf{t|(Hnk·S)t−(Hmk·S)t≥α}

so that we haveP[Tk<∞]≥α.

DefineLkasLk=Hnk (^1) [[ 0,Tk]]+Hmk (^1) ]]Tk,∞[[. The processLkis predictable
and it is 1-admissible. Indeed fort≤Tkwe have (Lk·S)t=(Hnk·s)t≥− 1
sinceHnkis 1-admissible. Fort≥Tkwe have
(Lk·S)t=(Hnk·S)Tk+(Hmk·S)t−(Hmk·S)Tk
≥(Hmk·S)t+α≥−1+α.
Denote limt→∞(Lk·S)tbyρk. From the preceding inequalities we deduce
thatρkcan be written asρk=φk+ψkwhere
φk=fnk (^1) {Tk=∞}+fmk (^1) {Tk<∞} and P[ψk≥α]≥α.
By assumptionφk →f 0 and by taking convex combination as in Lem-
ma 9.8.1 we may suppose thatψk →ψ 0 whereP[ψ 0 >0]>0. Therefore
convex combinations ofρk converge almost surely to an elementf 0 +ψ 0 ,
a contradiction to the maximality off 0.

Remark 9.4.7.Let us give an economic interpretation of the argument of the
proof. At timeTkwe know that the trading strategyHnkhas obtained the
result (Hnk·S)Tk, which is at leastαbetter than (Hmk·S)Tkon a set of
measure bigger thanα. On the other hand we know that, forkbig enough,
both strategies yield at time∞aresultclosetof 0. Having this information
the economic agent will switch from the strategyHnktoHmksince, starting
from a lower level,Hmkyields almost the same final result, i.e. the gain on
the interval ]]Tk,∞[[ is better forHmkthan forHnk.ThestrategyLkprecisely
describes this attitude.
The proof used convergence in probability. In the rest of the proof we will
make use of decomposition theorems, estimation of maximal functions etc.
These methods are easier when applied in an “L^2 -environment”. We therefore
replace the original measurePby a new equivalent measureQwe will now
construct.
First we observe that (Hn·S)tconverges uniformly int.Thevariable
q=supnsupt|(Hn·S)t|is therefore finite almost surely. ForQwe now take
a probability measure equivalent withPand such thatq ∈L^2 (Q) e.g. we
can takeQwith densityddQP=EPexp([exp(−q−)q)]. From the dominated convergence
theorem we then easily deduce that
lim
n,m→∞

∥

∥sup|(Hn·S)t−(Hm·S)t|

∥

∥

L^2 (Q)=0.