The Mathematics of Arbitrage

14.3 One-period Processes 287

Indeed:

EQ 1

[

exp

(

η(x, S 1 )−

)

(x, S 1 )

]

=−EQ 1

[

exp

(

η(x, S 1 )−

)

(x, S 1 )−

]

+EQ 1

[

(x, S 1 )+

]

<−EQ 1

[

(x, S 1 )−

]

+EQ 1

[

(x, S 1 )+

]

≤ 0.

The measureQ 2 does not necessarily satisfy the requirement thatEQ 2 [‖S 1 ‖]<
∞. We therefore make a last transformation and we define

dQ=

exp(−δ‖S 1 ‖)

EQ 2 [exp(−δ‖S 1 ‖)]

dQ 2.

Forδ>0 tending to zero we obtain that‖Q−Q 2 ‖tends to 0 andEQ[(x, S 1 )]
tends toEQ 2 [(x, S 1 )] which is strictly negative. So forδsmall enough we find
a probability measureQsuch thatQ∼P,‖Q−Q 1 ‖<ε,EQ[‖S 1 ‖]<∞
andEQ[(x, S 1 )]<0, a contradiction to the choice ofx.

Lemma 14.3.2 in conjunction with Lemma 14.3.1 implies in particular that,
given the stochastic processS=(St)^1 t=0withS 0 ≡0andF 0 trivial, we may
find a probability measureQ∼Psuch thatSis aQ-martingale. We obtained
the measureQin two steps: first (Lemma 14.3.1) we foundQ 1 ∼Pwhich
took care of theadmissible integrands, which means that

EQ 1 [(x, S 1 )]≤ 0 , forx∈Adm.

In a second step (Lemma 14.3.2) we foundQ∼Psuch thatQtook care
ofall integrands, i.e.,

(x,EQ[S 1 ]) =EQ[(x, S 1 )]≤ 0 , forx∈Rd

and therefore
EQ[S 1 ]=0,

which means thatSis aQ-martingale.
In addition, we could assert in Lemma 14.3.2 that‖Q 1 −Q‖<ε,aproperty
which will be crucial in the sequel.
The strategy for proving the main theorem will be similar to the above ap-
proach. Given a semi-martingaleS=(St)t∈R+defined on (Ω,F,(Ft)t∈R+,P)
we first replacePbyQ 1 ∼Psuch thatQ 1 “takes care of the admissible
integrands”, i.e.,

EQ 1 [(H·S)∞]≤ 0 , forH-admissible.

For this first step, the necessary technology has been developed in Chap. 9
and may be carried over almost verbatim.
The new ingredient developed in the present paper is the second step which
takes care of the “big jumps” ofS.Byrepeatedapplicationofanargumentas
in Lemma 14.3.2 above we would like to changeQ 1 into a measureQ,Q∼P,