The Mathematics of Arbitrage

14.3 One-period Processes 293

Defining

dQ

dP

=

∏T

t=1

Zt,

we obtain a probability measureQ,Q∼Psuch that (St)Tt=0is a martingale
underQ. Indeed,

EQ[∆St|Ft− 1 ]=EP

[

∆St

∏T

u=1

Zu

∣

∣

∣

∣

∣

Ft− 1

]

=

(t− 1

∏

u=1

Zu

)

EP

[

Zt∆St

∏T

u=t+1

Zu

∣

∣

∣

∣

∣

Ft− 1

]

=

(t− 1

∏

u=1

Zu

)

EQ(t)

[

∆St

∏T

u=t+1

Zu

∣

∣

∣

∣

∣

Ft− 1

]

=0

and

EQ

[

‖∆St‖Rd

]

≤

∥

∥

∥∥

∥

t∏− 1

u=1

Zu

∥

∥

∥∥

∥

∞

·EQ(t)

[∥

∥

∥∥

∥

∆St

∏T

u=t+1

Zu

∥

∥

∥∥

∥

Rd

]

<∞.

Finally we may estimate‖Q−P‖ 1 by

‖Q−P‖ 1 =EP

[∣

∣∣

∣

∣

∏T

t=1

Zt− 1

∣

∣∣

∣

∣

]

≤EP

[T

∑

t=1

∣

∣

∣

∣

∣

∏t

u=1

Zu−

t∏− 1

u=1

Zu

∣

∣

∣

∣

∣

]

≤

∑T

t=1

∥

∥

∥

∥

∥

t∏− 1

u=1

Zu

∥

∥

∥

∥

∥

L∞(P)

EP[|Zt− 1 |]

≤T· 4 Tε 4 −TT−^1 =ε.

The proof of the first part of Proposition 14.3.7 is thus finished and we have
shown in the course of the proof that we may findQsuch that, in addition
to the assertions of the proposition,ddQPis uniformly bounded.
As regards the final assertion, letP′be anyP-absolutely continuous mea-
sure. For givenε>0, first takeP′′∼Psuch that‖P′′−P′‖<ε.Now
apply the first assertion withP′′replacingP. As a result we get an equivalent
martingale measureQsuch that‖Q−P′′‖<ε, hence also‖Q−P′‖< 2 ε.
This finishes the proof of Proposition 14.3.7.