The Mathematics of Arbitrage

300 14 The FTAP for Unbounded Stochastic Processes

E[Y(ω, T(ω),∆ST(ω)) (^1) T<∞]
=E

[∫

R+×Rd

Y(ω, t, y)μ(ω, dt, dy)

]

=E

[∫

R+×Rd

Y(ω, t, y)ν(ω, dt, dy)

]

=E

[∫

R+

∫

Rd

Y(ω, t, y)Fω,t(dy)dAt(ω)

]

=E

[∫

R+

∫

Rd

Fω,t(Rd)dAt(ω)

]

=E

[∫

R+×Rd

ν(ω, dt, dy)

]

=E

[∫

R+×Rd

μ(ω, dt, dy)

]

=Q 1 [T<∞].

The processZcanalsobewrittenas

Z=Y(ω, T,∆ST) (^1) [[T,∞[[+ (^1) [[ 0,T[[,
from which it follows thatZis a process of integrable variation. The maximal
functionZ∗ofZis therefore integrable.
In order to show thatQ 2 is indeed a probability measure and thatZt=
dQ 2 |Ft
dQ 1 |Ftwe shall show that (Zt)t∈R+is a uniformly integrable martingale closed
byZ∞.
We may writeZ=(Zt)t∈R+as
Z=1+(Y(ω, t, x)−1)∗μ.
From the definition of the compensatorν([JS 87, II.1.8]) we deduce that
we may write the compensatorZpofZ
Zp=1+(Y(ω, t, x)−1)∗ν
=1+((Y(ω, t, x)−1)∗Fω,t)·A.
Noting that, fordA-almost each (ω, t)wehavethat(Y(ω, t, x)−1)∗Fω,t=
EFω,t[Y(ω, t, x)−1] = 0 we deduce that the compensatorZpis constant.
SinceZ−Zpis a martingale, by definition of the compensator of processes of
integrable variation, it follows thatZis a martingale as well.
To estimate the distance‖Q 2 −Q 1 ‖,notethat
‖Q 2 −Q 1 ‖=EQ 1

[∣

∣

∣

∣^1 −

dQ 2

dQ 1

∣

∣

∣

∣

]

≤EQ 1 [(|Y(ω, t, x)− 1 |∗ν)∞]

≤EQ 1 (‖Fω,t−Gω,t‖·A)∞≤

ε

2

EQ 1 [A∞]≤

ε

2

.