The Mathematics of Arbitrage

298 14 The FTAP for Unbounded Stochastic Processes

The boundedness property translates to the fact thatH=(Ht(ω))t∈R+is
anadmissible integrandfor the processXˇ. This follows from the definition of
the compensatorνin the following way. For each natural numbernwe have,
according to the definition of the compensator that

E

[(

〈Hω,t,∆ST(ω)〉−

)n

(^1) T<∞

]

=E

[∫

R+×Rd

(

〈Hω,t,y〉−

)n

μ(ω, dt, dy)

]

=E

[∫

R+×Rd

(

〈Hω,t,y〉−

)n

ν(ω, dt, dy)

]

=E

[∫

R+

∫

Rd

(

〈Hω,t,y〉−

)n

Fω,t(dy)dAt(ω)

]

≤E

[∫

R+

∫

Rd

1 Fω,t(dy)dAt(ω)

]

≤E

[∫

R+×Rd

1 ν(ω, dt, dy)

]

≤E

[∫

R+×Rd

1 μ(ω, dt, dy)

]

≤Q 1 [T<∞].

Since the inequality holds for eachnwe necessarily have that

(H·Xˇ)t≥− 1 a.s., for allt∈R+.

Noting thatMis a (locally bounded) local martingale andBis of lo-
cally bounded variation, we may find a sequence of stopping times (Uj)∞j=1
increasing to infinity, such that, for eachj∈N,

(1)MUjis a martingale, bounded in the Hardy spaceH^1 (Q 1 ), and
(2)BUjis of bounded variation.

Hence, for each predictable setP containedin[[0,Uj]] , f o r s o m e j∈N,we

have thatH (^1) Pis an admissible integrand forSandH (^1) P·Mis a martingale
bounded inH^1 (Q 1 ) and therefore
EQ 1 [(H (^1) P·M)∞]=0.
As by hypothesis
EQ 1 [(H (^1) P·S)∞]≤ 0
we obtain
EQ 1

[

(H (^1) P·(Xˇ+B))∞

]

≤ 0.