14.4 The GeneralRd-valued Case 299Using the identities
EQ 1[
(H (^1) P·(Xˇ+B))∞
]
=
∫
Ω×R+(Hω,t∗Fω,t+(Hω,t,bω,t)) (^1) PdA(ω, t)
=
∫
Ω×R+(Hω,t∗F ̃ω,t) (^1) PdA(ω, t)≤ 0
whichholdtrueforeachP ∈Pcontainedin[[0,Uj]] , f o r s o m e j ∈N,we
conclude that fordA-almost each (ω, t)wehave
Hω,t∗F ̃ω,t=EF ̃ω,t[(Hω,t,.)]≤ 0.
This inequality implies that assumption (2) in Lemma 14.3.4 is satisfied
and hence Fω,t satisfies the no-arbitrage property, i.e. the hypothesis of
Lemma 14.3.5 is satisfied.
Hence we may find a transition kernelG ̃ω,tas described by Lemma 14.3.5
— withεreplaced by ε 2 — and lettingGω,t= G ̃ω,tδ−b(ω,t) we obtain
a transition kernel satisfying (a) and (b) above.
We now have to translate the change of transition kernels fromFω,tto
Gω,tinto a change of measures fromQ 1 toQ 2 on theσ-algebraFTwhich will
be done by defining the Radon-Nikod ́ym derivativeddQQ^21. We refer to [JS 87,
III.3] for a treatment of the relevant version of Girsanov’s theorem for random
measures.
For (ω, t) fixed, denote byY(ω, t , .) the Radon-Nikod ́ym derivative of
Gω,twith respect toFω,t, i.e.
Y(ω, t, x)=
dGω,t
dFω,t
(x),x∈Rd,
which isFω,t-almost surely well-defined and strictly positive. We may and do
choose fordA-almost each (ω, t), a versionY(ω, t, x) such thatY(., ., .)is
P⊗B(Rd)-measurable.
We now define
dQ 2
dQ 1
(ω)=Z∞(ω)=Y(ω, T(ω),∆ST(ω)(ω)) (^1) {T<∞}+ (^1) {T=∞}
and Zt(ω)=Y(ω, T(ω),∆ST(ω)(ω)) (^1) {T≤t}+ (^1) {T>t}.
The intuitive interpretation of these formulas goes as follows: for fixedω∈
Ω we look at timeT(ω) which is the unique “big” jump of (St(ω))t∈R+.The
densityY(ω, T(ω),x) gives the density of the distribution of the compensated
jump measureGω,twith respect toFω,t, if the jump equalsxand therefore we
evaluateY(ω, T(ω),x)atthepointx=∆ST(ω)(ω) to determine the density
ofQ 2 with respect toQ 1 .IfT(ω)=∞the densityddQQ^21 (ω)issimplyequalto
- The variableY(ω, T(ω),∆ST(ω)) is certainly integrable. Indeed