238 12 Absolutely Continuous Local Martingale Measures
First note that, for each strictly positive operatorb 0 ,themapb→b−^1 is
continuous atb 0.
To calculatePawe simply take the limit
lim
ε↘ 0
a◦(a+εid)−^1.This constructive definition shows that the mapping a →Pais a Borel-
measurable mapping. The same trick is used to obtain the generalised inverse
a−^1 = lim
ε↘ 0a◦(a+εid)−^2.The processesσ−^1 andPσare therefore still predictable since they are the
composition of a predictable and a Borel-measurable mapping.
We will now describe a kind of absolute continuity of a vector measure
with respect to an operator-valued measure. Letνbe a measure defined on
theσ-ring of relatively compact Borel sets ofR+and taking values inRd.
Letμbe a measure defined on the sameσ-ring and taking values in Pos(Rd).
We say thatν μ, if wheneverf:R+→Rdis a Borel function such that
eitherf(t)=0or‖f(t)‖= 1, the expressiondμ f= 0 (as a vector measure)
impliesf′dν= 0 (as a scalar measure). (Heref′is the transpose off). One
can show that in this case the measureνhas a Radon-Nikod ́ym derivative
with respect toμ. Again we will need a predictable version of this theorem,
so we give details.
Suppose thatA:R+×Ω→Rdis predictable, cadlag and of finite variation
on finite intervals. Suppose thatA 0 =0.LetVbe as above, predictable, cadlag
taking values in Pos(Rd) and increasing. Suppose that for every predictable
processf:R+×Ω→Rd, such that‖f(t, ω)‖is either 0 or 1, the relation
dV f= 0 implies thatf′dA= 0. This means thatdA dVin a predictable
way. Letλ=trace(V)andletNbe a predictable null set forλ, i.e. (^1) Ndλ=0.
For such a predictable setNand for each predictablekwe have (^1) NdV k=0.
The hypothesis onAthen implies that (^1) Nk′dA= 0. This shows thatdA dλ
and the predictable Radon-Nikod ́ym theorem (applied for each coordinate)
shows the existence of a predictableRd-valued processgsuch thatdA=gdλ.
Now (id−σ◦σ−^1 )dV =dV(id−σ◦σ−^1 )=(id−σ◦σ−^1 )σdλ=0and
by the assumption onAwe have (id−σ◦σ−^1 )dA=0.Thisimpliesthat
(id−σ◦σ−^1 )gdλ= 0 and that up to null sets forλ,wehaveg∈R(σ). Now
leth=σ−^1 (g). Then obviouslyσ(h)=g(becauseg∈R(σ)),h∈R(σ)and
dA=σhdλ=dV h. The rangeR(σ) could have been called the infinitesimal
rangeR(dV) of the measureV. It is easy to show that it does not depend on
the control measure. We completed the proof of the following theorem.
Theorem 12.2.3.IfVis an increasing predictable cadlag process, taking val-
ues in the cone of the positive semi-definite operators onRd, then the vector
measure defined by the predictableRd-valued cadl ag processAof finite vari-
ation is of the formdA=dV h, for some predictableRd-valued processh,if