12.2 The Predictable Radon-Nikod ́ym Derivative 235complicated analytic proof had already been presented by the present au-
thors during the SPA conference in Amsterdam in June 1993 [DS 93] and in
the seminar of Tokyo University in September 93. Also [LS 94, Theorem 1]
corresponds to our Main Theorem 12.1.4 under the additional assumption
that the local martingale partMof the continuous semi-martingaleSis of
the formM=Σ·W,whereW is ad-dimensional Brownian motion defined
on its (saturated) natural filtration and Σ = (Σt) 0 ≤t≤ 1 an adapted matrix
valued process such that each Σtis invertible.
12.2 The Predictable Radon-Nikod ́ym Derivative
In this section we will prove Radon-Nikod ́ym theorems for stochastic mea-
sures. We first deal with the case of one-dimensional processes. A stochastic
measure onR+is described by a stochastic process of finite variation. In
our setting it is convenient to require that the measure has no mass at zero,
i.e. the initial value of the process is 0. If we have two predictable stochastic
measures defined by the finite variation processesAandB, respectively, we
can for almost everyωin Ω decompose theA-measure in a part absolutely
continuous with respect to theB-measure and a component that is singular
to it. We are interested in the problem whether such a decomposition can be
done in a measurable or even predictable way. Similar problems can be stated
for the optional and for the measurable case. For applications in Sect. 12.3,
we only need the case of continuous processes. However, the more general case
is almost the same and therefore we treat, at little extra cost, processes with
jumps.
Theorem 12.2.1.(i) IfA:R+×Ω→Ris a predictable, cadlag process of
finite variation on finite intervals, then the processV, defined by settingVt
equal to the variation ofAon the interval[0,t],iscadl ag and predictable.
(ii) IfA:R+×Ω→Ris a predictable, cadlag process of finite variation on
finite intervals, ifVis defined as in (i), there is a decomposition ofR+×Ω
into two disjoint, predictable subsets,D+andD−, such that
At=∫t0( (^1) D+− (^1) D−)dV.
(iii)IfA:R+×Ω→Ris a predictable, cadlag process of finite variation on
finite intervals, ifV is cadlag, predictable and increasing, then there are
predictableφ:R+×Ω→Rand a predictable subsetNofR+×Ωsuch
that
At=
∫
[0,t]φudVu+∫
[0,t](^1) N(u)dAu and
∫
R+(^1) NdVu=0.
Proof.(i) We give the proofs only in the caseA 0 =V 0 = 0. For the proof
we need some results from the general theory of stochastic processes (see