288 14 The FTAP for Unbounded Stochastic Processes
such thatSbecomes aQ-martingale. A glance at Example 14.2.3 above reveals
that this hope is, in the general setting, too optimistic and we can only try
to turnSinto aQ-sigma-martingale. This will indeed be possible, i.e., we
shall be able to findQand a strictly positive predictable processφ,such
that,for every — not necessarily admissible — predictableRd-valued process
Hsatisfying‖H‖Rd≤φ,wehavethatH·Sis aQ-martingale. In particular
φ·Swill be aQ-martingale.
In order to complete this program we shall isolate in Lemma 14.3.5 below,
the argument proving Lemma 14.3.2 in the appropriate abstract setting. In
particular we show that the construction in the proof of Lemma 14.3.2 may
be parameterised to depend in a measurable way on a parameterηvarying in
a measure space (E,E,π). The proof of this lemma is standard but long. One
has to check a lot of measurability properties in order to apply the measurable
selection theorem. Since the proofs are not really used in the sequel and are
standard, the reader can, at a first reading, look at the Definition 14.3.3,
convince herself that the two parameterisations given in Lemma 14.3.4 are
equivalent and look at Lemma 14.3.5.
Definition 14.3.3.We say that a probability measureμonRdsatisfies the
(NA) property if for everyx∈Rdwe haveμ({a|(x, a)< 0 })> 0 as soon as
μ({a|(x, a)> 0 })> 0.
We start with some notation that we will keep fixed for the rest of this
section. We first assume that (E,E,π) is a probability space that is saturated
for the null sets, i.e. ifA⊂B∈Eand ifπ(B)=0thenA∈E. The probability
πcan easily be repaced into aσ-finite positive measure, but in order not
to overload the statements we skip this straightforward generalisation. We
recall that a Polish spaceXis a topological space that is homeomorphic to
a complete separable metrisable space. The Borelσ-algebra ofXis denoted by
B(X). We will mainly be working in a spaceE×XwhereXis a Polish space.
The canonical projection ofE×XontoEis denoted bypr.IfA∈E⊗B(X)
thenpr(A)∈E, see [A 65] and [D 72]. Furthermore there is a countable family
(fn)n≥ 1 of measurable functionsfn:pr(A)→Xsuch that
(1) for eachn≥1 the graph offnis a selection ofA, i.e.{(η, fn(η))|η∈
pr(A)}⊂A,
(2) for eachη∈pr(A)theset{fn(η)|n≥ 1 }is dense inAη={x|(η, x)∈
A}.
We call such a sequence a countable dense selection ofA.
The setP(Rd) of probability measures onRdis equipped with the topology
of convergence in law, also called weak-star convergence. It is well-known that
P(Rd) is Polish. IfF:E→P(Rd) is a mapping, then the measurability ofF
can be reformulated as follows: for each bounded Borel functiong,wehave
that the mappingη→
∫
Rdg(y)dFη(y)isE-measurable. This is easily seen
using monotone class arguments. Using such a given measurable functionF
as a transition kernel, we can define a probability measureλFonE×Rdas