9.2 Definitions and Preliminary Results 1559.2 Definitions and Preliminary Results
Throughout the paper we will work with random variables and stochastic
processes which are defined on a fixed probability space (Ω,F,P). We will
without further notice identify variables that are equal almost everywhere.
The spaceL^0 (Ω,F,P), sometimes written asL^0 , is the space of equivalence
classes of measurable functions, defined up to equality almost everywhere.
The spaceL^0 is equipped with the topology of convergence in measure. It is
a complete metrisable topological vector space, a Fr ́echet space, but it is not
locally convex. The spaceL^1 (Ω,F,P) is the Banach space of all integrable
F-measurable functions. The dual space is identified withL∞(Ω,F,P)the
space of bounded measurable functions. The weak-star topology onL∞is the
topologyσ(L∞,L^1 ).
The existence of an equivalent martingale measure is proved using Hahn-
Banach type theorems. Central in this approach is the construction of a convex
weak-star-closed subset ofL∞. To prove that a set is weak-star-closed we will
use the following result. The proof essentially consists of a combination of the
classical Krein-Smulian theorem and the fact that the unit ball ofL∞under
the weak-star topology is an Eberlein compact. (see [D 75] or [G 54, Exercise
1, p. 321].
Theorem 9.2.1.IfCis a convex cone ofL∞thenCis weak-star-closed if
and only if for each sequence(fn)n≥ 1 inCthat is uniformly bounded by 1 and
converges in probability to a functionf 0 , we have thatf 0 ∈C.
The properties of stochastic processes are always defined relative to a fixed
filtration (Ft)t∈R+. This filtration is supposed to satisfy the usual conditions
i.e. the filtration is right continuous and contains all negligible sets: ifB⊂
A∈FandP[A]=0thenB∈F 0. We also suppose that theσ-algebraFis
generated by
⋃
t≥ 0 Ft. Stochastic intervals are denoted as [[T,S]] w h e r eS≤T
are stopping times and [[T,S]] ={(t, ω)|t∈R+,ω∈Ω,T(ω)≤t≤S(ω)}.
Stochastic intervals of the form ]]T,S]] etc. are defined in the same way. The
interval [[T,T]] is denoted by [[T]] and it is the graph of the stopping time
T,{(T(ω),ω)|T(ω)<∞}. We note that according to this definition the set
[[ 0,∞]] e q u a l sR+×Ω. Stochastic processes are indexed by a time set. In this
paper the time set will beR+. This will cover the case of infinite horizon
and indeed represents the general case since bounded time sets [0,t] can of
course be imbedded by requiring the processes to be constant after timet.It
also contains the case of discrete time sets, by requiring the processes and the
filtration to be constant between two consecutive natural numbers. A mapping
X:R+×Ω→Ris called an adapted stochastic process if for eacht∈R+the
mappingω→X(t, ω)=Xt(ω)isFt-measurable.Xis called continuous (right
continuous, left continuous), if for almost allω∈Ω, the mappingt→Xt(ω)
is continuous (right continuous, left continuous). Stochastic processes that
are indistinguishable are always identified. Other concepts such as optional
and predictable processes are also used in this paper and we refer the reader