The Mathematics of Arbitrage

290 14 The FTAP for Unbounded Stochastic Processes

The Crucial Lemma 14.3.5.Let(E,E,π)be a probability measure space
and let(Fη)η∈Ebe a family of probability measures onRdsuch that the map
η→FηisE-measurable.
Let us assume thatF satisfies the property that for each measurable map
x:E→Rd,η→xηwith the property that for everyη∈Ewe have(xη,y)≥
− 1 ,forFηalmost everyy, we also have that

∫

RdFη(dy)(xη,y)≤^0.
Letε:E→R+{ 0 }beE-measurable and strictly positive.
Then, we may find anE-measurable mapη→GηfromEto the probability
measures onB(Rd)such that, forπ-almost everyη∈E,

(i) Fη∼Gηand‖Fη−Gη‖<εη,
(ii)EGη[‖y‖]<∞andEGη[y]=0.

Proof.As observed above the setP(Rd) of probability measures onRd,en-
dowed with the weak-star topology is a Polish space. We will show that the set
{
(η, μ)

∣

∣

∣

∣

∫

Rd

‖x‖dμ <∞;

∫

Rd

xdμ=0;Fη∼μ;‖μ−Fη‖<ε(η)

}

is inE⊗B(P(Rd)). Since, by Lemma 14.3.4, for almost allη, the measureFη
satisfies the no-arbitrage assumption of Definition 14.3.3, we obtain that, for
almost allη, the vertical section is non-empty. We can therefore find a mea-
surable selectionGηand this will then end the proof.
The proof of the measurability property is easy but requires some argu-
ments.
First we observe that the setM={μ|

∫

Rd‖x‖dμ <∞}is inB(P(R

d)).

This follows from the fact thatμ→

∫

‖x‖dμis Borel-measurable as it is an
increasing limit of the weak-star continuous functionalsμ→

∫

min(‖x‖,n)dμ.

Next we observe thatM→Rd;μ→

∫

Rdxdμis Borel-measurable.
The third observation is that{(η, μ)|‖μ−Fη‖<ε(η)}is inE⊗B(P(Rd)).
Finally we show that{(η, μ)|μ∼Fη}is also inE⊗B(P(Rd)). This will
then end the proof of the measurability property.
We take an increasing sequence of finiteσ-algebrasDnsuch thatB(Rd)
is generated by

⋃

nDn.Foreachnwe see that the mapping (η, μ, x) →

dμ

dFη

∣

∣

∣

Dn

(x)=qn(η, μ, x)isE⊗B(P(Rd))⊗B(Rd)-measurable. The mapping

q(η, μ, x) = lim infqn(η, μ, x)

is clearlyE⊗B(P(Rd))⊗B(Rd)-measurable. By the martingale convergence
theorem we have that for eachμ, the mappingqdefines the Radon-Nikod ́ym
density of the part ofμthat is absolutely continuous with respect toFη.Now
we have that

{(η, μ)|μ∼Fη}

=

{

(η, μ)

∣

∣

∣

∣

∫

Rd

q(η, μ, x)dFη(x) = 1; lim

n

∫

Rd

(nq)∧ 1 dFη(x)=1

}

and this shows that{(η, μ)|μ∼Fη}is inE⊗B(P(Rd)).