204 9 Fundamental Theorem of Asset Pricing
Proof.We first take convex combinations of{fn−;n≥ 1 }that converge almost
surely. Since conv{fn−;n≥ 1 }is bounded inL^0 , the limit is finite almost
surely. We now apply the lemma to the same convex combination offn+.
This procedure yields convex combinations of the original sequence (fn)n≥ 1 ,
converging almost surely to a ]−∞,+∞]-valued function.
Remark 9.8.4.If in Remark 9.8.3 we only require that{fn−;n≥ 1 }is bounded
inL^0 then the conclusion breaks down. Indeed take (fn)n≥ 1 a sequence of 1-
stable (see [Lo 78] for a definition) independent random variables. If there were
convex combinations converging a.s. we could make the convex combinations
so thatgk ∈conv{fnk+1,...,fnk+1}wheren 1 <n 2 ....Thisimpliesthat
(gk)k≥ 1 is an independent sequence. Since convex combinations of independent
1-stable variables are 1-stable this would produce an i.i.d. sequence converging
almost surely, a contradiction.
Remark 9.8.5.If in the setting of Lemma 9.8.1 the sequence{fn;n≥ 1 }
is bounded in L^0 , but conv{fn;n ≥ 1 } not bounded in L^0 , then the
procedure used in the proof does not necessarily yield a functiongthat
is finite almost surely. The next example shows that there is a sequence
{fn;n≥ 1 }bounded inL^0 andsuchthateverygthat is a limit of func-
tionsgn∈conv{fn,fn+1,...}, is identically +∞.Beforewegivethecon-
struction let us recall some results from the theory of Brownian motion (see
[RY 91] for details). If (Bt) 0 ≤tis a standard 1-dimensional Brownian motion,
let us denote byTβthe stopping time defined asTβ=inf{t|Bt=β}.Itis
known (see [RY 91, p. 67]) that forβ>0,Tβ<∞a.s. and for eachu≥0:
E[exp(−uTβ)] = exp(−β
√
2 u). It follows that iffhas the same distribution as
Tβ,thenforλ>0,λfhas the same distribution asT(λ)^12 β.Iff 1 ,...,fNare in-
dependent and have the same distribution asTβ 1 ,...,TβNthenf 1 +···+fN
has the same distribution asTβ 1 +···+βN, (this follows easily from the inter-
pretation offnas the hitting time ofβn). Take now (fn)n≥ 1 a sequence of
independent identically distributed variables, each having the same distribu-
tion asT 1. Suppose thatgn∈conv{fn,fn+1,...}andgn→ga.e. We will
showg=+∞a.e. We can assume that the functionsgnare independent,
eventually we take subsequences. Eachgnhas a distribution of the form
λn 1 f 1 +···+λnNnfNnwhere (λn 1 ,...,λnNn) is a convex combination. From preceding considerations
it follows that the distribution ofgnisTαnwhereαn=
∑Nn
i=1√
λni ≥1. The
0-1-law gives us that eitherg=+∞or thatP[g<∞]=1.Inthiscasewe
conclude that there is a real numberαsuch thatαn→α≥1andghas the
same distribution asTα. From the 0-1-law it follows again that the distribution
ofgis degenerate, impossible ifα≥1. Thereforeg=+∞identically.
The following lemma is quite simple, it was used above in the proof of
Lemma 9.4.8 above.