8.2 The Crucial Lemma 137This result is a good illustration for our philosophy: supposewe know
already that the assumption of 8.2.4 implies thatSis a local martingale
under someQequivalent toP. Then the conclusion follows immediately from
known results: from Ansel-Stricker (Theorem 7.3.7) we know thatH·Sis a
super-martingale. AsH·Sis bounded from below, Doob’s super-martingale
convergence theorem (see, e.g., [W 91]) implies the almost sure convergence of
(H·S)tast→∞, to an a.s. finite random variable.
Our goal is to replace these martingale arguments by some arguments
relying only on(NFLVR). The nice feature is that these arguments also allow
for an economic interpretation.
Proof of Lemma 8.2.4.As in the usual proof of Doob’s super-martingale con-
vergence theorem we consider the number of up-crossings: to show almost sure
convergence of (H·S)t,fort→∞,weconsider,foranyβ<γ,theP-measure
of the set{ω|(H·S)t(ω) upcrosses ]β, γ[ infinitely often}.Weshallshowthat
it equals zero.
So suppose to the contrary that there isβ<γsuch that the set
A={ω|(H·S)tupcrosses ]β, γ[ infinitely often}satisfiesP[A]>0. The economic interpretation of this situation is the follow-
ing: an investor knows at time zero that, when applying the trading strategy
H, with probabilityP[A]>0 her wealth will infinitely often be less than or
equal toβand infinitely often be more than or equal toγ.Asmartinvestor
will realise that this offers a free lunch with vanishing risk, as she can modify
Hto obtain a very rewarding trading strategyK.
Indeed, define inductively the sequence of stopping times (σn)∞n=0 and
(τn)∞n=0byσ 0 =τ 0 = 0 and, forn≥1,
σn=inf{t≥τn− 1 |(H·S)t≤β},
τn=inf{t≥σn|(H·S)t≥γ}.The setAthen equals the set where,σnandτnare finite, for eachn∈N
(as usual, the inf over the empty set is taken to be +∞).
What every investor wants to do is to “buy low and sell high”. The
above stopping times allow her to do that in a systematic way: define
K =H (^1) {∪∞n=1]]σn,τn]]}, which is clearly a predictableS-integrable process.
A more verbal description ofKgoes as follows: the investor starts by doing
nothing (i.e., making a zero-investment into the risky assetsS^1 ,...,Sd)until
the timeσ 1 when the process (H·S)thas dropped belowβ(Ifβ≥0, we have
σ 1 = 0)). At this time she starts to invest according to the rule prescribed by
the trading strategyH; she continues to do so until timeτ 1 when (H·S)tfirst
has passed beyondγ.Notethat,ifτ 1 (ω) is finite, our investor following the
strategyKhas at least gained the amountγ−β.Attimeτ 1 (if it happens
to be finite) the investor clears all her positions and does not invest into the
risky assets until timeσ 2 , when she repeats the above scheme.