The Mathematics of Arbitrage

312 14 The FTAP for Unbounded Stochastic Processes

(1)Vt= lims↘t;s∈Q+limn→∞(Kn·S)sexists a.s., for allt≥0,
(2) (H^0 ·S)t−Vtis increasing,
(3)V 0 ≤0.

From this it follows that (H^0 ·S)t≥−V 0 +Vt≥Vt.SinceHnisw-admissible
(and hence (1,w)-admissible) we have thatKnis (1,w)-admissible and hence
we find thatVt≥−EQ[w|Ft] for allQ∈Meσsuch thatEQ[w]<∞.Itis
now clear thatH^0 isw-admissible. Since the sequenceαnis increasing, we
also obtain that for alltand allQ∈MeσwithEQ[w]<∞:

(Kn·S)t+αn+

1

n

≥EQ[(Kn·S)∞|Ft]+αn+

1

n

≥EQ[g∧n|Ft].

This yields that, for alltand alln,

(H^0 ·S)t+αn+

1

n

≥Vt+αn+

1

n

≥EQ[g∧n|Ft].

Ifttends to infinity this gives (H^0 ·S)∞+αn+^1 n≥g∧nfor alln.Bytaking
the limit overnwe finally find that

(H^0 ·S)∞+α≥g.

This shows the desired inequality and at the same time also shows that the
infimum is a minimum.

We are now ready to prove the duality results. We start with the case
of admissible integrands thus extending Theorem 11.3.4 to the case of non-
locally bounded processesS. Recall that we assume throughout this section
thatSis anRd-valued semi-martingale satisfying(NFLVR).

Theorem 14.5.20.For a non-negative random variablegwe have:

sup

Q∈Meσ

EQ[g]=inf{α|there isHadmissible andg≤α+(H·S)∞}.

Proof.From the previous lemmata it follows that we only have to show that

sup

Q∈Meσ

EQ[g]= sup

Q∈Mes

EQ[g].

This follows from Proposition 14.4.5 and the fact thatgis bounded from
below.

We now complete the proof for the case of feasible weight functionswand
w-admissible integrands:

Proof of Theorem 14.5.9.In this case we show thatMeσ=Mes,w.Wealready
observed thatMeσ⊂Mes,w.TakenowQ∈Mes,w.
Since w is now supposed to be a feasible weight function, we have
the existence of a strictly positive predictable function φsuch that (φ·