The Mathematics of Arbitrage

302 14 The FTAP for Unbounded Stochastic Processes

(ii) Qk+1|FTk−=Qk|FTk−anddQdQk+1k isFTk-measurable.

(iii)S(k)is a sigma-martingale underQk+1.

The condition in (i) above is chosen such that we may apply Lemma 14.4.3
to conclude that
Q= lim
k→∞

Qk

exists and is equivalent toP. From (ii) and (iii) it follows that eachS(k)
is a sigma-martingale underQl,foreachl≥k. It follows that eachS(k)is
aQ-sigma-martingale and henceS, being a local sigma-martingale is then
a sigma-martingale (see Corollary 14.2.7 above). This proves the first part of
Proposition 14.4.4.
As regards the final assertion of Proposition 14.4.4 note that, for any
predictable stopping timeU, the random times

Uk=

{

U ifTk− 1 <U≤Tk

∞ otherwise

are predictable stopping times, fork=1, 2 ,....Indeed, as easily seen, the set
{Tk<U≤Tk+1}is inFU−, showing thatUkis predictable.
By our construction and property (ii) above we infer that, fork=1, 2 ,...,

dQ|FUk

dQ 1 |FUk

=

dQ|F(Uk)−

dQ 1 |F(Uk)−

a.s.

which implies that
dQ|FU
dQ 1 |FU

=

dQ|FU−

dQ 1 |FU−

a.s..

The proof of Proposition 14.4.4 is complete now.

Proposition 14.4.4 contains the major part of the proof of the main the-
orem. The missing ingredient is still the argument for the predictable jumps
ofS. The argument for the predictable jumps given below will be similar to
(but technically easier than) the proof of Proposition 14.4.4.

Proof of the Main Theorem 14.1.1.LetSbe anRd-valued semi-martingale
satisfying the assumption(NFLVR). By Theorem 14.4.1 we may find a prob-
ability measureQ 1 ∼Psuch that,

EQ 1 [f]≤ 0 , forf∈C.

We also may find a sequence (Tk)∞k=1of predictable stopping times exhaust-
ing the accessible jumps ofS, i.e., such that for each predictable stopping time
TwithP[T=Tk<∞] = 0, for eachk∈N,wehavethatST−=STalmost
surely. We may and do assume that the stopping times (Tk)∞k=1are disjoint,
i.e., thatP[Tk=Tj<∞]=0fork=j.