188 9 Fundamental Theorem of Asset Pricing
L^2. Therefore there is a sequence of convex combinationsgn∈conv{hk;k≥n}
that converges tohinL^2 and therefore in probability. The sequencegnis
bigger than−1 and by the no-arbitrage propertygnis the final value of 1-
admissible integrandsHn(see Proposition 9.3.6). The property ofSnow says
thath=0.
The difference between(NFLVR)and(NFLBR)is now clear. In the no
free lunch with vanishing risk property we deal with sequences such that the
negative parts tend to 0 uniformly. In the no free lunch with bounded risk
property we only require these negative parts to tend to 0 in probability and
remain uniformly bounded!
If the case of an infinite time horizon the setK 0 was defined using general
admissible integrands. The infinite time horizon and especially strategies that
require action until the very end, are not easy to interpret. It would be more
acceptable if we could limit the properties(NFLBR)and(NFLVR)to be
defined with integrands having bounded support. The following proposition
remedies this. (We recall as already stated in the remark following Corol-
lary 9.3.4 that an integrandHis of bounded support ifHis zero outside
a stochastic interval [[0,k]] f o r s o m e r e a l n u m b e rk.)
Proposition 9.6.3.
(1)If the semi-martingaleSsatisfies (NFLBR) for integrands with bounded
support, then it satisfies (NA) for general admissible integrands.
(2)If the semi-martingaleSsatisfies (NFLVR) for integrands with bounded
support and (NA) for general integrands, then it satisfies (NFLVR) for
general integrands.
Proof.We start with the remark that ifSsatisfies(NFLVR)for integrands
with bounded support then from Theorem 9.3.3 it follows that for each ad-
missibleH, the limit (H·S)∞= limt→∞(H·S)texists and is finite almost
everywhere. We now show (1) of the proposition. Letg=(H·S)∞forH
1-admissible and suppose thatg≥0 almost everywhere. Letgn=(H·S)n.
Clearlyg−ntends to 0 in probability and eachgnis the result of a 1-admissible
integrand with bounded support. The property(NFLBR)for integrands with
bounded support shows thatg≥0 implies thatg= 0. The semi-martingale
therefore satisfies(NA)for admissible integrands.
We now turn to (2) of the proposition. Letgn=(Hn·S)∞withHn
admissible, be a sequence such that the sequenceg−ntends to 0 inL∞-norm.
Because the processSsatisfies(NA)it follows from Proposition 9.3.6 that
eachHnis‖q−n‖∞-admissible. For eachnwe taketnbig enough so thathn=
(Hn·S)tnis close tognin probability, e.g. such thatE[min(|hn−gn|,1)]≤^1 n.
Since eachhnis the result of a‖gn−‖∞-admissible integrand with bounded
support, the property(NFLVR)for integrands with bounded support implies
thathntends to 0 in probability. As a result we obtain that alsogntends to
0 in probability.