The Mathematics of Arbitrage

11.2 Basic Theorems 219

11.2 Basic Theorems

Before proving the main results of the paper we need to recall some definitions
and notations introduced in Chap. 9.

Definition 11.2.1.AnRd-valued predictable processHis calleda-admissible
if it isS-integrable, ifH 0 =0, if the stochastic integral satisfiesH·S≥−a
and if(H·S)∞= limt→∞(H·S)texists a.s.. We say thatHis admissible if
it isa-admissible for some numbera.

The following notations will be used:

K={(H·S)∞|His admissible}

Ka={(H·S)∞|Hisa-admissible}

C 0 =K−L^0 +

C=C 0 ∩L∞.

The basic Theorem 9.1.1 uses the concept of no free lunch with vanish-
ing risk. This is a rather weak form of no-arbitrage-type and it is stated in
terms ofL∞convergence. The(NFLVR)property is therefore independent of
the choice of equivalent probability measure. Only the class of negligible sets
comes into play.

Definition 11.2.2.We say that the locally bounded semi-martingaleSsatis-
fies the no free lunch with vanishing risk or property (NFLVR), with respect
to general admissible integrands, if

C∩L∞+={ 0 },

where the bar denotes the closure in the sup-norm topology ofL∞.
The locally bounded semi-martingaleSsatisfies the no-arbitrage or (NA)
property with respect to general admissible integrands, if

C∩L∞+={ 0 }.

The fundamental theorem of asset pricing can now be formulated as fol-
lows:

Theorem 11.2.3.The locally bounded semi-martingaleSsatisfies the prop-
erty (NFLVR), with respect to general admissible integrands, if and only if
there is an equivalent probability measureQsuch thatSis aQ-local mar-
tingale. In this case the setCis already weak-star (i.e.σ(L∞,L^1 ))closedin
L∞.

Remark 11.2.4.IfQis an equivalent local martingale measure forSand if
HsatisfiesH·S≥−athen the result of Ansel-Stricker [AS 94] shows that
H·S is still a local martingale and hence, being bounded from below, is
a super-martingale. It follows that the limit (H·S)∞exists a.s. and that
EQ[(H·S)∞]≤0.