The Mathematics of Arbitrage

142 8 Arbitrage Theory in Continuous Time: an Overview

taking its values in]0,∞[, such that theRd-valued stochastic integralφ·Sis
a martingale.

To motivate this definition we recallEmery’s example 7.3.4: we have seen ́
there that the processX=H·Mdefined in (7.15) fails to be a local martin-
gale. Nevertheless, the above definition gives us a tool to interpret the process
Xas “something which has the essential features of a martingale”: defining
φt=twe find a (deterministic and therefore predictable) processφsuch that
φ·X=(φH)·M=Mis a martingale. HenceXis a sigma-martingale.

The notion of sigma-martingales was introduced (using slightly different
notation) by Chou [C 77] and further analyzed byEmery [E 80]. It is tailor- ́
made for our present purposes for the following two reasons:

Fact 1:In the setting of Theorem 8.3.1 (i) it is unavoidable to pass to a

concept going beyond the notion of a local martingale.

Fact 2:For the purposes of hedging contingent claims the notion of a sigma-

martingale is just as useful as the notion of a local martingale (or even

that of a martingale).

To justify these two facts we start with the second one: assume that
S=(St)t≥ 0 is a sigma-martingale so that there is a ]0,∞[-valued predictable
processφsuch thatS ̃=φ·Sis a martingale. LetHbe anyRd-valued pre-
dictable process. ThenHisS-integrable, iffH ̃:=Hφ isS ̃-integrable and in

this case the processesH·SandH ̃·S ̃are identical. This follows from the
rather trivial formula

H ̃·S ̃=(H

φ

)

·(φ·S)=H·S.

As a consequence, the class of processes{H·S|HisS-integrable}and
{H ̃·S ̃|H ̃isS ̃-integrable}coincide. Every statement pertaining only to this
class (such as Theorem 8.3.1 (ii)) remains unaffected by the passage from the
sigma-martingaleSto the martingaleS ̃.
As regards Fact 1 above, we construct in Example 14.2.3 a slight variant of
Emery’s Example 7.3.4 with the following property: the process ́ Sis a sigma-
martingale (underP)but,foreachQP,Sfails to be a localQ-martingale.
Hence the processSsatisfies(NFLVR), but there is no probability measure
QPsuch thatSis a localQ-martingale.

We now try to give a sketch of the strategy for the proof of Theorem 8.3.1,
whereSis a general (not necessarily locally bounded) semi-martingale. As
usual the implication (i)⇒(ii) is the easy one: it follows from the discussion
of Fact 2 above and the Ansel-Stricker Theorem 7.3.7 that, ifSis a sigma-
martingale underQ,andHan admissible integrand forS, the processH·S
is a super-martingale underQ. HenceEQ[f]≤0, for allf∈C, which implies
Theorem 8.3.1 (ii) by the usual arguments.