The Mathematics of Arbitrage

14.2 Sigma-martingales 281

Definition 14.2.1.AnRd-valued semi-martingale X =(Xt)t≥ 0 is called
a sigma-martingaleif there exists anRd-valued martingaleMand anM-
integrable predictableR+-valued processφsuch thatX=φ·M.

We refer to [E 80, Proposition 2] for several equivalent reformulations of
this definition and we now essentially reproduce the basic example given by
M.Emery [E 80, p. 152] which highlights the difference between the notion of ́
a martingale (or, more generally, a local martingale) and a sigma-martingale.

Example 14.2.2 ([E 80]). A sigma-martingale which is not a local martingale.
Let the stochastic base (Ω,F,P) be such that there are two independent
stopping timesTandUdefined on it, both having an exponential distribution
with parameter 1.
DefineMby

Mt=

⎧

⎨

⎩

0fort<T∧U

1fort≥T∧UandT=T∧U

−1fort≥T∧UandU=T∧U.

It is easy to verify thatM is almost surely well-defined and is indeed
a martingale with respect to the filtration (Ft)t∈R+ generated byM.The
deterministic (and therefore predictable) processφt=^1 t isM-integrable (in
the sense of Stieltjes) andX=φ·Mis well-defined:

Xt=

⎧

⎨

⎩

0fort<T∧U

1

T∧U fort≥T∧UandT=T∧U

−T∧^1 U fort≥T∧UandU=T∧U.

ButX fails to be a martingale asE[|Xt|]=∞, for allt>0, and it
is not hard to see thatXalso fails to be a local martingale (see [E 80]), as
E[|XT|]=∞for each stopping timeTthat is not identically zero. But, of
course,Xis a sigma-martingale.

We shall be interested in the class of semi-martingalesSwhichadmit an
equivalent measure under which they are a sigma-martingale. We shall present
an example of anR^2 -valued processS which admits an equivalent sigma-
martingale measure (which in fact is unique) but which does not admit an
equivalent local martingale measure. This example will be a slight extension
ofEmery’s example. ́
The reader should note that inEmery’s Example 14.2.2 above one may ́
replace the measurePby an equivalent measureQsuch thatX is a true
martingale underQ. For example, chooseQsuch that under this new measure
TandUare independent and distributed according to a lawμonR+such that
μis equivalent to the exponential law (i.e., equivalent to Lebesgue-measure
onR+)andsuchthatEμ

[ 1

t

]

<∞.