282 14 The FTAP for Unbounded Stochastic Processes
Example 14.2.3. A sigma-martingaleSwhich does not admit an equivalent
local martingale measure.
With the notation of the above example define theR^2 -valued processS=
(S^1 ,S^2 ) by lettingS^1 =XandS^2 the compensated jump at timeT∧Ui.e.,
St^2 ={
− 2 t fort<T∧U
1 −2(T∧U)fort≥T∧U.(Observe thatT∧Uis exponentially distributed with parameter 2).
ClearlyS^2 is a martingale with respect to the filtration (Ft)t∈R+generated
byS.
Denoting by (Gt)t∈R+ the filtration generated byS^2 , it is a well-known
property of the Poisson-process (see, [J 79, p. 347]) that onGthe restriction of
PtoG=
∨
t∈R+Gtis the unique probability measure equivalent toPunder
whichS^2 is a martingale. It follows thatPis the only probability measure
onF =
∨
t∈R+Ftequivalent toPunder whichS=(S(^1) ,S (^2) ) is a sigma-
martingale.
AsSfails to be a local martingale underP(its first coordinate fails to be
so) we have exhibited a sigma-martingale for which there does not exist an
equivalent martingale measure.
Remark 14.2.4.In the applications to Mathematical Finance and in particu-
lar in the context ofpricing and hedging derivative securities by no-arbitrage
argumentsthe object of central interest is the set ofstochastic integralsH·S
on a given stock price processS,whereHruns through theS-integrable pre-
dictable processes such that the processH·Ssatisfies appropriate regularity
condition. In the present context this regularity condition is the admissibility
conditionH·S≥−Mfor someM∈R+(see [HP 81], Chap. 9 and Sect. 14.4
below). In different contexts one might impose anLp(P)-boundedness con-
dition on the stochastic integralH·S(see, e.g., [K 81], [DH 86], [Str 90],
[DMSSS 97]). In Sect. 14.5, we shall deal with a different notion of admis-
sibility, which is adjusted to the case of big jumps.
Now make the trivial (but nevertheless crucial) observation: passing from
Stoφ·S,whereφis a strictly positiveS-integrable predictable process,does
not change the set of stochastic integrals.Indeed,wemaywrite
H·S=(Hφ−^1 )·(φ·S)
and, of course, the predictableRd-valued processHisS-integrable iffHφ−^1
isφ·S-integrable.
The moral of this observation: when we are interested only in the set of
stochastic integralsH·Sthe requirement thatSis a sigma-martingale is just
as good as the requirement thatSis a true martingale.
We end this section with two observations which are similar to the results
in [E 80]. The first one stresses the distinction between the notions of a local
martingale and a sigma-martingale.