208 10 Counter-Example
applies to the more general situation of a locally bounded semi-martingaleS,
but in the situation of continuousS, perhaps the result can be proved more
simply (see e.g. Chap. 12 for the case of continuous processes and its relation
to noarbitrage). In particular, it is tempting to defineQby looking at the
decomposition (10.1) ofSand by setting
dQ
dP∣
∣
∣
∣
Ft=E(−α·M)tprovided the exponential local martingaleE(−α·M) is a true martingale. Is
it possible that there exist an equivalent local martingale measure forS,and
yet the exponential local martingaleE(−α·M) fails the martingale property?
The answer is yes; our example shows it. In the terminology of [FS 91], this
means that theminimal local martingale measurefor the processSdoes not
exist, althoughMe(S)isnon-empty.
A second question where our example finds interesting application is in
hedging of contingent claims in incomplete markets. The positive contingent
claimg, or more generally a function that is bounded below by a constant,
can be hedged ifgcan be written as
g=c+(H·S)∞, (10.2)wherecis a positive constant and whereHis some admissible integrand (i.e.
for some constanta∈R,(H·S)≥−a). In order to avoidsuicide strategies
we also have to impose that (H·S)∞is maximal in the set of outcomes
of admissible integrands (see Chaps. 11 and 13) for information on maximal
elements and [HP 81] for the notion ofsuicide strategies). We recall that an
outcome (H·S)∞of an admissible strategyH, is called maximal if for an
admissible strategyKthe relation (K·S)∞≥(H·S)∞implies that (K·
S)∞=(H·S)∞. S. Jacka [J 92], J.P. Ansel, Ch. Stricker [AS 94] and the
authors showed thatgcan be hedged if and only if there is an equivalent local
martingale measureQ∈Mesuch that
EQ[g]=sup{ER[g]|R∈Me}.Looking at (10.2) we then can show thatH·Sis aQ-uniformly integrable
martingale and hence thatc=EQ[g]. Also the outcome (H·S)∞is then
maximal. It is natural to conjecture that in fact for allR∈Me,wemight
haveER[g]=c, and the sup becomes unnecessary, which is the case for
bounded functionsg. However, our example shows that this too is false in
general.
To describe our example, suppose thatBandW are two independent
Brownian motions and letLt=exp(Bt−^12 t). ThenLis a strict local mar-
tingale. For information on continuous martingales and especially martingales
related to Brownian motion we refer to Revuz and Yor [RY 91]. Let us recall
that a local martingale that is not a uniformly integrable martingale is called