96 6 The Dalang-Morton-Willinger Theorem
6.6 A Utility-based Proof of the DMW Theorem forT=1
forT=1
We give another proof of the theorem of Dalang-Morton-Willinger (for the
caseT= 1) which is based on the ideas of utility maximisation. This proof is
due to C. Rogers [R 94]. The basic idea is — transferring the results on utility
maximisation, obtained in Chap. 3 for the case of finite Ω, to the present
situation — that for the optimal investmentX̂, the functionU′(X̂) should
define the density of an equivalent martingale measure, up to a normalising
constant. This was proved in Theorem 3.1.3 above for the case of finite Ω
and the hope is, of course, that this result should also hold in a more general
context. Rogers’ idea was to exploit this basic relation to find an equivalent
martingale measure in the context of the theorem of Dalang-Morton-Willinger.
Among other features such an approach has the advantage of being more
constructive than the mere existence result provided by the theorem of Kreps-
Yan. On the other hand we note that the present proof will not yield a bounded
densityddQP, i.e., we do not obtain assertion (iii) of Theorem 6.1.1.
Let us fix theRd-valued processS=(S 0 ,S 1 ) based on (Ω,(Ft)^1 t=0,P)and
assume that it is free of arbitrage. As utility function we useU(x)=−e−x
and as initial endowmentx 0 =0.
Our utility maximisation problem consists of finding the optimal element
̂h∈L^0 (Ω,F 0 ,P;Rd) solving the maximisation problem
E[U((h,∆S))]→max!,h∈L^0 (Ω,F 0 ,P,Rd). (6.5)In order to assure a well-defined solution to this problem, we need
some preliminary work. As a first step we may suppose that, for each
h∈L∞(Ω,F 0 ,P;Rd),
E[|U((h,∆S))|]<∞. (6.6)
To guarantee that this integrability condition holds, it suffices to pass from
Pto the equivalent probability measureP′defined by
dP′
dP=cexp(
−‖∆S‖^2 Rd)
,
wherec>0 is a normalising constant. Hence we may suppose that the original
Psatisfies (6.6).
The idea of the proof is that, for any maximising sequence (hn)∞n=1in
L∞(Ω,F 0 ,P;Rn) for the optimisation problem (6.5), the stochastic integrals
fn:= (hn,∆S) automatically converge in measure to the optimal functionf̂;
if, in addition, (hn)∞n=1is in the predictable range ofS 1 −S 0 ,then(hn)∞n=1will
also converge in measure to an optimal̂h∈L^0 (Ω,F 0 ,P;Rd). Having found̂h
we may conclude thatf̂=
(
̂h,∆S)
is a.s. finite, and that the formuladQ
dP=cU′(
f̂