3.2 The Incomplete Case 41now is, that the optimally investing economic agent is indifferent of first order
towards a marginal variation of the investment into the portfolioX̂T(x).
It now becomes clear that formulae (3.21) and (3.22) foru′(x)arejust
special cases of a more general principle: for eachf∈L∞(Ω,F,P)wehave
EQ[f]u′(x) = lim
h→ 0EP[U(X̂T(x)+hf)−U(X̂T(x))]
h. (3.25)
The proof of this formula again is along the lines of (3.23) and the in-
terpretation is the following: by investing an additional endowmenthEQ[f]
to finance the contingent claimhf, the increase in expected utility is of first
order equal tohEQ[f]u′(x); hence again the economic agent is of first order
indifferent towards an additional investment into the contingent claimf.
3.2 The IncompleteCase
We now drop the assumption that the setMe(S) of equivalent martingale
measures is reduced to a singleton (but we still remain in the framework of a
finite probability space Ω) and replace it byMe(S)=∅.
It follows from Theorem 2.4.2 that a random variableXT(ωn)=ξnmay
be dominated by a random variable of the formx+(H·S)TiffEQ[XT]=
∑N
n=1qnξn≤x,foreachQ=(q^1 ...,qN)∈M
a(S) (or equivalently, for each
Q∈Me(S)).
In order to reduce these infinitely many constraints, whereQruns through
Ma(S), to a finite number, make the easy observation thatMa(S)isa
bounded, closed, convex polytope inRN. Indeed,Ma(S) is a subset of the
probability measures on Ω defined by imposing finitely many linear con-
straints. ThereforeMa(S) equals the convex hull of its finitely many extreme
points{Q^1 ,...,QM}.For1≤m≤M,weidentifyQmwith the probabilities
(q 1 m,...,qmN).
Fixing the initial endowmentx∈dom(U), we therefore may write the util-
ity maximisation problem (3.1) similarly as in (3.2) as a concave optimisation
problem overRNwith finitely many linear constraints:
EP[U(XT)] =
∑N
n=1pnU(ξn)→max!EQm[XT]=∑N
n=1qmnξn ≤ x, for m=1,...,M.Writing againC(x)={
XT∈L^0 (Ω,F,P)|EQ[XT]≤x, for allQ∈Ma(S)}
we define the value function, forx∈dom(U),