The Mathematics of Arbitrage

34 3 Utility Maximisation on Finite Probability Spaces

We note that it is natural from an economic point of view to require that
the marginal utility tends to zero, when wealthxtends to infinity, and to
infinity when the wealthxtends to the infimum of its allowed values. The
infimum of the allowed values, i.e., of the domain{U>−∞}ofU,may
be finite or equal to−∞. In the former case we have assumed w.l.g. the
normalisation that this infimum equals zero.

We can now give a precise meaning to the problem of maximising expected
utility of terminal wealth. Define the value function

u(x):= sup

H∈H

EP[U(x+(H·S)T)],x∈dom(U), (3.1)

whereHruns through the familyHof trading strategies.
The optimisation of expected utility of wealth at a fixed terminal date
Tis a typical example of a larger family of portfolio optimisation problems,
where one can also include utility of intermediate consumption and many
other features. We only consider the prototypical optimisation problem (3.1)
above. The duality techniques developed for this case can easily be adapted
to variants of it.
The value functionu(x) is called theindirect utility function. Economically
speaking it indicates the expected utility of an economic agent at timeTfor
given initial endowmentx, provided she invests optimally in the financial
marketS.
We shall analyze the problem of finding, for given initial wealthx,the
optimiserĤ(x)∈Hin (3.1) at two levels of difficulty: first we consider the
case of an arbitrage-freecompletefinancial marketS. In a second step, we
generalise to arbitrage-free marketsS, which are not necessarily complete.

3.1 The CompleteCase......................................

We assume that the setMe(S) of equivalent probability measures under which
Sis a martingale, is reduced to a singleton{Q}. In this setting consider the

Arrow-Debreu assets (^1) {ωn}, which pay 1 unit of the num ́eraire at timeT,
whenωnturns out to be the true state of the world and pay out 0 otherwise.
In view of our normalisation of the num ́eraireS^0 t≡1, we get the following
relation for the price of the Arrow-Debreu assets at timet=0:
EQ

[

(^1) {ωn}

]

=Q[ωn]=:qn,

and by Corollary 2.2.12 each such asset (^1) {ωn}may be represented as (^1) {ωn}=
Q[ωn]+(Hn·S)T, for some predictable trading strategyHn∈H.
Hence, for fixed initial endowmentx∈dom(U), the utility maximisation
problem (3.1) above may simply be written as

EP[U(XT)] =

∑N

n=1

pnU(ξn)→max! (3.2)