3
Utility Maximisation
on Finite Probability Spaces
In addition to the modelSof a financial market, we now consider a function
U(x), modelling the utility of an agent’s wealthxat the terminal timeT.
We make the classical assumptions thatU:R→R∪{−∞}isincreasing
onR, continuouson{U>−∞},differentiable and strictly concaveon the
interior of{U>−∞}, and that the marginal utility tends to zero when wealth
tends to infinity, i.e.,
U′(∞) := lim
x→∞
U′(x)=0.
These assumptions make perfect sense economically. Regarding the be-
haviour of the (marginal) utility at the other end of the wealth scale we shall
distinguish two cases.
Case 1 (negative wealth not allowed): in this setting we assume thatU
satisfies the conditionsU(x)=−∞,forx<0, whileU(x)>−∞,forx>0,
and the so-calledInada condition
U′(0) := lim
x↘ 0
U′(x)=∞.Case 2 (negative wealth allowed): in this case we assume thatU(x)>
−∞, for allx∈R,andthat
U′(−∞) := lim
x↘−∞
U′(x)=∞.Typical examples for case 1 are
U(x) = ln(x),x> 0 ,or
U(x)=
xα
α,α∈(−∞,1)\{ 0 },x> 0 ,whereas a typical example for case 2 is
U(x)=−e−γx,γ> 0 ,x∈R.