The Mathematics of Arbitrage

36 3 Utility Maximisation on Finite Probability Spaces

form (3.5) of the Lagrangian and fixingy>0, the optimisation problem over
RNappearing in (3.7) splits intoNindependent optimisation problems overR

U(ξn)−yqpnnξn→max!,ξn∈R.

In fact, these one-dimensional optimisation problems are of a very conve-
nient form: recall (see, e.g., [R 70], [ET 76] or [KLSX 91]) that, for a concave
functionU:R→R∪{−∞},theconjugate functionVofU(which is just the
Legendre-transformofx→−U(−x)) is defined by

V(η)=sup

ξ∈R

[U(ξ)−ηξ],η> 0. (3.9)

Definition 3.1.1.We say that the functionV :R→R, conjugate to the
functionU, satisfies theusual regularity assumptions,ifV is finitely valued,
differentiable, strictly convex on]0,∞[, and satisfies

V′(0) := lim

y↘ 0

V′(y)=−∞. (3.10)

Regarding the behaviour ofVat infinity, we have to distinguish between case 1
and case 2 above:

case 1: lim

y→∞

V(y) = lim

x→ 0

U(x) and lim

y→∞

V′(y) = 0 (3.11)

case 2: lim

y→∞

V(y)=∞ and lim

y→∞

V′(y)=∞ (3.12)

We have the following well-known fact (see [R 70] or [ET 76]):

Proposition 3.1.2.IfUsatisfies the assumptions made at the beginning of
this section, then its conjugate functionV satisfies the inversion formula

U(ξ) = inf

η

[V(η)+ηξ],ξ∈dom(U) (3.13)

and it satisfies the regularity assumptions in Definition 3.1.1. In addition,
−V′(y)is the inverse function ofU′(x).
Conversely, ifV satisfies the regularity assumptions of Definition 3.1.1,
thenUdefined by (3.13) satisfies the regularity assumptions made at the be-
ginning of this section.

Following [KLS 87] we write−V′=I(for “inverse” function). We then
haveI =(U′)−^1. Naturally,U′has a nice economic interpretation as the
marginal utilityof an economic agent modelled by the utility functionU.
Here are some concrete examples of pairs of conjugate functions:

U(x)=ln(x),x> 0 ,V(y)=−ln(y)− 1 ,

U(x)=−e

−γx

γ ,x∈R,V(y)=

y

γ(ln(y)−1),γ>^0

U(x)=x

α

α,x>^0 ,V(y)=

1 −α

α y

αα− 1

,α∈(−∞,1)\{ 0 }.