The Mathematics of Arbitrage

38 3 Utility Maximisation on Finite Probability Spaces

∑N

n=1

qnξ̂n=x, (3.16)

and
∑N

n=1

pnU(̂ξn)=L(ξ̂ 1 ,...,ξ̂N,̂y(x)). (3.17)

In particular, we obtain that

u(x)=

∑N

n=1

pnU(̂ξn). (3.18)

Indeed, the inequalityu(x)≥

∑N

n=1pnU(
ξ̂n) follows from (3.16) and (3.8),
while the reverse inequality follows from (3.17) and the fact that, for all
ξ 1 ,...,ξNverifying the constraint (3.3), we have:

∑N

n=1

pnU(ξn)≤L(ξ 1 ,...,ξN,̂y(x))≤L(ξ̂ 1 ,...,ξ̂N,̂y(x)).

We shall writeX̂T(x)∈C(x) for the optimiserX̂T(x)(ωn)=ξ̂n,n=1,...,N.
Combining (3.15), (3.17) and (3.18) we note that the value functionsu
andvare conjugate:

inf

y> 0

(v(y)+xy)=v(ŷ(x)) +x̂y(x)=u(x),x∈dom(U).

Thus the relationv′(ŷ(x)) =−x, which was used to defineŷ(x), translates
into
u′(x)=ŷ(x), forx∈dom(U).
From Proposition 3.1.2 and the remarks after equation (3.14), we deduce
thatuinherits the properties ofUlisted at the beginning of this chapter.
Let us summarise what we have proved so far:

Theorem 3.1.3 (finiteΩ, complete market).Let the financial marketS=
(St)Tt=0be defined over the finite filtered probability space(Ω,F,(F)Tt=0,P)and
supposeMe(S)={Q}. Let the utility functionUsatisfy the above assump-
tions.
Denote byu(x)andv(y)the value functions

u(x)=supXT∈C(x)E[U(XT)],x∈dom(U),

v(y)=E

[

V

(

yddQP

)]

,y> 0.

(3.19)

We then have:

(i) The value functionsu(x)andv(y)are conjugate anduinherits the qual-
itative properties ofUlisted in the beginning of this chapter.