The Mathematics of Arbitrage

42 3 Utility Maximisation on Finite Probability Spaces

u(x)= sup

H∈H

E[U(x+(H·S)T)] = sup

XT∈C(x)

E[U(XT)].

The Lagrangian now is given by

L(ξ 1 ,...,ξN,η 1 ,...,ηM)

=

∑N

n=1

pnU(ξn)−

∑M

m=1

ηm

(N

∑

n=1

qmnξn−x

)

=

∑N

n=1

pn

(

U(ξn)−

∑M

m=1

ηmqnm

pn

ξn

)

+

∑M

m=1

ηmx,

where (ξ 1 ,...,ξN)∈dom(U)N, (η 1 ,...,ηM)∈RM+.

Writingy=η 1 +···+ηM,μm=ηym,μ=(μ 1 ,...,μM)and

Qμ=

∑M

m=1

μmQm,

note that, when (η 1 ,...,ηM) runs troughRM+, the pairs (y,Qμ) run through
R+×Ma(S). Hence we may write the Lagrangian as

L(ξ 1 ,...,ξN,y,Q)=EP[U(XT)]−y(EQ[XT−x]) (3.26)

=

∑N

n=1

pn

(

U(ξn)−

yqn

pn

ξn

)

+yx,

whereξn∈dom(U),y>0,Q=(q 1 ,...,qN)∈Ma(S).
This expression is entirely analogous to (3.5), the only difference now be-
ing thatQruns through the setMa(S) instead of being a fixed probability
measure. Defining again

Φ(ξ 1 ,...,ξn) = inf

y> 0 ,Q∈Ma(S)

L(ξ 1 ,...,ξN,y,Q),

and
Ψ(y,Q)= sup
ξ 1 ,...,ξN

L(ξ 1 ,...,ξN,y,Q),

we obtain, just as in the complete case,

sup

ξ 1 ,...,ξN

Φ(ξ 1 ,...,ξN)=u(x),x∈dom(U),

and

Ψ(y,Q)=

∑N

n=1

pnV

(

yqn

pn

)

+yx, y > 0 , Q∈Ma(S),