The Mathematics of Arbitrage

3.1 The Complete Case 35

under the constraint

EQ[XT]=

∑N

n=1

qnξn ≤ x. (3.3)

To verify that (3.2) and (3.3) are indeed equivalent to the original problem
(3.1) above (in the present finite, complete case), note that by Theorem 2.4.2
a random variable (XT(ωn))Nn=1=(ξn)Nn=1can be dominated by a random
variable of the formx+(H·S)T=x+

∑T

t=1Ht∆StiffEQ[XT]=

∑N

n=1qnξn≤
x. This basic relation has a particularly evident interpretation in the present

setting, asqnis simply the price of the asset (^1) {ωn}.
We have writtenξnforXT(ωn) to stress that (3.2) is simply a concave
maximisation problem inRN with one linear constraint which is a rather
elementary problem. To solve it, we form the Lagrangian
L(ξ 1 ,...,ξN,y)=

∑N

n=1

pnU(ξn)−y

(N

∑

n=1

qnξn−x

)

(3.4)

=

∑N

n=1

pn

(

U(ξn)−yqpnnξn

)

+yx. (3.5)

We have used the lettery≥0 instead of the usualλ≥0 for the Lagrange
multiplier; the reason is the dual relation betweenxandywhich will become
apparent in a moment.
Write

Φ(ξ 1 ,...,ξN) = inf

y> 0

L(ξ 1 ,...,ξN,y),ξn∈dom(U), (3.6)

and
Ψ(y)= sup
ξ 1 ,...,ξN

L(ξ 1 ,...,ξN,y),y≥ 0. (3.7)

Note that we have

sup

ξ 1 ,...,ξN

Φ(ξ 1 ,...,ξN)= sup

ξ 1 ,...,ξN

∑N

n=1qnξn≤x

∑N

n=1

pnU(ξn)=u(x). (3.8)

Indeed, if (ξ 1 ,...,ξN) is in the admissible region

{∑

N

n=1qnξn≤x

}

,then

Φ(ξ 1 ,...,ξN)=L(ξ 1 ,...,ξN,0) =

∑N

n=1pnU(ξn). On the other hand, if
(ξ 1 ,...,ξN)satisfies

∑N

n=1qnξn >x, then by lettingy→∞in (3.6) we
note thatΦ(ξ 1 ,...,ξN)=−∞.
Regarding the functionΨ(y) we make the following pleasant observation,
which is the basic reason for the efficiency of the duality approach: using the