The Mathematics of Arbitrage

3.2 The Incomplete Case 43

where (q 1 ,...,qN) denotes the probability vector ofQ∈Ma(S). The min-
imisation ofΨwill be done in two steps: first we fixy>0 and minimise over
Ma(S), i.e.,
Ψ(y):= inf
Q∈Ma(S)

Ψ(y,Q),y> 0.

For fixedy>0, the continuous functionQ→Ψ(y,Q) attains its minimum
on the compact setMa(S) and the minimiserQ̂(y) is unique by the strict
convexity ofV.WritingQ̂(y)=(̂q 1 (y),...,̂qN(y)) for the minimiser, it follows
fromV′(0) =−∞thatq̂n(y)>0, for eachn=1,...,N. Indeed, suppose that
̂qn(y)=0forsome1≤n≤Nand fix any equivalent martingale measure
Q∈Me(S). LettingQε=εQ+(1−ε)Q̂we have thatQε∈Me(S), for 0<

ε<1, andΨ(y,Qε)<Ψ(y,Q̂)forε>0 sufficiently small - a contradiction.
In other words,Q̂(y) is an equivalent martingale measure forS.
Defining the dual value functionv(y)by

v(y) = inf

Q∈Ma(S)

∑N

n=1

pnV

(

y

qn

pn

)

=

∑N

n=1

pnV

(

y

̂qn(y)

pn

)

we find ourselves in an analogous situation as in the complete case above:
defining againŷ(x)byv′(̂y(x)) =−xand

ξ̂n=I

(

ŷ(x)

̂qn(y)

pn

)

,

similar arguments as above apply to show that (̂ξ 1 ,...,̂ξN,̂y(x),Q̂(y)) is the
unique saddle-point of the Lagrangian (3.26) and that the value functionsu
andvare conjugate.
Let us summarise what we have found in the incomplete case:

Theorem 3.2.1 (finiteΩ, incomplete market).Let the financial market
S=(St)Tt=0be defined over the finite filtered probability space(Ω,F,(F)Tt=0,P)
and letMe(S)=∅. Let the utility functionUsatisfy the above assumptions.
Denote byu(x)andv(y)the value functions

u(x)=supXT∈C(x)E[U(XT)],x∈dom(U), (3.27)

v(y) = infQ∈Ma(S)E

[

V

(

yddQP

)]

,y> 0. (3.28)

We then have:

(i) The value functionsu(x)andv(y)are conjugate andushares the quali-
tative properties ofUlisted in the above assumptions.