Option Pricing via Utility Maximization in the presence of Transaction Costs: an Asymptotic Analysis

Option Pricing via Utility Maximization

in the presence of Transaction Costs :

an Asymptotic Analysis



Bruno Bouchard

CEREMADE, Université Paris Dauphine and CREST

bouchard@ensae.fr

This version

January 20, 2000

Abstract

We consider a multivariate nancial market with proportional transaction costs as in Kabanov (1999). We study the problem of contingent claim pricing via utility max-imization as in Hodges and Neuberger (1989). Using an exponential utility function, we derive a closed form characterization for the asymptotic price as the risk aversion tends to innity. We prove that it is reduced to the super-replication cost if the initial endowment is only invested in the non-risky asset, as it was conjectured in Barles and Soner (1996). We do not make use of the dual formulation for the super-replication price obtained in Kabanov (1999).

Key words: Transaction costs, option pricing, dynamic programming, viscosity solutions, stochastic games.

JEL subject classications: G13, G11, C73.

MSC 1991 subject classications: 90A09, 90A10, 49L20, 49L25, 49J35.

I am grateful to Professor Nizar Touzi for a careful reading of this paper which considerably improved

its presentation.

1 Introduction

In the presence of proportional transaction costs, no perfect replication strategy is in general available. It is then necessary to dene other pricing criteria. In recent papers, Bouchard and Touzi (1999) and Touzi (1999) provide an explicit solution to the multivariate super-replication problem under proportional transaction costs. Using the framework introduced by Kabanov (1999), they extend the result of Cvitanic, Pham and Touzi (1997) and prove that the super-replication cost can be expressed as the concave envelope of some modied payo function. This implies that super-replication prices are highly expensive and are not acceptable for practical purpose. In Hodges and Neuberger (1989), the price of the con-tingent claim is dened by a utility maximization argument. In their model, they consider the dierence between the maximal expected utility from terminal wealth with and without option liability, and they dene the price of the option as the unique cash increment (in the case of option liability) which osets the dierence.

Adapting the framework of Hodges and Neuberger (1989) to the multivariate case and using an exponential utility function, we prove that the price tends to a super-replication related cost as the risk aversion tends to innity. This result was rst conjectured by Barles and Soner (1996) in the one dimensional case. (see Rouge 1996 for a similar study in the case of incomplete markets without transaction costs).

Our approach for the derivation of the large risk aversion asymptotics is inspired from Bouchard and Touzi (1999) and does not rely on the dual formulation obtained by Kabanov (1999). By introducing a large set of ctitious nancial markets without transaction costs, we dene naturally a stochastic dierential game which majorizes the value function of the utility maximization problem. Using some duality argument and a suitable transformation of the above stochastic control problem we reduce it to an auxiliary stochastic control prob-lem and we provide a lower bound for the pricing function. Then, we prove a dynamic programming principle for the auxiliary value function which allows us to characterize it as a viscosity solution of a suitable Hamilton-Jacobi-Bellman partial dierential equation. Finally, we use some stability results for general Hamiltonians to study the asymptotic be-haviour. We shall notice that the construction of the set of ctitious markets is slightly dierent from the one of Bouchard and Touzi (1999). Here the modied price process needs to be a martingale under a suitable equivalent probability measure up to the time horizon

T and not only up to a stopping time as it was the case in the previous paper.

After setting some notations in section 2, we describe the model and the pricing problem in section 3. The main results of the paper are stated in section 4 with a partial argument in section 5 ; the proof is concluded in section 8 after some preparation in the sections in between. In section 6, we introduce the ctitious nancial market without transaction costs and we dene the auxiliary stochastic control problem. Viscosity properties of this value

function and its asymptotic properties as the risk aversion tends to innity are then reported in section 7.

2 Notations

To avoid some heaviness in the computations, we denote by
the natural scalar product in IR

n and

k
k the associated norm. Given a vector x 2 IR n, its

i-th component is denoted

by x i.

IM

n;p denotes the set of all real-valued matrices with

n rows and pcolumns. Given a

matrix M 2 IM

n;p, we denote by M

ij the component corresponding to the

i-th row and the j-th column. IM

n;p

+ denotes the subset of IM

n;p whose elements have non-negative entries

and IM n;p

p the subset of IM

n;p whose elements have range equal to

p. If n = p, we simply denoteIM n and IM n + for IM n;n and IM n;n + . Since IM

n;p can be identied with IR

np, we dene

the norm onIM

n;pas the norm of the associated element of IR

np. Transposition is denoted by . Given a square matrix

M 2 IM n, we denote by Tr[ M] := P n i=1 M

ii the associated trace.

Given n scalars x 1 ;::: ;x n, we denote by Vect[ x i ; i = 1;::: ;n] the vector of IR n dened by the components x 1 ;:::;x n. For all x 2 IR n, diag[

x] denotes the diagonal matrix of IM n

whosei-th diagonal element isx

i. Given a matrix M 2IM n;p, we denote by  M the matrix in IM n+1;p obtained from

M by adding a rst row of 1. The same notation prevails for vectors

inIR n. We denote by 1 i the vector of IR n dened by 1 j i = 1 if

j =iand 0 otherwise and by 1

i;j the matrix of IM d+1 dened by 1 l k i;j = 1 if ( i;j) = (l;k) and 0 otherwise.

Given a smooth function ' mapping IR n into

IR

p, we denote by

D' the Jacobian matrix

of ', i.e. (D') ij = @' i =@x j. If x = (y;z), D y

' denotes the (partial) Jacobian matrix of '

with respect to the y variable. In the case p = 1, we denote by D 2

'the Hessian matrix of ', i.e. (D 2 ') ij = @ 2 '=@x i @x j. If

x = (y;z), we dene the matrices D 2 yy ', D 2 zz ' and D 2 yz ' accordingly.

Given a ltered probability space (;F;P;fF t

; 0t Tg) and a scalar p 0, we denote

by L p(

t) the set of all F

t-measurable random variables with nite L

p norm. This notation

is extended naturally to stopping times. For p = 0, L 0(

t) is the set of all F

t-measurable

random variables. Finally, we will use the convention inff;g:= +1.

3 The model

LetT be a nite time horizon and (;F;P) be a complete probability space supporting ad

-dimensional Brownian motionfB(t), 0t Tg. We shall denote by IF =fF t, 0

t Tg

the P-augmentation of the ltration generated by B.

3.1 The nancial market

We consider a nancial market which consists of one bank account, with constant price process S

0, normalized to unity, and

d risky assets S := (S 1

;::: ;S

d). The price process S = fS(t), 0  t  Tg is an (0;1)

d-valued stochastic process dened by the following

stochastic dierential system :

dS(t) = diag[S(t)][(t;S(t))dt+(t;S(t))dB(t)]; 0 < t  T : (3.1)

Here (:;:) and (:;:) are respectively IR d and

IM d

d-valued functions. We shall assume all

over the paper that the functions diag[s](t;s) and diag[s](t;s) satisfy the usual Lipschitz

and linear growth conditions in order for the process S to be well-dened, and that (t;s)

is strongly non-degenerate for all (t;s) 2 [0;T](0;1)

d. As a consequence, for all borel

subset B of (0;1)

d, we have

P(S(t)2B j S(u)) > 0 P ?a.s. (0  u < t  T). Notice

that, with our notations, S := (S 0

;S 1

;:::;S d).

We also assume that function , dened as : (t;s) :=  ?1(

t;s)(t;s) ; is Lipschitz and

satises the Novikov condition :

E  exp  1 2 Z T 0 jj (t;S(t))jj 2 dt  <+1 ;

whereE[:] denotes the expectation operator under P. We introduce the probability measure P  equivalent P : dP  dP := E  ? Z T 0 (t;S(t))
dB(t)  ;

whereE stands for the Doléans-Dade exponential operator. In the following, we will denote

by E

 the expectation operator under P

.

Remark 3.1

As usual, the assumption that the interest rate of the bank account is zero could be easily dispensed with by discounting.

3.2 Trading strategies and wealth process

A trading strategy is aIM d+1

+ -valued process

L, with initial valueL(0?) = 0, such thatL ij is IF?adapted, right-continuous, and nondecreasing for alli;j = 0;:::;d. Here, L

ij describes

the cumulative amount of funds transferred from assetito asset j. Proportional transaction

costs in this nancial market are described by matrix2 IM d+1

+ . This means that transfers

from assetito assetj are subject to proportional transaction costs 

ij for all

i;j = 0;::: ;d.

Then, given an initial holdings vector x 2 IR

d+1 and a strategy

L, the portfolio holdings

X L x = ( X i;L x )

i=0;:::;d are dened by the dynamics : X i;L x (0 ?) = x i dX i;L x ( t) = X i;L x ( t) dS i( t)  S i( t) + d X j=0  dL ji( t)?(1 + ij) dL ij( t)  ; 0tT for alli = 0;::: ;d.

Following Kabanov (1999), we dene the solvency region :

K := ( x2IR d+1 : 9a2IM d+1 + ; x i + d X j=0 (a ji ?(1 + ij )a ij )  0; i= 0;::: ;d )

The elements of K can be interpreted as the vectors of portfolio holdings such that the

no-bankruptcy condition is satised : the liquidation value of the portfolio holdings x;through

some convenient transfers, is nonnegative. This means that the portfolio holdings ?x can

be reached from zero initial portfolio holdings through some convenient transfers.

Clearly, the set K is a closed convex cone containing the origin. We introduce the partial

ordering induced by K, dened by :

for allx 1 ;x 2 2IR d+1 ; x 1 x 2 if and only if x 1 ?x 2 2K : The relation x 1  x

2 means that starting with portfolio holding x

1, it is possible to reach,

at least, portfolio x

2, by means of a suitable transfer strategy.

A trading strategy L is said to be admissible if there exists some  >0 (possibly depending

onL) such that :

X L

x(

t)?S(t) ; P ?a.s.; 0t T : (3.2)

We shall denote by A the set of such processes. Given some x 2 IR

d+1, we dene X(x) :=  X 2L 0( T) : 9L2 A; X =X L x( T) P ?a.s.

Remark 3.2

LetLbe an admissible strategy for the initial holdingx2IR

d+1, then, possibly

by changing the constant , it is also admissible for initial holding y, for all y 2 IR d+1.

Therefore, A is independent of the initial holding.

Remark 3.3

Notice that for all y 2 IR, x 2 IR

d and for all trading strategy

L in A : X L x + y1 0 = X L x+y1

0 . This follows from the normalization of the non-risky asset  S 0 = S 0 to unity. 5

3.3 Liquidation function

The liquidation function ` is a mapping from IR

d+1 into

IR dened by : `(x) := supfw2IR : xw1

0 g:

`(x) must be interpreted as the maximal cash endowment (endowment in terms of non risky

asset) that we can get from x when clearing all the positions invested in risky assets.

Remark 3.4

For all x 2IR

d+1 and for all

c2 IR, we have : `(x+c1 0) =

`(x) +c. Notice

that, in general, given some arbitrary (x;x 0) 2IR 2d+2, we only have : `(x+x 0) `(x)+`(x 0).

Following Bouchard and Touzi (1999), we introduce the set :  :=fr 2IR

d :

8x2K; x
r0g:

First, notice that f1g is the section of the positive polar cone of K by the hyperplane fx2IR

d+1 : x

0 = 1

gand that the partial ordering can be characterized in terms of  by x 1  x 2 if and only if  r
(x 1 ?x 2) 0 for all r2:

Moreover  is compact and non-empty since 1 = P d i=1 1 i 2 .

Remark 3.5

It is easily checked that the supremum in the denition of `(x) is attained, so

that ` can be written alternatively :

`(x) := maxfw2IR : xw1 0

g:

In other words, starting with the initial holding x, we can nd a transfer matrix a which

enables to replicate `(x)1 0.

To see this, it sucies to consider some maximizing sequencew k

!`(x) and notice that from

the previous characterization of the partial ordering, we have for allk : r
(x?w k

1

0) 0

for all r 2 . Then, passing to the limit and using the previous characterization of the

partial ordering  again, we get :x`(x).

Lemma 3.1

For all x in IR d+1 :

`(x) = inf r2



r
x:

Proof.

Fix x2 K. Then, by denition,x`(x)1

0. This implies that

x?`(x)1 0

2 K and

then, for allr 2 , r
(x?`(x)1 0)

0 and, since r

0 = 1, 

r
x`(x).

Conversely, for all r 2 , we have r
(x?(inf r2 r
x)1 0) = r
x?inf r2 r
x  0 . This

proves that : x(inf r2 r
x)1 0 and therefore `(x)inf r2 r
x : 2 6

3.4 The problem

A contingent claim is a (d+ 1)-dimensional F

T-measurable random variable

G = g(S(T)) in L 2( T). Here, g maps IR d into IR d+1 and satises

g is lower semicontinuous and 9 c>0 s.t. g(s)?cs for all s2IR d

: (3.3)

For alli= 0;::: ;d, the random variableG i =

g i(

S(T)) represents a target position in asseti.

Following Barles and Soner (1996), we dene onIR the utility function : U : w7!1?exp(?w) :

for some>0 which is interpreted as the absolute risk aversion. We also introduce onIR d+1

U ` :

x7!1?exp(?`(x)) :

Next, we introduce the value functions :

u(x) := sup X2X(x) EU `( X) , u g( x) := sup X2X(x) EU `( X?G) ; (3.4)

and the pricing function p  : p ( x) := inffp2IR : u g( x+p1 0) u(x)g :

Here, we do not write the dependence of U,u and u g on

 to alleviate the notations.

Notice that : p ( x) = 1  ln(1?u g( x))? 1  ln(1?u(x)) (3.5)

This is a direct consequence of the following

Lemma 3.2

For all x 1, x 2 2IR d inffp2IR : u g( x 1+ p1 0) u(x 2) g = 1  ln 1?u g( x 1) 1?u(x 2) : (3.6)

Proof.

Fix x 1, x 2 2 IR d and denote by

p the left-hand side term of (3.6). If p = 1 the

result is trivial. If p<1, then, since u g(

x) is increasing and continuous in thex

0 variable : sup X2X(x 2 ) EU `( X) = sup X2X(x 1 +p1 0 ) EU `( X?G) :

Using Remark 3.2 and Remark 3.3 we obtain : sup X2X(x 2 ) EU `( X) = sup X2X(x 1 ) EU `( X+p1 0 ?G) = sup X2X(x 1 ) e ?p EU `( X?G) :

The desired result is then obtain by denition of u g( x 1) and u(x 2). 2 7

4 The main results

4.1 Probability of missing the hedge

We rst provide a very simple upper bound for the probability of missing the hedge by a constant k when starting with an initial endowment equal to p

 0 := p (0) 1 0.

Proposition 4.1

For all  >0 and k >0 :

inf X2X(p  0 ) P [`(X?G)?k]  exp [?k] :

Notice that, for all positive k, this probability tends to zero as  tends to +1. This

is in agreement with Theorem 4.1 below, which states that p

(0) converges to the

super-replication cost of G.

Proof.

We follow the proof of Theorem 3.2 in Barles and Soner (1996). Fix some  > 0

and k >0. Then, clearly

inf X2X(p  0 ) P [`(X?G)?k] = inf X2X(p  0 ) E1 f`(X?G) ?kg  inf X2X(p  0 ) Eexp (?[`(X?G) +k]) : and by denition of p  0, inf X2X(p  0 ) Eexp (?[`(X?G) +k]) = inf X2X(0) Eexp (?[`(X) +k]) ;

which implies that inf X2X(p  0 ) P [`(X?G)?k]  inf X2X(0) Eexp (?[`(X) +k]) ;  Eexp (?[0 +k]) ;

where the last inequality is obtained by taking L(t)=0 for all 0tT. 2

Barles and Soner (1996) provide a similar majoration but it requires the resolution of a non-linear Black-Scholes equation.

4.2 Large risk aversion asymptotic

Before stating the main result of this subsection, we recall some notations and results con-tained in Bouchard and Touzi (1999) and Touzi (1999).

We rst dene : ^

g(S(0);x) := inffw2IR : 9X 2X(w1 o+

x); X G P ?a:s:g :

Given the initial holding x 2 IR d+1, ^

g(S(0);x) is the minimal cash increment needed to

construct a self-nancing super-replicating strategy. Under the condition

 ij+

 ji

>0 for all i;j = 0;:::;d; i6=j ; (4.1)

the super-replication price is explicitly given by ^ g(S(0);x) = sup r2 ~ g conc(diag[ r]S(0))?r
x where ~ g(z) := sup r2  r
g ? diag[r] ?1 z
for all z in (0;1) d ; and ~g

conc is the concave envelope of ~ g.

Remark 4.1

If ^g(S(0);x)<1, Bouchard and Touzi (1999) proved that we can nd some X 2X(^g(S(0);x)1

o+

x) such that X G P ?a:s:

We can now state the main result of this subsection.

Theorem 4.1

Under condition (4.1), for all x in IR d+1 : lim !+1 p  (x) = ^g(S(0);x) +`(x):

The proof of the last result will be provided in the subsequent sections of the paper after some partial argument in section 5. Notice that in the case where the initial holdingxis only

invested in non-risky asset (x=x 0

1

0), the asymptotic pricing function is equal to the

super-replication cost ^g(S(0);0). This follows from the fact that, by construction : ^g(S(0);x 0 1 0)+ `(x 0 1 0) = ^ g(S(0);0).

5 Basic idea of the proof of Theorem 4.1

In this section, we provide some partial arguments for the proof of Theorem 4.1. Remember that u

g and

u depend on.

We start by the following easy observation : 9

Proposition 5.1

For all x 2 IR d+1 and  >0 : 1  ln(1?u(x))  ?`(x) :

Proof.

Fixx 2 IR d+1,

 >0 and consider the strategy : L(t) := a ; 0t T

where a is a transfer matrix satisfying x i+ d X j=0 ? a ji ?(1 + ij) a ij
=`(x)1I i=0 ; i= 0;::: ;d :

Existence of a follows from Remark 3.5. ClearlyL 2A andX L

x(

T) =`(x)1

0. By denition

of the control problemu, we see that :

u(x)  1?exp(?`(x)) ;

which provides the required result. 2

The next result is the dicult step in the proof of Theorem 4.1. We shall provide its proof in section 8 after some preparation in sections 6 and 7.

Proposition 5.2

For all x 2 IR d+1 : liminf !+1 1  ln(1?u g( x))  g^(S(0);x) : (5.1)

In particular, applying Proposition 5.2 to g 0 and using Proposition 5.1, we get :

Corollary 5.1

For all x 2 IR d+1 : lim !+1 1  ln(1?u(x)) = ?`(x) : (5.2)

We can now prove Theorem 4.1.

Proof of Theorem 4.1

Fix some x 2 IR

d+1. From (3.5) : liminf !+1 p  (x) = liminf !+1 1  [ln(1?u g (x))?ln(1?u(x))]

Then, from Proposition 5.2 and Corollary 5.1 , we get liminf !+1 p ( x)  ^g(S(0);x) +`(x) (5.3) 10

Moreover, applying Corollary 5.1 to x and `(x) and using Remark 3.4, we get : lim !+1 1  ln(1?u(x)) = lim !+1 1  ln(1?u(`(x)1 0)) : Then, limsup !+1 p ( x) = limsup !+1 1  [ln(1?u g( x))?ln(1?u(x))] ; = limsup !+1 1  [ln(1?u g( x))?ln(1?u(`(x)1 0))] : (5.4)

We now claim that for all >0 :

1  ln(1?u g( x))? 1  ln(1?u(`(x)1 0))  g^(S(0);x) +`(x): (5.5)

Then, the required result is a direct consequence of the last inequality and (5.3)-(5.4). We now prove (5.5). If ^g(S(0);x) = 1, the result is trivial. Otherwise from Remark 4.1, we

can nd some ~X 2 X(x+ ^g(S(0);x)1

0) such that : ~

X ?G  0 P ?a.s. : Then, using

Remark 3.2, for all X 2 X(`(x)1

0), we can nd some ^

X 2 X(x+ (^g(S(0);x) +`(x))1 0)

such that ^X ?G  X P ? a.s. : From the arbitrariness of X 2 X(`(x)), this proves

that : `(x) + ^g(S(0);x)  inffp2IR : u g( x+p1 0) u(`(x)1 0)

g. Finally, the required

result is obtained by Lemma 3.2. 2

6 Fictitious nancial market without transaction costs

To conclude the proof of Theorem 4.1, it remains to prove Proposition 5.2. In this section, we introduce a ctitious nancial market with no transaction costs but with a modied price process. On this market, we dene an auxiliary control problem and we show that the auxiliary value function provides a lower bound for the left-hand side term of (5.1) (see Proposition 6.1 and Corollary 6.1). We also prove a dynamic programming principle for the auxiliary value function. Finally, in subsection 6.3, we introduce a parameterization of the ctitious market.

6.1 An auxiliary control problem

We rst introduce some notations.

Let P be the set of pairs (Q;) such that :

(P1)  is a -valued IF-adapted continuous process.

(P2) Q is a probability measure equivalent to P dened by the density H(T) := dQ dP =E  ? Z T 0 (s)
dB(s)  11

for some IF-adated process  satisfying the Novikov condition, such that there exists some IM

d

d-valued,

IF-adapted process ? satisfying : E Q Z T 0 jj?(t)jj 2 dt  < +1

and such that :

Z(t) := diag[(t)]S(t) =E Z t 0 ?(s)dB Q( s)  ; 0tT ; (6.1) where B Q is a

d-dimensional Brownian motion under Q and E

Q denotes the expectation

operator under Q.

Notice that if (Q;)2P then diag[]Sis a square integrableQ-martingale. Also notice that P 6=; since (P



;1) 2P.

Let r be a vector in . We denote byR(0;r) the set of all processes  satisfying :

(R1) (0) =r.

(R2) There exists some probability measure Qsuch that (Q;)2 P.

We now x r 2 ,  2 R(0;r) and Q as in the last denition and we set Z := diag[]S as

in (6.1).

Let  be a progressively measurable process valued in IR

d+1 satisfying d X i=0 Z T 0 j i( t)j 2 d  Z i( t)  <1 P ?a.s. (6.2)

Then, givenw2 IR ; we introduce the process W  w dened by W  w( t) =w+ Z t 0 (r)
dZ(r); 0tT; (6.3)

and we denote by B(0;S(0)) the set of all such processes  satisfying the additional

condi-tion :

W 

w(

t)?M(t) ; P ?a.s. 0t T (6.4)

for some IF-adapted Q-martingaleM.

Remark 6.1

For all  2 B(0;S(0)), W 

w is a

Q-local martingale which is bounded from

below by a Q-martingale. Then, W 

w is a

Q-supermartingale.

Before stating the main result of this section, we provide an economic interpretation of the newly introduced processes. The process Z describes the price process of d+ 1 assets on a

ctitious nancial market without transaction costs. The process  is a portfolio strategy

on the ctitious nancial market : 

i is the number of shares of asset

i held at each time,

for i = 0;::: ;d. The process W 

w describes the wealth induced by portfolio strategy  and

initial wealth w, under the self-nancing condition.

Example 6.1

Letd= 1. Then, clearly  = [1=(1 + 10)

;1 +

01]. This implies that for all

0tT : 1 1 + 10  S 1( t) Z 1( t) S 1( t)(1 + 01) :

This means that the price process of the risky asset on the ctitious market lies in between the bid and the ask price of the initial nancial market with transaction costs.

This example provides an intuitive justication of the following Proposition.

Proposition 6.1

For all x 2 IR d+1 : u g (x)  sup 2B(0;S(0)) EU  W   (0)
x( T)?(T)
G  : (6.5)

Proof.

Let x 2 IR d+1, set

w = (0)
x, and consider some portfolio strategy L 2 A. By

the same argument as in Bouchard and Touzi (1999), there exists some IF-adapted process  with bounded variations such that :

W  w( t)  X L x(

t)
(t) P ?a.s., for all 0tT :

Then by Lemma 3.1 and the fact that  is -valued, we get

 (T)
? X L x( T)?G
 ` ? X L x( T)?G
P ?a.s. and nally U ? W  w( T)?(T)
G
 U ?  (T)
[X L x( T)?G]
 U ` ? X L x( T)?G
P ?a.s.

In order to conclude the proof, it sucies to prove that  2 B(0;S(0)). To see this, notice

that condition (6.4) is satised by denition of A and R(0;r). Also, from the denition of R(0;r) and the fact that  is a bounded variation process, condition (6.2) is satised. 2

Moreover, we have the following result for the stochastic control problem dened by the right-hand side term of (6.5).

Lemma 6.1

Fix w 2 IR. Then, there exists some ^  2 B(0;S(0)) such that : sup 2B(0;S(0)) EU ? W  w( T)?(T)
G
= EU  W ^   w ( T)?(T)
G  :

The proof of the last Lemma is left to Appendix. 13

6.2 Dynamic programming

We rst extend the denition of process S to the case where the time origin is dened by

some stopping time  valued in [0;T]: Let  be a (0;1)

d-valued random variable in L

2( ),

we dene the process S

; by the dynamics (3.1), and the initial condition S

;(

) = . We

accordingly dene B(;). Let % be a -valued random process. We extend the denition

of R(;%) by replacing (R1) by (R1') :

(R1') (t^) =%(t^); for all 0t T :

For all r 2 , we dene similarlyR(;r) by identifyingr with the corresponding constant

process. Set :=diag[%

]

. Given some  2 B(;),  2 R(;%) and some real-valued random

vari-able in L 2(

), we dene the processes  Z  ; ;W ; ; ; 

by the dynamics (6.1)-(6.3) and the initial conditionsZ  ;( ) =andW ; ; ;(

)= . We nally use the notationG ; :=

g(S ;(

T)):

Next, we introduce the auxiliary control problem :

v(;%;) := ess sup 2R(;%) ess inf 2B(;) 1  lnh 1?E  U  W ; ; ;( T)?(T)
G ;  i + where E

 denotes the expectation operator conditionnally on F

. We do not write the

dependence of v with respect to in order to alleviate the notations. Notice that v is

inde-pendent of by denition of U.

As a direct consequence of Proposition 6.1, we get a lower bound for the left-hand side term of (5.1) :

Corollary 6.1

For all x 2 IR

d+1 and r 2  : 1  ln(1?u g( x))  v(0;r;diag[r]S(0))?r
x

Finally, we have the following dynamic programming principle on the auxiliary value func-tion :

Proposition 6.2

(Dynamic programming) Fix some (t;r;z) 2 [0;T)(0;1)

d. Then,

for all (Q;) 2 P such that  2 R(t;r) and for all [t;T]-valued stopping time  : v(t;r;z)  E Q t  v ? ;();Z  t;z( )
? 1  lnH() 

where H(T) :=dQ=dP dened on [t;T] and H() :=E 

H(T).

We rst prove the following Lemma :

Lemma 6.2

Fix r 2 . Let  be some [0;T]-valued stopping time, and, let ;  be random

variables in L 2(

) respectively valued in IR and (0;1)

d. Then for all  0 2R(0;r) : ( J  := ess sup 2B(;) E  U  W ; ; ;( T)?(T)
G ;  ; 2R(; 0) )

is directed downwards, i.e. for all  1 and



2 in

R(;

0) there exists some

 in R(; 0) such that : J  minfJ  1 ; J  2 g.

Proof.

Without loss of generality we x = 0, =s and =w. Dene J  := sup 2B(0;s) EU  W ; 0;w;s( T)?(T)
G 0;s  Consider some  1 and  2 2 R(0; 0). Set : := 11I fJ  1 J  2 g+  21I fJ  1 >J  2 g Clearly, 1IfJ  1 J  2 g 2 F

0 and, by Lemma 10.1 in Appendix,

 2R(0; 0). Then, clearly : J 1I fJ  1 J  2 g = J  11I fJ  1 J  2 g :

Using the same argument to compute J 1I fJ  1 >J  2 g, we nally obtain : J  = J 11I fJ  1 J  2 g+ J 21I fJ  1 >J  2 g = min fJ 1; J 2 g : 2

Proof of Proposition 6.2

Fix some (t;w;r;s)2[0;T)IR(0;1)

d. Consider some

stopping time  such that t T P-a.s. Set z:= diag[r]s. By Lemma 6.1, there exists

some ^  2B(t;s) such that : U(w?v(t;r;z)) = inf 2R(t;r) E t h U  W ^ ; t;w;s( T)?(T)
G t;s  i

Fix some (Q;~) 2 P such that ~ 2 R(t;r) and notice that  2 R(;~) implies  2 R(t;r).

Then U(w?v(t;r;z))  inf 2R(;~ ) E t h U  W ^ ; t;w;s( T)?(T)
G t;s i and U(w?v(t;r;z))  inf 2R(;~ ) E t " ess sup 2B(;S t;s ()) E   U  W ; ;W ^ ;~ t;w ;s ();S t;s () (T)?(T)
G ;S t;s ()   #  sup ~ 2B(t;s) inf 2R(;~ ) E t " ess sup 2B(;St;s()) E   U  W ; ;W ~ ;~ t;w ;s ();S t;s () (T)?(T)
G ;S t;s ()  # (6.6) 15

From Lemma 6.2 and Neveu (1975) p121, for all ~ 2 B(t;s) there exists a minimizing

sequence f n

g inR(;~) such that, P ?a.s. :

lim n!1 #ess sup 2B(;St;s()) E   U  W ;n ;W ~ ;~ t;w ;s ();S t;s () (T)? n( T)
G ;S t;s ()  = U  W ~ ;~ t;w;s( )?v(;~();Z ~  t;z( ))  :

Now, observe that u(
)1. Then, by monotone convergence,

lim n!1 # E t " ess sup 2B(;S t;s ()) E   U  W ; n ;W ~ ;~ t;w ;s ();St;s() (T)? n( T)
G ;St;s()  # = E t h U  W ~ ;~ t;w;s( )?v(;~();Z ~  t;z( )) i

Let H(T)=dQ=dP on [t;T]. Then, plugging the previous equality in (6.6), we get : U(w?v(t;r;z))  sup ~ 2B(t;s) E t h U  W ~ ;~ t;w;s( )?v(;~();Z ~  t;z( )) i = sup ~ 2B(t;s) E Q t h 1?exp  ?W ~ ;~ t;w;s( ) +v(;~();Z ~  t;z( ))?lnH()  i

by denition of U and the fact that E 

H(T) = H(). Then, using Remark 6.1 as well as

the increase and the concavity of x!?e

?x, it follows from Jensen's inequality that : U(w?v(t;r;z))  sup ~ 2B(t;s) h 1?exp  ?w+E Q t h v(;~();Z ~  t;z( ))?lnH() ii ; = U  w?E Q t  v(;~();Z ~  t;z( ))? 1  lnH()  :

The required result is obtained by the fact that U is increasing and the arbitrariness of

(Q;~)2 P such that ~ 2R(t;r). 2

6.3 Parameterization of the auxiliary control problem

From Bouchard and Touzi (1999), there exists a function f mapping (0;1) n into IR d such that  = ff(y) : y2(0;1) n g ;

and satisfying under (4.1) :

range(Df(y)) = d (6.7) 1=(1 + i0) f i( y)(1 + 0i) and kDf(y)diag[y]k<K (6.8) 16

for ally2(0;1)

n and for all

i= 0;::: ;n, where K is some real constant.

In order to alleviate the notations, we introduce the matrix-valued function :

F(y) := diag[f(y)] ; y 2(0;1) n

:

We will now construct a subset of processes satisfying conditions (R1') and (R2). LetD be

the set of all bounded progressively measurable processes (a;b) = f(a(t);b(t)); 0  t  Tg

valued in IM n;d d IR n. For all (t;s;y) in [0;T]  (0;1) d+n and (

a;b) in D; we introduce the controlled process Y

(a;b)

t;y;F(y)s dened on [

t;T] as the solution of the stochastic dierential equation : dY() = diag[Y()] ?  (a;b) (;Y();F(Y())S t;s( ))a() +b()
dt+a()dB()  Y(t) = y: where 

(a;b) is a Lipschitz function of (

t;s), locally Lipschitz iny, dened as : (t;F(y) ?1 z)F(y) (a;b)( t;y;z) := F(y)(t;F(y) ?1 z) +Df(y)diag[y]b +12Vect Tr? D 2 f i( y)diag(y)aa diag[ y]
;i= 1;::: ;d  +Vect? Df(y)diag[y]a ( t;F(y) ?1 z)
ii ;i= 1;::: ;d  ;

The reason for introducing this rather complex function will appear in Remark 6.3 and Lemma 6.3 .

For all (t;y;z) 2[0;T] (0;1)

n+d and (

a;b) inD , we dene the process Z (a;b) t;y;z by Z (a;b) t;y;z = F(Y (a;b) t;y;z) S t;F(y) ?1 z ; (6.9)

and we introduce the sets ~ D(t;y;z) := n (a;b)2D : lnY (a;b) t;y;z is bounded o

Remark 6.2

Fix (t;y;z) 2 [0;T]  (0;1) n+d and (

a;b) in D. Set " >0 such that 1=" < jjyjj<" and dene the positive stopping time :

 := inf  s >t : jjY (a;b) t;y;z ( s)jj2= [1 " ;"]  :

Then clearly (a;b)1I [t;]

2D~(t;y;z).

For all (a;b) 2 D~(t;y;z), we introduce the exponential : H (a;b) t;y;z( ) = E  ? Z  t  (a;b)( r;Y (a;b) t;y;z ( r);Z (a;b) t;y;z( r))
dB(r)  ; t  T : 17

Notice that by assumption on , denition of and ~D(t;y;z),H (a;b)

t;y;z is a martingale and we

can dene on [t;T] the probability measure P (a;b) t;y;z equivalent to P by : dP (a;b) t;y;z dP := H (a;b) t;y;z( T) :

By Girsanov's Theorem, the process :

B (a;b) t;y;z( ) := B() + Z  t  (a;b)( r;Y (a;b) t;y;z( r);Z (a;b) t;y;z( r))dr; t T: is a P (a;b)

t;y;z-Brownian motion. We shall denote by E

(a;b)

t;y;z the expectation operator under the

probability measure P (a;b)

t;y;z.

Remark 6.3

Fix (t;y;z)2 [0;T] (0;1)

n+d and (

a;b) 2 D~(t;y;z). Then, direct

compu-tation shows that the dynamics of Y (a;b) t;y;z, Z (a;b) t;y;z and ln H (a;b) t;y;z on [ t;T] are given by : dY (a;b) t;y;z ( r) = diag[Y (a;b) t;y;z ( r)]  b(r)dr+a(r)dB (a;b) t;y;z( r)  dZ (a;b) t;y;z( r) = diag[Z (a;b) t;y;z( r)]? a(r)  r;Y (a;b) t;y;z ( r);Z (a;b) t;y;z( r)  dB (a;b) t;y;z( r) dlnH (a;b) t;y;z( r) = +12[ (a(r);b(r))( r;Y (a;b) t;y;z( r);Z (a;b) t;y;z( r))] 2 dr ? (a(r);b(r))( r;Y (a;b) t;y;z( r);Z (a;b) t;y;z( r))
dB (a;b) t;y;z( r): where ?a( t;y;z) := (t;F(y) ?1 z) +F(y) ?1 Df(y)diag[y]a:

Remark 6.4

Denition of processes (a;b) is slightly dierent from the one used in Bouchard

and Touzi (1999). This comes from the fact that the modied price process Z (a;b)

t;y;z needs to

be a martingale under P (a;b)

t;y;z up to time

T to make use of the dual formulation (see proof of

Lemma 6.1). In the previous paper, this property was needed to hold only up to a stopping time.

The reason for introducing the previous objects is the following :

Lemma 6.3

Fix (t;y;z) 2 [0;T]  (0;1) n+d. Then, ff(Y (a;b) t;y;z) ; ( a;b)2D~(t;y;z)gR(t;f(y)):

Proof.

Fix (t;y;z) 2 [0;T]  (0;1) n+d and ( a;b) 2 D~(t;y;z). Condition (R1') is clearly

satised by denition of f. Condition (R2) is also satised and the equivalent probability

measure Qis given by P (a;b)

t;y;z. Under P

(a;b)

t;y;z, the dynamic of Z

(a;b)

t;y;z is given by Remark 6.3. By

(6.8), ?a(

:;:;:) is a Lipschitz function ofF(y) ?1

z. Moreover, by (6.7), assumption on (:;:)

and denition of ~D(t;y;z), ? a( :;:;:) is valued in IM d d. Finally, by construction of ~ D(t;y;z), and assumption on , 

(a;b) satises the Novikov condition.

2

We nally adapt the denition of the auxiliary control problem to our parameterization. For all (t;y;z)2[0;T](0;1)

n+d, we write :

v(t;y;z) for v(t;f(y);z) :

Then, we have the following dynamic programming principle on our parameterized auxiliary control problem :

Corollary 6.2

Fix (t;y;z)2[0;T](0;1)

n+d. Then, for all (

a;b) 2 D~(t;y;z) and for all

[t;T]-valued stopping time  : v(t;y;z) sup (a;b)2 ~ D(t;y;z) E (a;b) t;y;z  v  ;Y (a;b) t;y;z ( );Z (a;b) t;y;z( )  ? 1  lnH (a;b) t;y;z( )  :

Proof.

This is a direct consequence of Proposition 6.2 and Lemma 6.3. 2

7 Viscosity properties of the parameterized auxiliary

control problem

7.1 Viscosity property of the auxiliary control problem

Let v

 denote the lower-semicontinuous envelope of v : v ( t;y;z) = liminf (t 0 ;y 0 ;z 0 )!(t;y;z) v(t 0 ;y 0 ;z 0 ) ; (t;y;z)2[0;T](0;1) n+d :

Then, we have the following characterization of v .

Proposition 7.1

Functionv

 is a lower-semicontinuous viscosity supersolution of the

Hamilton-Jacobi-Bellman equation : inf (a;b)2IM n;d d IR n ?L a ' ?G a;b '+ 12  ( (a;b))2 = 0 on [0 ;T)(0;1) n+d (7.1) where L a '(t;y;z) = D t '(t;y;z) + 12Tr  ?a diag[z]D 2 zz 'diag[z]? a  (t;y;z) G a;b '(t;y;z) = diag[y]b
D y '(t;y;z) + 12Tr  D 2 yy '(t;y;z)diag[y]aa diag[ y]  +Tr diag[z]? a( t;y;z)a diag[ y]D 2 yz '(t;y;z)  : 19

Proof.

Fix some (t;y;z)2[0;T)(0;1) n+d. Let 'be an arbitraryC 2([0 ;T)(0;1) n+d)

function such that

0 = (v 

?')(t;y;z) = min(v 

?'):

Fix (a;b)2D~(t;y;z) such that the process (a;b) is constant on a neighborhood of t (this is

possible by Remark 6.2). Dene :

N :=f(t 0 ;y 0 ;z 0) : jj(t 0 ;y 0 ;z 0) ?(t;y;z)jjg

for some  such that 0 <  < y i ( i = 1;::: ;n). Let (t k ;y k ;z k)k1 be a sequence in N satisfying (t k ;y k ;z k) !(t;y;z) and v(t k ;y k ;z k) !v ( t;y;z) as k !+1 Set  k := v(t k ;y k ;z k) ?'(t k ;y k ;z

k) and observe that 

k

!0 as k !+1:

First case : If the set fk  1 :  k = 0

g is nite, then there exists a subsequence renamed

(

k)k1 such that 

k

6

= 0 for all k1: So we may assume that  k 6 = 0 for all k1: Then dene :  k:= inf n s>0 :  t k+ s;Y (a;b) t k ;y k ;z k( t k+ s);Z (a;b) t k ;y k ;z k( t k+ s)  = 2N o :

Dene similarly for the initial data (t k

;y

k ;z

k) = (

t;y;z). We introduce the stopping time h k :=  k ^ p  k :

Observe that for all >0 P ?a.s. and that

liminf k!+1  k  1 2 > 0: (7.2)

This follows from the fact that (Y (a;b) t k ;y k ;z k ;Z (a;b) t k ;y k ;z k) ! (Y (a;b) t;y;z ;Z (a;b) t;y;z) for P-a.e. ! 2 ,

uni-formely on compact subsets (see Protter (1990) Theorem 37 p246). For ease of notations we write E

k for E (a;b) t k ;y k ;z

k. Then, from Corollary 6.2, it follows that v(t k ;y k ;z k)  E k  v  t k+ h k ;Y (a;b) t k ;y k ;z k( t k+ h k) ;Z (a;b) t k ;y k ;z k( t k+ h k)  ? 1  lnH (a;b) t k ;y k ;z k( t k+ h k)  : Since v  v 

 ', we may replace v by ' in the previous inequality and we get by Itô's

lemma and Remark 6.3 :

 k  E k Z t k +h k t k ? L a '+G (a;b) '
 r;Y (a;b) t k ;y k ;z k( r);Z (a;b) t k ;y k ;z k( r)  dr  ? E k Z t k +h k t k 1 2   (a;b)  r;Y (a;b) t k ;y k ;z k( r);Z (a;b) t k ;y k ;z k( r)  2 dr  20

Notice that we used the fact that by denition ofh

k, assumption on

and , and denition

of ~D(t;y;z), all diusion terms are in fact martingales.

Dividing by p 

k and sending

k to innity, we get by dominated convergence and the right

continuity of the ltration 0liminf k!+1 1 p  k Z t k +h k t k (L a '+G a;b ')(r;Y (a;b) t k ;y k ;z k( r);Z (a;b) t k ;y k ;z k( r))dr ? 1 p  k Z t k +h k t k 1 2   (a;b)  r;Y (a;b) t k ;y k ;z k( r);Z (a;b) t k ;y k ;z k( r)  2 dr:

The required result is a direct consequence of (7.2) and the fact that for all locally Lipschitz function : [0;T)(0;1) n+d !IR 1 p  k Z t k +h k t k h (r;Y (a;b) t k ;y k ;z k( r);Z (a;b) t k ;y k ;z k( r))? (t;y;z) i dr !0 as k !+1 P-a.s.

(See Lemma 8.1 in Bouchard and Touzi 1999). Second case : If the set fk  1 : 

k = 0

g is not nite, then there exists a subsequence

renamed (

k)k1 such that 

k = 0 for all

k1:So we may assume that 

k = 0 for all k 1

and the required result is obtained as in First case by dening :

h k :=  k ^ 1 k ; and replacingp  k by k ?1 in the proof. 2

7.2 Asymptotic analysis

We now emphasize the dependence of v  on

 by writting v 

. Using this new notations, we

dene on [0;T](0;1) n+d : v 1  ( t;y;z) := liminf (t 0 ;y 0 ;z 0 )!(t;y;z) !+1 v  ( t 0 ;y 0 ;z 0) :

Then, we have the following result :

Proposition 7.2

Functionv 1

 is a lower-semicontinuous viscosity supersolution of the

equa-tion : inf (a;b)2IM n;d d IR n ?L a ' ?G a;b ' = 0 on [0;T)(0;1) n+d (7.3) where L a ' and G a;b

' are dened as in Proposition 7.1.

Moreover, for all (y;z) in (0;1) n+d ; we have liminf t%T v 1  ( t;y;z)f(y)
g ? F(y) ?1 z
: (7.4) 21

Proof.

We rst prove (7.3). Consider some function ' 2 C 2

b([0

;T) (0;1)

n+d) such

that (t;y;z) is a local (strict) minimizer of (v 1



?'). Then by Lemma A.3 in Barles

and Perthames (1987), there exists a sequence f n g and a sequence f(t n ;y n ;z n) g of local minimizers of (v  n  ?') such that  n ?! +1 (t n ;y n ;z n) ?! (t;y;z) v n  ( t n ;y n ;z n) ?! v 1  ( t;y;z) (7.5) as n ! 1.

From Proposition 7.1, for alln,v n

 is a viscosity supersolution on [0

;T)(0;1)

n+dof (7.1).

Then, for all n, 'satises :  ?L a ' ?G a;b '+ 1 2 n ( (a;b))2  (t n ;y n ;z n)  0 (7.6) for all (a;b) 2 IM n;d d IR

n. Notice that we can always assume that v



 is locally bounded

uniformly in  (otherwise we could use an increasing transformation of v 

). Then, the

de-sired result is obtained by sending n to 1 in the previous inequality, using (7.5) and the

continuity of 

(a;b) with respect to (

t;y;z).

We now prove (7.4). Set (a;b) = (0;0). From Proposition 6.2 and the fact thatv ( T;y;z) =  f(y)
g(F(y) ?1 z), we have : v ( t;y;z)  ? 1  E (0;0) t;y;z h lnH (0;0) t;y;z( T) i +E (0;0) t;y;z h  f(y)
g(F(y) ?1 Z (0;0) t;y;z( T)) i : (7.7)

Using standard arguments, it is easily checked that :

Z (0;0) t 0 ;y 0 ;z 0( T) ?! z as (t 0 ;y 0 ;z 0) ?!(T;y;z) P ?a.s. E t 0 h lnH (0;0) t 0 ;y 0 ;z 0( T) i ?! 0 as (t 0 ;y 0 ;z 0) ?!(T;y;z)

along some subsequence. Then, by sending (t 0 ;y 0 ;z 0) to (

T;y;z) and using Fatou's Lemma as well as the

lower-semicontinuity of g, we get : liminf (t 0 ;y 0 ;z 0 )!(T;y;z) !+1 v  ( t 0 ;y 0 ;z 0)  f(y)
g(F(y) ?1 z): 2

Corollary 7.1

Under condition (4.1), function v 1

 is independent of

y, nonincreasing in t

and concave in z.

Proof.

This is a consequence of Proposition 7.2 and Lemma 8.2 in Bouchard and Touzi (1999) (this is mainly obtained by sending b to1 component by component in (7.3)).

2

8 Proof of Proposition 5.2

We emphasize the fact that u

g depends on

 by writtingu g;.

Fixx2IR

d+1. Following the argument in section 9 in Bouchard and Touzi (1999) and using

Corollary 7.1 as well as (7.4) in Proposition 7.2, we get :

v 1  (0 ;y;z)  g~ conc (z) for all (y;z)2(0;1) n+d :

Now from Corollary 6.1, Lemma 6.3 and denition of v 1  , we get : liminf !+1 1  ln(1?u g;( x))  sup y2(0;1) n v 1  (0 ;y;F(y)S(0))?f(y)
x  sup y2(0;1) n ~ g conc( F(y)S(0))?f(y)
x = ^g(S(0);x): 2

9 Conclusion

Using an exponential utility function, we have proved the conjecture of Barles and Soner (1996) in a multivariate framework : in a market with proportional transaction costs, the price dened as in Hodges and Neuberger (1989) tends to a super-replication related cost as the risk aversion tends to innity. The proof is direct and does not make use of the dual formulation of Kabanov (1999). We have also provided a very simple upper-bound for the probability of missing the hedge as a function of the risk aversion. Finally, we shall notice that a simpler proof of Proposition 5.2 may be found and easily extended to the continuous semi-martingale case by using the dual formulation of Kabanov (1999).

10 Appendix

Lemma 10.1

LetA be an F

0-measurable event. Fix

r 2 and  1 and  2 2R(0;r). Then :  :=  11IA+  21IA c is in R(0;r): 23

Proof.

First notice that condition (R1) is obtained by construction. For all i= 1;2, dene : H

i( T) := dQ i dP and H i( s) :=E s[ H i( T)], 0 sT, where Q i is the

equivalent probability measure associated with

i. Then, by (R2), there exists some adapted

process

i satisfying the Novikov condition such that : H i( t) = E(? R t 0  i( s)
dB(s))P?a.s.

on [0;T]. By Girsanov Theorem, this implies that we can dene a Brownian motion under Q i by dB i( t) :=dB(t)+ i(

t)dt,t2[0;T]. Then, by (P2), there exists some adapted process  i such that : diag[ i( t)]S(t) = Z t 0  i( s)dB(s) + i( s) i( s)ds P ?a.s.; 0tT ; E Z T 0 jj i( s)jj 2 ds  < +1: Now dene : (s) :=  1( s)1I A+  2( s)1I A c (s) :=  1( s)1I A+  2( s)1I A c

on [0;T]. Clearly,  is adapted and satises the Novikov condition. Moreover,  is IM d

d

-valued, IF-adapted and satises E h R T 0 jj(s)jj 2 ds i < +1. Next, by introducing H(t) = E(? R t 0

(s)
dB(s)) P ?a.s. on [0;T], we can dene an equivalent probability measure Q

satisfying H(T) = dQ

dP and such that (R2) is satised. Finally, (R1) and (R2) are satised

and  2 R(0;r). 2

Proof of Lemma 6.1

The proof is similar to Rouge (1996). We provide it for completness.

Fix (w;r) 2 IR  and  2 R(0;r). Then, by denition of R(0;r), there exists some

probability measure Q  P such that diag[]S is a Q-martingale. Dene H(T) := dQ

dP and H(t) :=E

t[

H(T)] for 0tT. Let ~U be the Legendre Transform ofU :

~

U : (0;+1) ?! IR y ?! max

x2IR

[U(x)?xy] ;

and letI denote the inverse of the derivative ofU so that for all y 2(0;+1) :

~

U(y) = U(I(y))?yI(y):

Notice that for all (x;y) 2 IR(0;1) :

U(I(y))  U(x)?xy+yI(y):

We also introduce the functions :

W : (0;1) ?! IR

x ?! E[H(T)I(xH(T))] Y : IR ?! (0;1)

x ?! exp(?E[H(T)lnH(T)?x]):

Notice that for allx 2 IR :W(Y(x)) = x.

Set ~w:=w?E[H(T)(T)
G] . Then, direct computation shows that for all2B(0;S(0)) : U(I(Y( ~w)H(T)))  U ? W  w( T)?(T)
G
? ? W  w( T)?(T)
G
Y( ~w)H(T) +Y( ~w)H(T)I(Y( ~w)H(T)) :

Since Y(:)>0 and since, by Remark 6.1, W  w is a Q-supermartingale on [0;T], we get : E[U(I(Y( ~w)H(T)))]  E  U ? W  w( T)?(T)
G
 ? E ? W  w( T)?(T)
G
Y( ~w)H(T)  + E[Y( ~w)H(T)I(Y( ~w)H(T))] :  E  U ? W  w( T)?(T)
G
 + Y( ~w)[W(Y( ~w))?w+E[(T)
GY( ~w)H(T)]] :  E  U ? W  w( T)?(T)
G
 + Y( ~w)[ ~w?w+E[(T)
GY( ~w)H(T)]] ;

and by construction of ~w, for all 2 B(0;S(0)) : E[U(I(Y( ~w)H(T)))]  E  U ? W  w( T)?(T)
G
 Moreover, since I(Y( ~w)H(T)) + (T)
G = 1  E[H(T)lnH(T)] + ~w? 1  lnH(T) + (T)
G is in L 1( Q) and E[H(T)I(Y( ~w)H(T)) +H(T)(T)
G] = w ;

by construction of R(0;r), there exists some process ^ 2 B(0;S(0)) such that : W

^



w(

T) = I(Y( ~w)H(T)) + (T)
G P ?a.s. (10.1)

This concludes the proof. 2

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