Remarks on the pricing of contingent claims under constraints

under Constraints

A. Bensoussan

University of Paris-Dauphine Place du Mar¶echal de Lattre de Tassigny

Paris Cedex 16, 75775, France (e-mail: [email protected])

March 27, 2003

1

Introduction

In this paper we consider the results of I. Karatzas, S.E. Shreve [5] related to the problem of contingent claims with constraints, see also I. Karatzas, S.G. Kou [4]. We propose a di®erent approach, which is more analytic, and also more elementary and straightforward. However, it is limited to the case when the Dynamic Programming approach is possible

Our approach has also the advantage of working without any consumption process, which turns out to be somewhat arti¯cial and slightly confusing.

2

SETTING OF THE PROBLEM

2.1

ASSUMPTIONS

Let

-;A = Borel ¾ ¡ algebra on -; P;

a probability space, equipped with a ¯ltration and a n dimensional standard-ized Wiener process for this ¯ltration

Fs_{; w(s):}

We consider

b(t); bounded adapted process in Rn (2.1) r(t) a bounded continuous deterministic scalar function (2.2) and

¾(t) n£ n bounded continuous with bounded inverse matrix; a(t) = ¾(t)¾¤(t)

(2.3) We consider a market with n stocks, governed by the usual Ito equations

dSi(s) = Si(s)(bi(s)ds + §j¾ij(s)dwj); i; j = 1;¢ ¢ ¢ ; n; S(0) ¸ 0; 6= 0

(2.4) We set

a(s) = ¾(s)¾(s)¤ (2.5)

There is also a risk-free money market, given by

dS0(s) = S0(s)r(s)(s)ds; S0(0) = 1 (2.6) We de¯ne °0(s) = 1 S0(s) (2.7) µ(s) = ¾¡1(s)(b(s)¡ r(s)1I) (2.8) Since µ is bounded, we can de¯ne a new probability on -;A, by setting

dP0 dP jFs = expf¡ Z s 0 µ(¿ )dw(¿)_¡1 2 Z s 0 jµ(¿ )j 2_d¿ g: (2.9) From the Girsanov theorem, if we introduce the process

w0(s) = w(s) + Z s

0

then the system -;A; Fs_{; P}

0; w0(s) forms a probability system in which w0(s)

is aFs _{standardized Wiener process. Note that from (2.4) one has :}

dSi(s) = Si(s)(r(s)ds + §j¾ij(s)dwj0); i; j = 1;¢ ¢ ¢ ; n (2.11)

A portfolio of assets is a vector of adapted processes

¼0(s); ¼(s) = (¼1(s);¢ ¢ ¢ ; ¼n(s)) (2.12) such that Z T 0 k¼(s)k 2 ds < +1; a.s. (2.13) We next de¯ne the Wealth as the process

X(s) = ¾n

i=0¼i(s)Si(s) (2.14)

We assume the self ¯nancing property, which amounts to the fact that the Wealth process has the following Ito di®erential

dX(s) = ¾n_i=0¼i(s)dSi(s) (2.15)

and, as well known, using (2.11)

dX(s) = r(s)X(s)ds + §i;j¼i¾ij(s)dw0j (2.16)

The initial wealth will be given by

X(0) = x (2.17)

We shall denote by

X¼ x(s)

the Wealth process, corresponding to an initial Wealth x and a portfolio ¼ We de¯ne the set of portfolios

A =_{f¼j¼adapted;} Z T 0 k¼(s)k 2 ds < +_{1; a.s.} Xx¼(s)¸ ¡¤¼x;8s 2 [0; T]; ¤¼x ¸ 0; E0¤¼x < 1; 1 < p < +1g (2.18) We note that the set A does not depend on x, which explains the notation. It is easy to check that, whenever ¼ 2 A, the process X¼

x(s)°0(s) is aFs; P0

supermartingale, and in particular

2.2

CONTINGENT CLAIMS

Let h(S) be a function such that

h continuous; 0 · h(S) · C(1 + jSj¯0_{);8S ¸ 0; h(S) > 0; 8S ¸ 0; 6= 0}

(2.20) A Contingent Claim is the variable

B = h(S(T )) (2.21)

which, thanks to the assumption (2.20) and (2.4) is a.s. strictly positive and _FT _{measurable. The concept of arbitrage opportunity is the following:}

we say that a price u represents an opportunity of arbitrage in selling, if 9 x < 0; x + u > 0 such that 9 ¼ 2 A, with

Xx+u¼ (T )¸ B a.s. (2.22)

Similarly, we say that a price u represents an opportunity of arbitrage in buying, if9 x < 0 such that 9 ¼ 2 A, with

X_x¼_¡u(T ) + B_{¸ 0 a.s.} (2.23) A price presents an arbitrage opportunity if it presents an opportunity of arbitrage in selling or in buying. An opportunity of arbitrage is a very favourable situation, since it leads to the possibility of an in¯nite wealth. Indeed, consider the selling case, for instance, we sell k > 1 contingent claims, and use the portfolio

~¼ = x + ku x + u ¼

which also belongs to A and keeps the same proportion of wealth for each stock, then one has

(X_x+ku~¼ (T )_{¡ kB)°}0(T ) = x + ku x + u (X ¼ x+u(T )¡ B) + (1¡ k)x x + u B ¸ (1¡ k)x x + u B and since x < 0, this tends to +_{1 as k ! 1.}

2.3

CONSTRAINTS

Let K+; K¡ be two non empty convex closed subsets of Rn. We shall write

K for K+ or K_¡, to simplify the notation, whenever there is no confusion.

We set for x¸ 0 A(x; K+) = 8 < : ¼ 2 A; ¤¼x = 0; p(s) = ¼(s) X¼ x(s) 2 K+; if X_x¼(s) > 0 ¼(s) = 0; if X¼ x(s) = 0 (2.24) Suppose there exists ¼2 A(x; K+) such that

X_x¼(T )¸ B a.s. (2.25) then for x0_{¸ x one considers X}¼0

x0(s) with ¼0(s) = x0 x¼(s) and clearly X_x¼00(s) = x0 xX ¼ x(s): Therefore ¼0_{2 A(x}0; K+) and X_x¼00(T ) ¸ B a.s.

This leads to the de¯nition

hup(K+) = inffx ¸ 0j9¼ 2 A(x; K+) with Xx¼(T ) ¸ Ba.s.g (2.26)

Remark 2.1. Because of (2.19), one cannot hope to have (2.26) with x < 0, even without the constraint

The quantity hup(K+) is called the upper hedging price.

We proceed similarly to de¯ne the lower hedging price. We set for x¸ 0 A(x; K_¡) = 8 < : ¼ _{2 A; X¡x}¼ (s)_{· 0; p(s) =} ¼(s) X_¡x¼ (s) 2 K¡; ifX ¼ ¡x(s) < 0 ¼(s) = 0; ifX_¡x¼ (s) = 0 (2.27)

Suppose there exists ¼2 A(x; K¡) such that

X_¡x¼ (T ) + B¸ 0a.s. (2.28) Considering x0< x, and again

¼0(s) = x 0 x¼(s) then X_¡x¼00(s) = x0 xX ¼ ¡x(s): Therefore, necessarily ¼02 A(x0_{; K} ¡) and X_¡x¼00(T ) + B ¸ 0a.s.

This leads to the de¯nition

hlow(K_¡) = supfx ¸ 0j9¼ 2 A(x; K_¡) with X_¡x¼ (T ) + B ¸ 0 a.s.g

(2.29) Remark 2.2. Note that 02 A(0; K¡), and satis¯es the condition (2.30).

The quantity hlow(K_¡) is called the lower hedging price. We shall check

the elementary property

Lemma 2.3. We have the property

hlow(K_¡)· hup(K+) (2.30)

PROOF: We may assume

hlow(K_¡) > 0; hup(K+) < +1

otherwise the property is trivial. If there exists

¼2 A(x; K+) with Xx¼(T ) ¸ B a.s.

then, using the supermartingale property (2.19), we have x¸ E0Xx¼(T )°0(T ) ¸ E0B°0(T ):

Setting

u0= E0B°0(T ) (2.31)

then we deduce

hup(K+)¸ u0 (2.32)

and u0 is ¯nite. Next, considering ², with

² < hlow(K_¡)

then, setting x²= hlow(K_¡)¡ ²

9¼ 2 A(x²; K¡) with X_¡x¼ ²(T ) + B¸ 0

and from the supermartingale property again,we get

¡x² ¸ E0X_¡x¼ _²(T )°0(T )¸ ¡E0B°0(T ) (2.33)

therefore

x²· u0 (2.34)

and letting ² tend to zero, we obtain :

hlow(K_¡)· u0 (2.35)

which, compared to (2.32)completes the proof. ¥

Remark 2.4. It is an immediate consequence of the de¯nitions that a price u > hup(K+) leads to an opportunity of arbitrage in selling, and a positive

price u < hlow(K_¡) leads to an opportunity of arbitrage in buying. On the

other hand a price u_{2 [h}low(K¡); hup(K+)] cannot o®er any opportunity of

3

CHARACTERIZATION OF THE UPPER

HEDGING PRICE

3.1

STOCHASTIC CONTROL PROBLEM

We shall consider the upper hedging price, and thus to simplify the notation, we write K = K+: Let us set ±(º) = sup fp2Kg(¡p ¤_{º) (3.1)}

which is a lower semicontinuous (lsc), proper, convex function, ¯nite on its e®ective domain

~

K =_{fºj±(º) < +1g} (3.2) Using R.T. ROCKAFELLAR [7], the set ~K is a convex cone, and

0_{2 ~}K ; with ±(0) = 0: Moreover ± is positively homogeneous, i.e.

±(¸º) = ¸±(º); if ¸_{¸ 0} and subadditive, i.e.

±(º + ¹) · ±(º) + ±(¹); 8º; ¹: Relying on ROCKAFELLAR again, we state the property

p2 K , p¤_{º + ±(º)¸ 0; 8º 2 ~K (3.3)}

In the sequel, we shall assume

±(º) ¸ ±0;8º (3.4)

Consider the Hilbert space

L2

of stochastic processes º(s) adapted to the ¯ltrationFs _{such that} E0 Z T 0 jjº(s)jj 2 ds < +1: We then de¯ne D = fº(:) 2 L2_F(0; T ; Rn)jº(s; !) 2 ~K; Z T 0 j±(º(s))jds < +1a.s.g (3.5) Let next, writing º for º(:)

Zº(s) = exp[¡ Z s 0 ¾¡1º(¿ )dw0_(¿)¡1 2 Z s 0 j¾ ¡1_º(¿)j2_{d¿ ] (3.6)} °º(s) = exp[¡ Z s 0 (r(¿ ) + ±(º(¿ )))d¿ ] (3.7) These two processes are well de¯ned for any º(:)_{2 D. We set}

D(m)_{=fº(:) 2 DjZ}

º(:)is a martingale forP0;Fsg (3.8)

It is well known that if

sup_{jº(s; !)j < +1} (3.9) then º(:)2 D(m)_{. We call}

D(b)_{=fº(:) 2 D boundedg (3.10)}

The setD(b)_{, hence alsoD}(m) _{is dense inD. Indeed, we can approximate º(:)}

by ºk(s) = ¯ ¯ ¯¯ ¯¯ º(s) if _{jº(s)j · k} k jº(s)jº(s) if jº(s)j ¸ k (3.11) Since ±(ºk(s)) = ¯ ¯¯ ¯¯ ¯ ±(º(s)) if _{jº(s)j · k} k jº(s)j±(º(s)) if jº(s)j ¸ k (3.12)

then clearly ºk(:) belongs to D(b) if º(:) belongs to D. Moreover

ºk(:)! º(:) 2 L2(0; T ; Rn); a.s. as k! 1: (3.13)

±(ºk(:))! ±(º(:)) 2 L1(0; T ; Rn); a.s. as k! 1: (3.14)

We shall call in the sequel

Hº(s) = Zº(s)°º(s) (3.15) and we de¯ne uº = E0(Hº(T )B) ^ u = sup º (:)_2D (3.16) If º(:) = 0, we recover the de¯nition (2.31).

We state the easy result

Lemma 3.1. We have the property ^ u = sup º(:)_2D(b) = sup º(:)_2D(m) (3.17) PROOF: Clearly ^ u¸ sup º(:)2D(b) (3.18) On the other hand, if for any º_{2 D, we consider the approximation º}k, then

thanks to (3.13),(3.14), we can state that

Hºk(T )! Hº(T ); a.s.

and thus , by Fatou's Lemma

E0(Hº(T )B) · lim inf k!1 E 0_(H ºk(T )B)· sup º(:)_2D(b) which implies ^ u_{· sup} º(:)_2D(b)

3.2

THE MAIN RESULT

The main result is the following

Theorem 3.2. Assuming (3.4), (2.20), and then one has

hup(K) = ^u (3.19)

We begin by proving that

hup(K) ¸ ^u (3.20)

PROOF of (3.20)

If we consider º 2 D(m)_{, then, since Z}

º(s) is a P0;Fs martingale, then we

can de¯ne a new probability measure Pº on -;A, by setting dPº

dP0jFs = Zº(s): (3.21)

From the Girsanov theorem, if we introduce the process wº(s) = w0(s) +

Z s 0

¾¡1º(¿ )d¿; (3.22) then the system -;A; Fs_{; P}º_{; w}º_{(s) forms a probability system in which}

wº_{(s) is a F}s _{standardized Wiener process. Note that from (2.11) one has :}

dSi(s) = Si(s)((r(s)¡ ºi)ds + §j¾ij(s)dwjº); i; j = 1;¢ ¢ ¢ ; n (3.23)

We can then write

uº = Eº(°º(T )B) (3.24)

Suppose now that there exists , for x¸ 0

¼ 2 A(x; K) with Xx¼(T )¸ Ba.s.

then , using Ito's formula one has d(X¼ x(s)°º(s)) = °º(s)¼¤(s)¾(s)dwº¡ ¡°º(s)Xx¼(s)( ¼¤(s) X¼ x(s) º(s) + ±(º(s)))

and from (3.3)

· °º(s)¼¤(s)¾(s)dwº:

It follows easily that x¸ Eº_(X¼

x(T )°º(T ))¸ Eº(B°º(T )): (3.25)

Therefore we have

x_{¸ u}º

and since º is arbitrary, we obtain (3.20). ¥

We next introduce the function

u(S; t) = u(S; t; h) = E0h(SS;t(T ))°0(t; T ) (3.26)

where SS;t(T ) represents the solution of (2.11) with initial data S at time t,

namely Si;S;t(T ) = Siexpf Z T t (r(s)¡ 1 2aii(s))ds + Z T t §j¾ij(s)dw0jg (3.27) and °0(t; T ) = °0(T ) °0(t) (3.28) Clearly, we have (Markov property)

SS;0(T ) = SS(t);t(T ) (3.29)

and

u(S; 0) = u0

It is useful to note the following change of variables: Si = exp yi, and

v(y; t) = v(y; t; h) = u(¢ ¢ ¢ ; exp yi;¢ ¢ ¢ ; t) (3.30)

with @v @t ¡ r(t)v + (r(t) ¡ 1 2aii(t)) @v @yi +1 2ai;j(t) @2v @yi@yj = 0 v(y; T ) = h(¢ ¢ ¢ ; exp yi;¢ ¢ ¢ ) (3.31)

4

PROOFS

4.1

PRELIMINARIES

We shall use results related to the Cauchy problem (3.31). More precisely, we consider the problem

@z @t ¡ r(t)z + (r(t) ¡ 1 2aii(t)) @ z @ yi +1 2aij(t) @2_z @yi@yj = 0 z(y; T ) = Á(y) (4.1)

where the initial condition Á(y) satis¯es

Á measurable ; 0· Á(y) · c(1 + exp ¯0jyj); a.e. (4.2)

Introduce the functional spaces

L2¯ = L2¯(Rn) =fÃ(y) exp ¡¯jyj 2 L2(Rn)g

H_¯1(Rn) =_{fà 2 L}2_{¯jDà 2 L}2_¯g W¯(0; T ) =fÃ(y; t)jà 2 L2(0; T ; H¯1); @à @t 2 L 2 (0; T ; H¯¡1)g (4.3)

where H_¯¡1represents the dual of H1 ¯.

From J.L. LIONS [6], it follows that

W¯(0; T ) ½ C0([0; T ]; L2¯) (4.4)

with continuous injection.

The function Á(y) belongs to L2

¯,for any ¯ > ¯0,. We recall the equations

(3.29), and introduce the stochastic processes yi;y;t(s) = yi+ Z s t (r(¿)_¡ 1 2aii(¿ ))d¿ + Z s t §j¾i;j(¿ )dwj0 (4.5) We denote by

yy;t(s) the solution of (4.5)

and to simplify the notation write ½(t; s) for the vector of coordinates ½i(t; s) =

Z s t

(r(¿ )¡ 1₂aii(¿))d¿:

Theorem 4.1. Assuming (2.2), (2.3), (4.2), there exists one and only one solution of (4.1), z2 W¯(0; T ), for any ¯ > ¯0. Moreover ,

z; Dz; D2z;@z

@t are continuous on R

n

£ [0; T) ; z(y; t)_{¸ 0:} (4.6) 0_{· z(y; t) · C(1 + exp ¯}0jyj); 8y; t (4.7)

If

Á is continuous (4.8)

then one has

z is continuous on Rn_{£ [0; T]: (4.9)}

One has the probabilistic interpretation

z(y; t) = E0Á(yy;t(T ))°0(t; T ) (4.10)

PROOF:

We do not detail everything, and refer to classical results, see A. FRIEDMAN [2]. Whenever (4.8) holds, then we have (4.6) and (4.1)holds on Rn£ [0; T). Moreover, we have (4.9), and the probabilistic interpretation (4.10).

We proceed with a priori estimates. We test (4.1) with z exp¡2¯jyj; ¯ > ¯0,

and integrate over Rn_{. We obtain easily the estimate}

jjzjjfW¯(0;T )g· C¯jjÁjjfL2¯g (4.11)

and thus also, from (4.4) jjzjj_fC0_{([0;T ];L}2

¯)g · C¯jjÁjjfL2¯g= (4.12)

To extend the results when (4.8) is no longer valid, we ¯rst notice, from the variational theory of linear parbolic P.D.E. of J. L. LIONS [4], that there exists one and only one solution of (4.1), z 2 W¯(0; T ), and z ¸ 0. The

estimates (4.11) and (4.12) still hold. We can then consider independently, the function denoted for the time being by

which makes sense, even when Á is not continuous. From the relation (4.5) and the notation hereafter, we have

yy;t(T ) = y + ½(t; T ) + Z T t ¾dw0_: Let ¡(t; T ) = Z T t a(¿)d¿:

Since yy;t(T ) is gaussian, with mean y+½(t; T ) and covariance matrix ¡(t; T ),

we can write the formula ³(y; t) = Z Rn Á(x)exp[¡ 1 2¡(t; T )¡1(x¡ y ¡ ½(t; T )):(x ¡ y ¡ ½(t; T))] (2¼)n2j¡(t; T)j 1 2 dx°0(t; T ) (4.14) Performing easy majorations, one gets , for all ¯0> ¯0 the inequality

³(y; t)· jjÁjjfL2 ¯0g(exp ¯ 0_{jyj)K(t; T) (4.15)} where K(t; T ) = exp ¯0_{j½(t; T )j} ³R Rn exp[¡j´j2+ 2¯0j¡(t; T) 1 2´j] d´ ´1 2 (2¼)n2j¡(t; T)j 1 4 °0(t; T ) (4.16) and chosing ¯ > ¯0_{, we obtain, after easy calculations}

jj³(:; t)jjL2 ¯ · jjÁjjfL2_¯0gK(t; T ) µ (n_{¡ 1)!} (2(¯¡ ¯0₎₎n¡1 ¶1 2 (4.17) We next show that, for any Á, satisfying (4.2), there exists a sequence of continuous functions Á¹, such that

0· Á¹(y)· c(1 + exp ¯0jyj) (4.18)

and for any ¯ > ¯0, and for any ², there exists ¹¯(²), with

This is done by the classical procedure of cutting and regularization. Let BR

be the ball of center 0 and radius R. The function ÁR(y) = Á(y)1IBR

1 + exp ¯0jyj

is positive, bounded by c, and vanishes outside BR. Clearly

Á1IBR ! Á in L

2

¯; as R ! +1:

Consider next the smoothing procedure by the function µ¸(y) = 1 ¸nµ( y ¸) with µ(y) = ¯¯ ¯¯ ¯¯ 1 aexp¡ 1 1¡ jyj2 if jyj · 1 0 if jyj ¸ 1 and the scalar a is such that

Z

Rn

µ(y)dy = 1: For each ¯xed R, the function

(1 + exp ¯0jyj)µ¸¤ ÁR ! Á1IBR in L

2

(Rn) as ¸ ! 0: Set

¹ = (¸; R) and

Á¹(y) = (1 + exp ¯0jyj)µ¸¤ ÁR(y):

Then Á¹is continuous, and recalling that ÁR(y)· c

Á¹(y)· c(1 + exp ¯0jyj):

Moreover, it is easy to check, that by choosing ¯rst R, then ¸, one can ¯nd ¹¯(²) such that (4.18) holds.

These approximations allow us to assert that the probabilistic representation (4.10) still holds, even when the initial condition Á is not continuous. Indeed,

let z¹; ³¹ be the functions corresponding to Á¹, in (4.1) and (4.13). Then,

since the probabilistic representation holds for Á¹, we have

z¹ = ³¹:

Also, from (4.12) and (4.17) we have jjz ¡ z¹jjL2

¯ · C¯jjÁ ¡ Á¹jjL2¯

and

jj³(:; t) ¡ ³¹(:; t)jjL2

¯ · C¯;¯0;tjjÁ ¡ Á¹jjL2¯0:

Collecting results, the property (4.10) follows.

From this probabilistic representation, and the growth assumption (4.2), the growth on z, (4.7) follows. The regularity properties (4.6) follow also from the representation formula, and (4.14).

The proof has been completed.

¥

In spite of the fact that Á is not continuous, we have the following important continuity property. Let us write

y(t) = yy;0(t) (4.20)

then we state the following result Corollary 4.2.

z(y(t); t)! Á(y(T)); as t ! T; in L2_{(-;A; P}0_):

PROOF:

We ¯rst notice that

z(y(t); t) = E0[Á(y(T ))jFt]°0(t; T ) (4.21)

which follows from the Markov property y(T ) = yy(t);t(T )

and the representation formula (4.10). Hence z(y(t); t)°0(t) is an Ft; P0

martingale. Therefore, from classical martingale estimates, see N. IKEDA, S. WATANABE [3], we have

E0 sup

f0·t·Tg(jz(y(t); t)j°0(t)) 2

· 4E0(Á(y(T ))°0(T ))2 (4.22)

and , since y(T ) is gaussian, see also (4.14), we get easily E0 sup

f0·t·Tgjz(y(t); t)j 2

· CT;¯jjÁjj2L2

¯ (4.23)

Considering the continuous approximation Á¹ of Á, and using (4.23),we can

write the inequality

E0jz(y(t); t) ¡ Á(y(T ))j2 · CT ;¯jjÁ ¡ Á¹jj2L2 ¯ + 3E

0

jz¹(y(t); t)¡ Á¹(y(T ))j2:

Since z¹(y; t) is continuous on Rn£ [0; T], we have for ¯xed ¹, and thanks

to (4.23)

E0_jz¹(y(t); t)¡ Á¹(y(T ))j2! 0; as t " T:

The result follows immediately. ¥

4.2

ADDITIONAL ASSUMPTIONS and

COMPLE-TION of PROOFS

We introduce the function ^h(S) = sup

fº2 ~K_g

h(¢ ¢ ¢ ; Siexp¡ºi;¢ ¢ ¢ ) exp ¡±(º) (4.24)

and its transformation ^h(y) = sup

fº2 ~K_g

h(¢ ¢ ¢ ; exp(yi¡ ºi);¢ ¢ ¢ ) exp ¡±(º) (4.25)

We make on ^h(S) the same growth assumption as for h(S), see (2.20), for-mulated on ^h(y), as follows

Introduce next the function ~v(y; t) solution of the Cauchy problem @ ~v @t ¡ r(t)~v + (r(t) ¡ 1 2aii(t)) @~v @yi +1 2aij(t) @2~v @yi@yj = 0 ~ v(y; T ) = ^h(y) (4.27)

which is, with the notation (3.30),(3.31),

~v(y; t) = v(y; t; ^h) (4.28) We check easily that

^h is lower semicontinuous (4.29) We state the

Lemma 4.3. We assume (2.2), (2.3),and (4.26), then one has the property D~v(y; t):º + ~v(y; t)±(º)¸ 0; forally; t < T; 8º 2 ~K (4.30) PROOF:

Let µ > 0, and º; º0_{2 ~K. By de¯nition of ^y, and for any ¹ we have}

^h(y + µº) ¸ h(¢ ¢ ¢ ; exp(yi+ µºi¡ ¹i);¢ ¢ ¢ ) exp ¡±(¹):

Since

¹ = º0+ µº belongs to ~K, we can assert that we can deduce

^h(y + µº) ¸ h(¢ ¢ ¢ ; exp(yi¡ ºi0);¢ ¢ ¢ ) exp ¡±(º0) exp¡±(µº):

hence, since º0 is arbitrary in ~K ,

^h(y + µº) ¸ ^h(y)exp ¡±(µº): This can be written as

^h(y + µº) ¡ ^h(y) + ^h(y)(1 ¡ exp ¡µ±(º)) ¸ 0: (4.31) Now , we apply Theorem (4.1) to write the probabilistic representation

therefore also

~v(y + µº; t) = E0h(y^ y+µº;t(T ))°0(t; T ) (4.33)

Since

yy+µº;t(T ) = yy;t(T ) + µº

we may apply (4.31, with y changed by yy;t(T ) to obtain

~v(y + µº; t)¡ ~v(y; t) + ~v(y; t)(1 ¡ exp ¡µ±(º)) ¸ 0: (4.34) Dividing by µ and letting µ tend to 0, making use of the di®erentiability of ~v, we obtain the property (4.30) .

¥

Remark 4.4. It follows from (3.3)and (4.30), noting that, thanks to the as-sumption (4.26) ~v(y; t) > 0, the property

D~v(y; t) ~

v(y; t) 2 K; 8y; t (4.35) We then deduce the property

Proposition 4.5. Under the assumptions of Lemma (4.3), one has the prop-erty

~v(y; 0) = v(y; 0; ^h) _{¸ h}up(K ) (4.36)

PROOF:

From equation (4.27), one checks easily that, recalling the process y(s), see (4.20)

d(~v(y(s); s)°0(s)) = °0(s)D~v(y(s); s):¾(s)dw0(s) (4.37)

and thus, for t < T , one has ~v(y(t); t)°0(t) = ~v(y; 0) + Z t 0 °0(s)D~v(y(s); s):¾(s)dw0(s) (4.38) and also E0_{(~v(y(t); t)°} 0(t))2= (~v(y; 0))2+ E0 Z t 0 (°0(s))2j¾¤(s)D~v(y(s); s)j2ds: (4.39)

Using Corollary (4.2), we can let t tend to T in the preceding equation, and obtain E0(^h(y(T ))°0(T ))2= (~v(y; 0))2+ E0 Z T 0 (°0(s))2j¾¤(s)D~v(y(s); s)j2ds: (4.40) We may then set

~

¼(s) = D~v(y(s); s) and, recall (2.16)

X~v(y;0)¼~ (s) = ~v(y(s); s):

Thanks to (4.35) and (4.40) one may assert that ~

¼2 A(~v(y; 0); K) and

Xv(y;0)~~¼ (T ) = ^h(y(T ))¸ h(y(T )):

Therefore (4.36) follows.

¥ From (4.36) and (3.20) we deduce

~v(y; 0) = v(y; 0; ^h) ¸ hup(K )¸ ^u (4.41)

We can also check directly the inequality

~v(y; 0)¸ ^u (4.42)

PROOF:

First recall Lemma 3.1, and equation (3.16). We can write uº = Eº[h(¢ ¢ ¢ ; exp(yi + Z T 0 (r(s)_¡1 2ai;i(s))ds + Z T 0 ¾i;j(s)dwº(s)¡ Z T 0 ºi(s)ds;¢ ¢ ¢ )°º(T )] (4.43) But _Z T 0 ±(º(s))ds¸ ±( Z T 0 º(s)ds):

Since _Z T 0 º(s)ds2 ~K we can state uº · Eº[^h(¢ ¢ ¢ ; exp(yi+ Z T 0 (r(s)_¡ 1 2ai;i(s))ds + Z T 0 ¾i;j(s)dwº(s);¢ ¢ ¢ )°0(T )] (4.44) which can also be written

uº · E0[^h(¢ ¢ ¢ ; exp(yi + Z T 0 (r(s)_¡1 2ai;i(s))ds + Z T 0 ¾i;j(s)dw0(s);¢ ¢ ¢ )°0(T )] (4.45) which means uº · ~v(y; 0)

and thus (4.42) follows.

We can also give a proof more in line with the standard Dynamic Program-ming argument. Consider

d(~v(y(s); s)°º(s) =¡°º(s)(D~v(y(s); s):º(s) + ~v(y(s); s)±(º(s)))ds+

+°º(s)D~v(y(s); s):¾(s)dwº(s)

then from (4.30) we deduce

d(~v(y(s); s)°º(s)· °º(s)D~v(y(s); s):¾(s)dwº(s):

Therefore, integrating and using (4.40) as well as the continuity of ~v(y(s); s), as s" T it follows

~

v(y; 0)¸ Eº_{(^h(¢ ¢ ¢ ; exp y}

i;y;0(T );¢ ¢ ¢ )°º(T ))

¸ Eº(h(_{¢ ¢ ¢ ; exp y}i;y;0(T );¢ ¢ ¢ )°º(T )) = uº

4.3

THE REVERSE INEQUALITY

The objective here is to prove the reverse inequality

~v(y; 0)· ^u (4.46)

which will imply from (4.41) the property (3.19),and complete the proof of Theorem 3.2.

We shall use the function, which extends (4.25) ^v(y; t) = sup

fº2 ~K_g

v(y_{¡ º; t) exp ¡±(º)} (4.47) Of course ^v(y; t) is di®erent from ~v(y; t). However, we can state the property ^v(y; t)· ~v(y; t) · C(1 + exp ¯0jyj) (4.48)

PROOF of (4.48):

The second inequality is just a consequence of Theorem (4.1), see (4.7), and assumption (4.26). To prove the ¯rst one, we notice that

v(y_{¡ º; t) exp ¡±(º) = E}0h(yy;t(T )¡ º)) exp ¡±(º)°0(t; T )

· E0^h(yy;t(T ))°0(t; T ) = ~v(y; t):

¥

Since the supremum in (4.47) may not be attained, we use an approximation as follows ^v²(y; t) = sup fº2 ~Kg [v(y¡ º; t) exp ¡±(º) ¡ ² 2jºj 2_{] (4.49)}

Then we can state the property: there exists ^º²(y; t), measurable, such that ^v²(y; t) = v(y¡ ^º²_{; t) exp¡±(^º}²_)¡ ²

2j^º

²_j2 _(4.50)

for any y 2 Rn_{; t < T .}

PROOF of (4.50): Consider the function

©²(y; t; º) = v(y¡ º; t) exp ¡±(º) ¡ ² 2jºj

which is continuous in y; t and u.s.c. in º. To ¯nd the supremum, we can restrict º so that

² 2jºj

2 _{· v(y ¡ º; t) exp ¡±(º) · ^v(y; t)}

and from (4.48)

² 2jºj

2

· C(1 + exp ¯0jyj):

Consider the ball BR, of center 0 and radius R, and the function ©²(y; t; º),

restricted to BR£ (0; T ). Then the supremum is obtained on the compact

set

² 2jºj

2_{· C(1 + exp ¯} 0R):

Therefore, using classical selection properties, see I. EKELAND, R.TEMAM [1], there exists a selection ^º_R²(y; t) , such that

^

v²(y; t) = v(y¡ ^ºR²; t) exp¡±(^ºR²)¡

² 2j^º ² Rj2 for any y 2 BR; t < T . If we de¯ne ^ º²(y; t) = +1 X fn=1g 1IBn¡Bn¡1(y)^º ² n(y; t)

this function satis¯es the properties. ¥ We then de¯ne ^ º² t(s) = 8 < : 0 if s < t ^ º²_{(y(t); t)} T ¡ t ; if s¸ t (4.51) We can check that

^

º_t² 2 Dfmg: Indeed, considering Zº^²

t(s), then one has

Zº^² t(s) = 1; if s < t Zº^² t(s) = exp[¡ (^º²)¤(y(t); t) T ¡ t Z s t (¾¡1)¤dw0 ¡1 2 (^º²₎¤_{(y(t); t)} T _{¡ t} Z s t a¡1(¿)d¿^º ²_{(y(t); t)} T _{¡ t} ] (4.52)

Therefore, one has, as easily seen E0[Zº^²

t(T )jF

t_{] = 1}

and the martingale property follows . We can then assert that

^ u¸ u^º²

t: (4.53)

But we have, see (4.43) u^º² t = E ^ º² t_{[h(y(t)+½(t; T )+} Z T t ¾(s)dwº^t²_(s)¡^º²_{(y(t); t))°} 0(T ) exp¡±(^º²(y(t); t))]:

Taking ¯rst the conditional expectation with respect to _Ft _yields

u^º² t = E

0_{[v(y(t)¡ ^º}²_{(y(t); t))°}

0(t) exp¡±(^º²(y(t); t))] (4.54)

and recalling the de¯nition of ^º²_{(y; t), we can write}

u^º² t ¸ E

0

^v²(y(t); t)°0(t) (4.55)

But, it is easy to check that

v²_{(y; t)" v(y; t)}

By Fatou's Lemma, we deduce ^

u¸ E0^v(y(t); t)°0(t) (4.56)

Now, we let t_{" T, in the right hand side of (4.56). By Fatou's Lemma again,} we state ^ u¸ E0lim inf t_"T v(y(t); t)°^ 0(T ) (4.57) But lim inf

t_"T v(y(t); t)^ ¸ limt_"T v(y(t)¡ º; t) exp ¡±(º); 8º

¸ h(¢ ¢ ¢ ; exp(yi(T )¡ ºi);¢ ¢ ¢ ) exp ¡±(º); 8º

hence

lim inf

t_"T v(y(t); t)^ ¸ ^h(¢ ¢ ¢ ; exp yi(T );¢ ¢ ¢ ):

From (3.30), it then follows ^

u_{¸ E}0^h(¢ ¢ ¢ ; exp yi(T );¢ ¢ ¢ )°0(T ) = ~v(y; 0): (4.58)

which is (4.46). ¥

5

CHARACTERIZATION OF THE LOWER

HEDGING PRICE

We just state the counterpart of the previous section. Setting, to simplify the notation

K = K_¡ and de¯ning successively

³(º) = inf p_2K(¡p ¤_{º) (5.1)} ~ K =fºj³(º) > ¡1g (5.2) ·h(y) = inf fº2 ~Kgh(y ¡ º) exp ¡³(º) (5.3)

¹v(y; t) = E0·h(yy;t(T ))°0(t; T ) = v(y; t; ·h) (5.4)

·

u = inf

fº(:)2D(m)_gE

º

(h(y(T ))°º(T ) (5.5)

then we have the property ·

u = hlow(K) = ¹v(y; 0) (5.6)

References

[1] I. EKELAND, R. TEMAM, Convex Analysis and Variational Problems, North Holland, Amsterdam (1976)

[2] A. FRIEDMAN, Partial Di®erential Equations of parabolic type, Prentice Hall, N.J. (1964)

[3] N. IKEDA, S. WATANABE, Stochastic Di®erential Equations and Di®usion Processes North Holland, Amsterdam (1981)

[4] I. KARATZAS, S.G. KOU, On the Pricing of Contingent Claims with Constraints, The Annals of Applied Probability, Vol 6, N 2, pp 321-369

[5] I. KARATZAS, S.E. SHREVE, Methods of Mathematical Finance, Applications of Mathematics, Springer Verlag, New York (1998)

[6] J. L. LIONS, Contr^ole optimal des systµemes gouvern¶es par des ¶equations aux d¶eriv¶ees Dunod, Paris (1969)

[7] R.T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, N.J.