[springer optimization and its applications] modern stochastics and applications volume 90 ||...

Comparing Brownian Stochastic Integralsfor the Convex Order

Francis Hirsch and Marc Yor

Abstract We show that, in general, inequalities between integrands with respectto Brownian motion do not lead to majorization in the convex order for thecorresponding stochastic integrals. Particular examples and counterexamples arediscussed.

1 Introduction

In this chapter, we are interested in the following general question. Let X and Y

be square integrable Brownian centered random variables given by their predictablerepresentations:

X DZ 1

0

Ht dBt Y DZ 1

0

Kt dBt

withZ 1

0

EŒH 2t � dt < 1 and

Z 1

0

EŒK2t � dt < 1: (1)

F. HirschLaboratoire d’Analyse et Probabilités, Université d’Évry-Val d’Essonne,23 Boulevard de France, F-91037 Evry Cedex, Francee-mail: [email protected]

M. Yor (�)Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI et VII,4 Place Jussieu, Case 188, F-75252 Paris Cedex 05, France

Institut Universitaire de France, Paris, Francee-mail: [email protected]

V. Korolyuk et al. (eds.), Modern Stochastics and Applications, Springer Optimizationand Its Applications 90, DOI 10.1007/978-3-319-03512-3__1,© Springer International Publishing Switzerland 2014

3

mailto:[email protected]

mailto:[email protected]

4 F. Hirsch and M. Yor

Is it possible to give conditions on H and K ensuring that X � Y in the convexorder?

We recall that two integrable random variables X and Y are said to satisfy X � Y

in the convex order, which will be denoted in the sequel by X.c/� Y , if, for every

convex ' W R �! R,

�1 < EŒ'.X/� � EŒ'.Y /� � C1:

There is an obvious necessary condition:

X.c/� Y H)

Z 1

0

EŒH 2t � dt �

Z 1

0

EŒK2t � dt: (2)

This condition is far from being sufficient. In Sect. 2 we present an easycounterexample.

Then, in Sect. 3, we consider the elementary case where .Ht / or .Kt / isdeterministic. In this case, there is a simple sufficient condition.

In the three following sections, we study particular families .Xf .a/I a � 0/

defined by

Xf .a/ DZ 1

0

f .a; s; Bs/ dBs;

where f denotes a nonnegative Borel function on RC � RC � R.In Sect. 4, f .a; s; x/ D 1.0;1/.s/ 1.a;C1/.x/. Then f is decreasing with respect

to a, and the map: a �! Xf .a/ also is decreasing in the convex order.In Sect. 5, f .a; s; x/ D 1.0;1/.s/ 1.�1;a/.x/. Then f is increasing with respect to

a, and the map: a �! Xf .a/ also is increasing in the convex order.In Sect. 6, f .a; s; x/ D a 1.0;1/.s/ C 1.1;2/.s/ 1.�1;0/.x/. Then f is increasing

with respect to a, and the map: a �! Xf .a/ is not monotone in the convexorder. More precisely, a �! EŒ.Xf .a//2� is obviously increasing, but a �!EŒexp.Xf .a//� is strictly decreasing on Œ0; a0� for some a0 > 0.

2 A Simple Example

In this section, we show that, in general, the converse of (2) does not hold, even ifH is deterministic.

Proposition 1. We set, for � � 0,

X� DZ 1

0

exp.� Bs/ dBs:

Stochastic Integrals and Convex Order 5

Obviously, for any � � 0,

Z 1

0

EŒexp.2� Bs/� ds DZ 1

0

exp.2�2s/ ds � 1:

However, there exists � > 0 such that B1

.c/� X� does not hold.

Proof. For every C 1-function ' with bounded first derivative, we have

d

d�EŒ'.X�/�

ˇ̌ˇ̌�D0

D 1

2EŒ' 0.B1/ .B2

1 � 1/�:

Suppose that for every � > 0, one has B1

.c/� X�. Then, for every convexC 1-function ' with bounded first derivative, EŒ' 0.B1/ .B2

1 � 1/� � 0. In particular,we obtain for the convex function: '.x/ D .xC1/2 1.�1;0/.x/C.1C2x/ 1.0;C1/.x/,

2EŒ..B1 C 1/C ^ 1/ .B21 � 1/� � 0:

Now,

EŒ..B1 C 1/C ^ 1/ .B21 � 1/� D 1p

2�.e�1=2 � 1/ < 0;

which yields a contradiction. ut

3 Case Where H Or K Is Deterministic

The following proposition partially extends a result of Pagès [8, Proposition 2.4]with a different method.

Proposition 2. Let .Ht / be an adapted process and k be a deterministic Borelfunction such that

Z 1

0

EŒH 2t � dt < 1 and

Z 1

0

k2.t/ dt < 1:

We set

X DZ 1

0

Ht dBt Y DZ 1

0

k.t/ dBt:


One has

1. ifZ 1

0

H 2t dt �

Z 1

0

k.t/2 dt a.s., then X.c/� Y ;

2. ifZ 1

0

k.t/2 dt �Z 1

0

H 2t dt a.s., then Y

.c/� X .

Proof. By the Dubins–Schwarz theorem (see, e.g., Revuz-Yor [9, Chap. V]), thereexist a filtration .Gu/ and a G-Brownian motion .ˇu/ such that: X D ˇT , withT WD R1

0H 2

t dt a .Gu/-stopping time. By hypothesis, T is integrable. Denote bys the deterministic time

R10

k.t/2 dt . Let also QY D ˇs . Since k is deterministic,

we have: Y.law/D QY . Now, if T � s, then X D EŒ QY jGT �, and if s � T , then

QY D EŒX jGs�. The desired result then follows from Jensen’s inequality. utRemark 1. We shall show in Sect. 6 that in the above proposition, thehypothesis that k is deterministic cannot be deleted. Likewise, the hypothesis:R1

0EŒH 2

t � dt < 1 cannot be deleted as shown by the following example. Supposethat k.t/ D 1.0;1/.t/ and Ht D 1.0;d1/.t/ with d1 D infft � 1I Bt D 0g. Then, since1 � d1, one has 0 � k.t/ � Ht . But, X D Bd1 D 0, Y D B1 and, obviously,

B1

.c/� 0 does not hold. On the contrary, we have 0.c/� B1, which shows that, in

general, the implication (2) does not hold if the condition (1) is not fulfilled.

4 A Decreasing Family

We set, for t � 0 and a � 0,

Xt .a/ DZ t

0

1.Bs>a/ dBs

and we denote X1.a/ simply by X.a/. We also denote as usual by Ta the entrancetime of .Bt / in Œa; C1Œ.

Proposition 3. The map: a � 0 �! X.a/ is decreasing in the convex order.

Proof. Denote by B.a/ the Brownian motion defined by B.a/t D BtCTa � a.

It is independent of FBTa

(where .FBt / denotes the natural filtration of B) and, in

particular, it is independent of Ta. We set, for t � 0,

X.a/t D

Z t

0

1.B

.a/s >0/

dB.a/s :

We clearly have

X.a/ D X.a/

.1�Ta/C : (3)


Let QB be an independent copy of B . We deduce from (3) that

X.a/.law/D QX.1�Ta/C.0/; (4)

where QXt .a/ is defined as Xt.a/ from QB in place of B . We set, for a � 0,

OFa D �fBs; s � 0I QBs^.1�Ta/C ; s � 0g:

Then, the above family of �-algebras is decreasing with respect to a. We have,if a � b,

QX.1�Tb/C.0/ D EŒ QX.1�Ta/C.0/ j OFb�:

The desired result follows from (4) and Jensen’s inequality. utRemark 2. Proposition 3 says, in the terminology of Hirsch et al. [3], that.X.a/I a � 0/ is an inverse peacock. By the general theorem of Kellerer (see [2, 5]and also [3, Exercise 1.6]), there exists an inverse martingale which is associatedto .X.a/I a � 0/, which means that both processes have the same 1-marginals (wealso say that .X.a/I a � 0/ is a 1-inverse martingale). In the previous proof, weshowed that we may take as associated inverse martingale: . QX.1�Ta/C.0/; a � 0/.

In the sequel of this section, we shall give another proof, more analytic, ofProposition 3, from a computation of the law of X.a/.

Proposition 4. The law of X.a/ is

r2

�

��Z a

0

e�u2=2 du

�ı0 C ha.z/ dz

�;

where ı0 denotes the Dirac measure at 0, and

ha.z/ D 4

3exp

�� .a � 2z/2

2

�1.�1;0/.z/ C 1

3exp

�� .a C z/2

2

�1.0;C1/.z/:

Proof. Denote by �a the law of Ta. One has

�a.dy/ D ap2�

1.0;C1/.y/ y�3=2 exp

�� a2

2y

�dy: (5)

By (4), we have for every nonnegative ',

EŒ'.X.a//� D '.0/

Z C1

1

�a.dy/ CZ 1

0

EŒ'.X1�y.0//� �a.dy/: (6)


By (5) and the change of variable y D a2 u�2, we obtain

Z C1

1

�a.dy/ Dr

2

�

Z a

0

e�u2=2 du: (7)

By Tanaka’s formula, Xt .0/ D BCt � 1

2Lt where Lt denotes the local time at 0

of the Brownian B . According to [9, Exercise 2.18], the joint law of .Bt ; Lt / has adensity given by

1p2�t3

�.a C b/ exp

�� .a C b/2

2t

��for a; b � 0 and

1p2�t3

�.�a C b/ exp

�� .�a C b/2

2t

��for a � 0; b � 0:

Consequently, the density of the law of Xt.0/ is

r2

� t

�4

3exp

��2u2

t

�1.u<0/ C 1

3exp

��u2

2t

�1.u>0/

�: (8)

Thus, by (5) and (8),

Z 1

0

EŒ'.X1�y.0//� �a.dy/ D a

3�

Z 1

0

Ia.z/ Œ2'.�z=2/ C '.z/� dz (9)

with

Ia.z/ DZ 1

0

y�3=2.1 � y/�1=2 exp

�� a2

2y

�exp

�� z2

2.1 � y/

�dy:

The change of variable: y D a2u.1 C a2u/�1 yields

Ia.z/ D a�1 exp

�� .a2 C z2/

2

�Z 1

0

u�3=2 exp

�� 1

2u

�exp

��a2z2u

2

�du

D p2� a�1 exp

�� .a2 C z2/

2

�E

�exp

�� .az/2T1

2

��

and hence

Ia.z/ D p2� a�1 exp

�� .a C z/2

2

�: (10)

Finally, gathering (6), (7), (9), and (10), we obtain the announced result, after anobvious change of variable. utThe next corollary follows easily from Proposition 4.


Corollary 1. Let ' be a suitably integrable function. Then, for any a � 0,

d

daEŒ'.X.a//�

Dr

2

�

Z 1

0

.a C z/ exp

�� .a C z/2

2

��'.0/ � 2

3'.�z=2/ � 1

3'.z/

�dz:

In particular, if moreover ' is convex, then

d

daEŒ'.X.a//� � 0

and the inequality is strict if and only if ' is not an affine function.

Clearly, the above corollary entails Proposition 3.By Remark 2, .X.a/I a � 0/ is a 1-inverse martingale. One may wonder whether

it also is a 2-inverse martingale, that is, whether it has the same 2-marginals as aninverse martingale. We answer this question in the next proposition.

Proposition 5. For every a > 0, EŒX.0/ X.a/2� > EŒX.a/3�.Consequently, .X.a/I a � 0/ is not a 2-inverse martingale.

Proof. Set:

E.a/ D EŒX.0/ X.a/2� � EŒX.a/3�:

By Itô’s formula:

E.a/ DZ 1

0

E

��Z t

0

1.0<Bs<a/ dBs

�1.Bt >a/

�dt:

We set, for t > 0,

U.t/ D E

��Z t

0

1.0<Bs<1/ dBs

�1.Bt >1/

�(11)

By scaling,

E

��Z t

0

1.0<Bs<a/ dBs

�1.Bt >a/

�D a U.a�2t/:

Hence,

E.a/ D a

Z 1

0

U.a�2t/ dt D a3

Z a�2

0

U.t/ dt: (12)

The result will then follow from the next lemma.


Lemma 1. For every t > 0,

U.t/ D 1

2p

2�

Z 1

t�1

e�u=2 u�1=2 du;

where U.t/ is defined in (11).

Proof. By Tanaka’s formula,

U.t/ D E

��1 C 1

2L1

t � 1

2L0

t

�1.Bt >1/

�;

where L1t (resp. L0

t ) denotes the local time of the Brownian motion at time t in 1

(resp. 0). Using the classical property

Lxt D lim

"!0

1

"

Z t

0

1.x<Bs<xC"/ ds

we obtain

EŒL1t 1.Bt >1/� D 1

2p

2�

Z t

0

e�1=2s s�1=2 ds; (13)

EŒL0t 1.Bt >1/� D 1

2�

Z t

0

e�1=2s .t � s/1=2 s�3=2 ds: (14)

By (14),

d

dtEŒL0

t 1.Bt >1/� D 1

4�

Z t

0

e�1=2s .t � s/�1=2 s�3=2 ds

and the change of variable s D t .2vt C 1/�1 yields

d

dtEŒL0

t 1.Bt >1/� D 1

2p

2�e�1=2t t�1=2;

which is equal tod

dtEŒL1

t 1.Bt >1/� by (13). Thus, EŒL0t 1.Bt >1/� D EŒL1

t 1.Bt >1/�, and

consequently U.t/ D PŒBt > 1�, which is the announced result. utThe proposition follows then from the above lemma and (12). utRemark 3. By Lemma 1, limt!1 U.t/ D 1

2and U.t/ is equivalent to 1p

2�e�1=2t t1=2

when t tends to 0. Therefore, by (12), E.a/ is equivalent to a2

when a tends to 0 and

is equivalent toq

2�

e�a2=2a�2 when a tends to 1.


5 An Increasing Family

We set, for t � 0 and a � 0,

Yt.a/ DZ t

0

1.Bs<a/ dBs

and we denote Y1.a/ simply by Y.a/.

Proposition 6. The law of Y.a/ isr

2

�

�Aa.z/ 1.�1;a/.z/ C Da.z/ 1.a;C1/.z/

�dz

with

Aa.z/ D�

1

2exp

�� z2

2

�� 1

6exp

�� .2a � z/2

2

��

and Da.z/ D 4

3exp

�� .2z � a/2

2

�:

Proof. We shall follow the same lines as in the proof of Proposition 4, of which wekeep the notation.We set, for t � 0,

Y.a/

t DZ t

0

1.B

.a/s <0/

dB.a/s :

We clearly have

Y.a/ D 1.Ta>1/ B1 C 1.Ta<1/

�a C Y

.a/

.1�Ta/

(15)

and

1.Ta<1/ B1 D 1.Ta<1/

�a C B

.a/

.1�Ta/

: (16)

Since Y .a/ and B.a/ are independent of Ta, by (15) and (16), we have for everynonnegative ',

EŒ'.Y.a//� D EŒ'.B1/� CZ 1

0

�EŒ'.a C Y1�y.0//� � EŒ'.a C B1�y/�

��a.dy/:

(17)

Clearly, Yt .0/.law/D �Xt.0/. Consequently, by (8), the density of the law of Yt .0/ is

r2

� t

�1

3exp

��u2

2t

�1.u<0/ C 4

3exp

��2u2

t

�1.u>0/

�: (18)


By (18) and (5),

Z 1

0

EŒ'.aCY1�y .0//� �a.dy/ D a

3�

Z 1

0

Ia.z/ Œ2'.aCz=2/C'.a�z/� dz; (19)

where Ia.z/ is given in (10). Likewise,

Z 1

0

EŒ'.a C B1�y/� �a.dy/ D a

2�

Z 1

0

Ia.z/ Œ'.a C z/ C '.a � z/� dz: (20)

Finally, gathering (17), (19), (20), and (10), we obtain the announced result, afterobvious changes of variable. utThe next corollary follows easily from Proposition 6.

Corollary 2. Let ' be a suitably integrable function. Then, for any a � 0,

d

daEŒ'.Y.a//�

Dr

2

�

Z 1

0

.aCz/ exp

�� .aCz/2

2

��2

3'.aCz=2/C1

3'.a�z/�'.a/

�dz:

In particular, if moreover ' is convex, then

d

daEŒ'.Y.a//� � 0

and the inequality is strict if and only if ' is not an affine function.

Clearly, the above corollary entails the following proposition.

Proposition 7. The map: a � 0 �! Y.a/ is increasing in the convex order.

Proposition 7 says, in the terminology of Hirsch et al. [3], that .Y.a/I a � 0/

is a peacock. By the general aforementioned theorem of Kellerer, there exists amartingale which is associated to .Y.a/I a � 0/ (in the sense that both processeshave the same 1-marginals). We shall exhibit such a martingale, using the stochasticdifferential equation method (see [3, Chap. 6] and [2]).

We first introduce some further notation. We set, for .a; x/ 2 RC � R,

p.a; x/ D 1p2�

�exp

��x2

2

�� 1

3exp

�� .2a � x/2

2

��if x < a

and p.a; x/ D 1p2�

8

3exp

�� .2x � a/2

2

�if x � a:


Thus, by Proposition 6, the law of Y.a/ is p.a; x/ dx. We then define the callfunction C by

8.a; x/ 2 RC � R; C.a; x/ D EŒ.Y.a/ � x/C�:

We also set, for x 2 R,

N.x/ D 1p2�

Z 1

x

exp

��u2

2

�du:

Lemma 2.

1. For every .a; x/ 2 RC � R,

p.a; x/ � 2

3p

2�exp

�� Œ.2x � a/ _ .2a � x/�2

2

�and

p.a; x/ � 8

3p

2�exp

�� Œ.2x � a/ _ x�2

2

�:

2. For .a; x/ 2 RC � R,

@

@aC.a; x/ D 2

3N..2x � a/ _ .2a � x//:

3. For .a; x/ 2 RC � R,

@

@aC.a; x/ �

p2�

2p.a; x/:

Proof. The first point follows from the above definition of p.The second point is a direct consequence of Corollary 2.For the third point, we remark that .2x � a/ _ .2a � x/ � a � 0 and

supfexp.u2=2/N.u/I u � 0g D N.0/ D 1=2: utWe now set, for .a; x/ 2 RC � R,

�.a; x/ D

2

@@a

C.a; x/

p.a; x/

!1=2

:

This definition of � comes from Dupire [1].


Proposition 8. The stochastic differential equation:

Mt D M0 CZ t

0

�.s; Ms/ dBs; M0.law/D Y.0/ (21)

admits a weak solution which is unique in law. Such a solution is a continuous,strong Markov martingale, which is associated to the peacock .Y.a/I a � 0/.

Proof. We shall first prove the existence of a weak solution to (21). We remarkthat, by Lemma 2, one has 0 < � � .2�/1=4. However, � is not continuous onRC � R, but only continuous on the complement of f.a; a/I a � 0g. So, we need toapproximate � . We set, for " > 0 and .t; x/ 2 RC � R,

p".t; x/ DZ 1

0

p.t; x C " u/ du:

Thus, p".t; x/ dx is the law of Y "t WD Y.t/ � " U , where U denotes a uniform

variable on Œ0; 1�, independent of Y.t/. Clearly, p" is continuous and > 0 on RC�R.We set:

C".t; x/ D EŒ.Y "t � x/C� D

Z 1

0

C.t; x C " u/ du:

Consequently, by Lemma 2, for .t; x/ 2 RC � R,

0 <@

@tC".t; x/ D

Z 1

0

@

@tC.t; x C " u/ du �

p2�

2p".t; x/:

We then set, for .t; x/ 2 RC � R,

�".t; x/ D

2

@@t

C".t; x/

p".t; x/

!1=2

:

Thus, �" is continuous and, for every .t; x/ 2 RC � R,

0 < �".t; x/ � .2�/1=4: (22)

Therefore, the stochastic differential equation

Mt D M0 CZ t

0

�".s; Ms/ dBs; M0.law/D Y "

0 (23)

admits a weak solution, and, using M. Pierre’s uniqueness theorem ([3, Theo-rem 6.1]), one sees as in the proof of Theorem 6.2 in [3] that such a solution M " isunique in law and M " is a continuous martingale, which is associated to the peacock.Y "

t I t � 0/. Besides, by (22) and BDG inequalities, for every � > 0, there existsc� > 0 such that


8" > 0; 8s; t � 0; EŒjM "t � M "

s j� � � c� jt � sj�=2: (24)

We denote by P" the law of M " on C.RCIR/. We deduce from (24) and

Kolmogorov’s criterion (see, e.g., [9, Theorem 1.8, Chap. XIII]) that the family oflaws fP"I " > 0g is weakly relatively compact. Therefore, there exists a sequence."n/ tending to 0 and a probability P on C.RCIR/ such that P"n weakly convergesto P when n tends to infinity. We denote by M a continuous process with law P.Obviously, the law of Mt is p.t; x/ dx. To show that M is a weak solution to (21),we shall prove that P is a solution to the corresponding martingale problem. Letf be a C 2-function with compact support and let 0 � s � t and g be a boundedcontinuous function on C.Œ0; s�IR/. By (23), for every n,

E"n

��f .yt / � f .ys/ � 1

2

Z t

s

�2"n

.u; yu/ f 00.yu/ du

�g.yjŒ0;s�/

�D 0:

We set

Rn D E"n

�Z t

s

j�2"n

.u; yu/ � �2.u; yu/j du

�:

Then,

Rn DZR

dx

Z 1

0

dv

Z t

s

du j�2"n

.u; x/ � �2.u; x/j p.u; x C "nv/:

Since, for every u � 0, x �! p.u; x/ is right-continuous, we obtain by dominatedconvergence (see point 1 in Lemma 2): limn!1 Rn D 0. We define the boundedfunction H on C.RCIR/ by

H.y/ D�

f .yt / � f .ys/ � 1

2

Z t

s

�2.u; yu/ f 00.yu/ du

�g.yjŒ0;s�/:

We obtain by what precedes: limn!1 E"n ŒH � D 0. On the other hand, H is

continuous at any y such that u 6D yu du-a.e. Now, since the law of yu under Padmits a density, namely, p.u; x/, one has

Z 1

0

Z1fuDyug P.dy/ du D 0:

Therefore, P-a.s., u 6D yu du-a.e. Thus, H is continuous at every point of the com-plement of a P-negligible set. By a classical result, this entails limn!1 E

"n ŒH � DEŒH �, and therefore, EŒH � D 0 which means

E

��f .yt / � f .ys/ � 1

2

Z t

s

�2.u; yu/ f 00.yu/ du

�g.yjŒ0;s�/

�D 0:


So, P is a solution to the martingale problem corresponding to .Y.0/; �2/, which isequivalent to say that M is a weak solution of (21).

Now, by Lemma 2, � satisfies the conditions allowing to apply M. Pierre’suniqueness theorem [3, Theorem 6.1] with a D 1

2�2. Consequently, we may show,

as in the proof of Theorem 6.2 in [3], the uniqueness in law of the weak solutionof (21), and the strong Markov property follows. utRemark 4. By the results of Lowther [6], the martingale M , weak solutionof (21), is the only (in law) continuous, strong Markov martingale associated to.Y.a/I a � 0/. It also follows from Lowther [6, 7] (see also [4, Theorem 4.4])that there exists a continuous inverse martingale associated to .X.a/I a � 0/.This inverse martingale is therefore different, in law, of the one proposed inRemark 2, which is not continuous. Note that the above method does not applyto .X.a/I a � 0/, since, as seen in Proposition 4, the law of X.a/ is not absolutelycontinuous.

Here again, one may wonder whether .Y.a/I a � 0/ is a 2-martingale, that is,whether it has the same 2-marginals as a martingale. We answer this question in thenext proposition.

Proposition 9. For every a > 0, EŒY.0/2 Y.a/� < EŒY.0/3�.Consequently, .Y.a/I a � 0/ is not a 2-martingale.

Proof. We follow the same lines as in the proof of Proposition 5. Set:

F.a/ D EŒY.0/2 Y.a/� � EŒY.0/3�:

By Itô’s formula:

F.a/ DZ 1

0

E

��Z t

0

1.0<Bs<a/ dBs

�1.Bt <0/

�dt:

We set, for t > 0,

V.t/ D E

��Z t

0

1.0<Bs<1/ dBs

�1.Bt <0/

�(25)

By scaling,

E

��Z t

0

1.0<Bs<a/ dBs

�1.Bt <0/

�D a V.a�2t/:

Hence,

F.a/ D a

Z 1

0

V .a�2t/ dt D a3

Z a�2

0

V .t/ dt: (26)

The result will then follow from the next lemma.


Lemma 3. For every t > 0,

V.t/ D �1

4p

2�

Z 1

t�1

.1 � e�2u/ u�3=2 du;

where V.t/ is defined in (25).

Proof. By Tanaka’s formula,

V.t/ D 1

2E�

L1t � L0

t

�1.Bt <0/

�;

where L1t (resp. L0

t ) still denotes the local time of the Brownian motion at time t in1 (resp. 0). Using again

Lxt D lim

"!0

1

"

Z t

0

1.x<Bs<xC"/ ds

we obtain

EŒL0t 1.Bt <0/� D t1=2

p2�

EŒL1t 1.Bt <0/� D 1

4�

Z t

0

e�1=2s s�1=2

Z t�s

0

e�1=2u u�3=2 du ds: (27)

By (27),

d

dtEŒL1

t 1.Bt <0/� D 1

4�

Z t

0

exp

�� t

2

1

s .t � s/

�.t � s/�3=2 s�1=2 ds

and the change of variable s D t .v C 1/�1 yields

d

dtEŒL1

t 1.Bt <0/� D 1

2p

2�e�2=t t�1=2:

Thus, we obtain

V.t/ D �1

4p

2�

Z t

0

.1 � e�2=s/ s�1=2 ds

which yields the desired result. utThe proposition follows then from the above lemma and (26). ut


Remark 5. By Lemma 3, limt!1 V.t/ D � 12

and V.t/ is equivalent to � 1

2p

2�t1=2

when t tends to 0. Therefore, by (26), F.a/ is equivalent to � a2

when a tends to 0,and lima!1 F.a/ D � 1

3p

2�.

6 An Example of Non-monotony

The following example (which is inspired from Pagès [8, Example 2.4.3]) showsthat there exist functions f and g, satisfying

8s � 0; 8x 2 R; 0 � f .s; x/ � g.s; x/ � 1

and such thatR

f .s; Bs/dBs andR

g.s; Bs/dBs are not comparable in the convexorder. In particular, Proposition 2 does not hold in general, if neither H nor K isdeterministic.

Proposition 10. Set, for a � 0,

Z.a/ D a B1 CZ 2

1

1.Bs<0/ dBs:

Then, there exists a0 > 0 such that a �! EŒexp.Z.a//� is strictly decreasingon Œ0; a0�. In particular, if 0 � a1 < a2 � a0, then Z.a1/ and Z.a2/ are notcomparable in the convex order.

Proof. We begin with the following lemma.

Lemma 4. We set, for x 2 R,

E.x/ D E

�exp

�Z 1

0

1.Bs<x/ dBs

��:

Then, for every x 2 R, E0.x/ > 0, and hence E is strictly increasing on R.

Proof. For x � 0, E.x/ D EŒexp.Y.x//�, and, for x � 0, E.x/ D EŒexp.�X.�x//�.The result follows then from Corollaries 1 and 2. ut

We have:

Z.a/ D a B1 CZ 1

0

1. QBs<�B1/ d QBs

with QBs D B1Cs � B1. Since . QBs/0�s�1 is independent of B1, one obtains:

EŒexp.Z.a//� D EŒexp.aB1/E.�B1/�:


Hence,

d

daEŒexp.Z.a//�

ˇ̌ˇ̌aD0

D EŒB1 E.�B1/�:

Since x �! E.�x/ is strictly decreasing by Lemma 4 and x �! x is strictlyincreasing,

d

daEŒexp.Z.a//�

ˇ̌ˇ̌aD0

< EŒB1�EŒE.�B1/� D 0:

This entails the desired result. ut

References

1. Dupire, B.: Pricing with a smile. Risk Mag. 7, 18–20 (1994)2. Hirsch, F., Roynette, B.: A new proof of Kellerer’s theorem. ESAIM: PS 16, 48–60 (2012)3. Hirsch, F., Profeta, C., Roynette, B., Yor, M.: Peacocks and Associated Martingales, with

Explicit Constructions. Bocconi and Springer Series, vol. 3. Springer, New York (2011)4. Hirsch, F., Roynette, B., Yor, M.: Kellerer’s theorem revisited, Prépublication Université dÉvry,

vol. 361 (2012)5. Kellerer, H.G.: Markov-Komposition und eine Anwendung auf Martingale. Math. Ann. 198,

99–122 (1972)6. Lowther, G.: Fitting martingales to given marginals. http://arxiv.org/abs/0808.2319v1 (2008)7. Lowther, G.: Limits of one-dimensional diffusions. Ann. Probab. 37–1, 78–106 (2009)8. Pagès, G.: Convex order for path-dependent American options using the Euler scheme of

martingale jump diffusion process. http://hal.archives-ouvertes.fr/hal-00767885 (2012)9. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, New York

(1999)

http://arxiv.org/abs/0808.2319v1

http://hal.archives-ouvertes.fr/hal-00767885

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