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3 Applications to (Brownian) Functional Peacocks and Option Pricing

3 Applications to (Brownian) Functional Peacocks and Option Pricing

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Convex Order for Path-Dependent Derivatives: A Dynamic Programming Approach

. /



47



. /



also [31]). Then, .St /t2Œ0;T is a true .FtW /t2Œ0;T -martingale satisfying

. /



St



Z

D s0 exp



t

0



.s; Ss. / /dWs.



/



Z



1

2



t



2



0



Á

.s; Ss. //ds



. /



so that St > 0 for every t 2 Œ0; T. One introduces likewise the local volatility

. /

model .St /t2Œ0;T related to the bounded volatility function  W Œ0; T RC !

R, still starting from s0 > 0. Then, the following proposition holds which

appears as a functional

non-parametric extension of the peacock property shared

Á

2t

RT

(see e.g. [6, 13]).

by 0 e Bt 2 dt

0



Proposition 3 (Functional Peacocks) Let and  be two real valued bounded

continuous functions defined on Œ0; T R. Assume that S. / is the unique weak

solution to (7) as well as S. / for its counterpart involving Â. If one of the following

additional conditions holds:

.i/ Convex Partitioning function: there exists a function Ä W Œ0; T

such that, for every t 2 Œ0; T,



RC ! R C



x 7! x Ä.t; x/ is convex on RC and 0 Ä .t; :/ Ä Ä.t; :/ Ä Â.t; :/ on RC ,

or

.ii/ Convex Domination property: for every t 2 Œ0; T the function x 7! x Â.t; x/ is

convex on RC and

j .t; :/j Ä Â.t; :/;

then, for every convex (hence continuous) function f W R ! R with polynomial

growth

ÂZ

Ef



T

0



Ss. /



Ã



ÂZ



.ds/ Ä E f



T

0



Ss. /



Ã

.ds/



where is a signed (finite) measure on .Œ0; T; Bor.Œ0; T//. More generally,

for every convex functional F W C .Œ0; T; RC / ! R with .r; k : ksup /-polynomial

growth,

E F S.



/



Ä E F S. / :



(8)



Proof We focus on the setting .i/. The second one can be treated likewise. First

note that Ä is bounded since  is. As a consequence, the function x 7! x Ä.t; x/ is

zero at x D 0 and can be extended into a convex function on the whole real line

by setting x Ä.t; x/ D 0 if x Ä 0. One extends x .t; x/ and x Â.t; x/ by zero on R

likewise. Then, this claim appears as a straightforward consequence of Theorem 1



48



G. Pagès



applied to the diffusion whose coefficients are given by the extension of x .t; x/ and

x Â.t; x/ on the whole real line. As above, the sup-norm continuity follows from the

convexity and polynomial growth. In the end, we take advantage of the a posteriori

positivity of S. / and S. / when starting from s0 > 0 to conclude.

Applications to Volatility Comparison Results The corollary below shows that

comparison results for vanilla European options established in [9] appear as

consequences of Proposition 3.

Corollary 3 Assume



2 C .Œ0; T







min .t/



R; RC /,



Ä .t; :/ Ä



min ;



max 2



max .t/;



C .Œ0; T; R/ satisfy



t 2 Œ0; T;



then, for every convex functional F W C .Œ0; T; RC / ! R with .r; k : ksup /polynomial growth (r 1),

E F Ss.



min /



Ä E F Ss.



/



Ä E F Ss.



max /



:



(9)



Proof We successively apply the former Proposition 3 to the couple . min ; / and

the partitioning function Ä.t; x/ D min .t/ to get the left inequality and to the couple

. ; max / with Ä D max to get the right inequality.

Note that the left and right hand side of the above inequality are usually

considered as quasi-closed forms since they correspond to a Hull-White model (or

even to the regular Black-Scholes model if min , max are constant). Moreover, let

us emphasize that no convexity assumption on is requested.



2.4 Counter-Example (Discrete Time Setting)

The above comparison results for the convex order may fail when the assumptions

of Theorem 1 are not satisfied by the diffusion coefficient. In fact, for simplicity, the

counter-example below is developed in a discrete time framework corresponding to

;x

Proposition 1. We consider the 2-period dynamics X D X ;x D .X0W2

/ satisfying

X1 D x C Z1



and



X2 D X1 C



p

2v.X1 /Z2



L



where Z1W2

N .0I I2 /,

0, and v W R ! RC is a bounded C 2 -function such

that v has a strict local maximum at x0 satisfying v 0 .x0 / D 0 and v 00 .x0 / < 1. So is

2

the case if v.x/ D v.x0 / .x x0 /p

Co..x x0 /2 /, 0 < < 12 , in the neighbourhood

of x0 . Of course, this implies that v cannot be convex.

Let f .x/ D ex . It is clear that

'.x; / WD Ef .X2 / D ex E e



Z1 Cv.xC Z1 /



:



Convex Order for Path-Dependent Derivatives: A Dynamic Programming Approach



49



Elementary computations show that

Á

' 0 .x; / D ex E e Z1 Cv.xC Z1 / 1 C v 0 .x C Z1 / Z1

h

Á

2

' 002 .x; / D ex E e Z1 Cv.xC Z1 / 1 C v 0 .x C Z1 / Z12

Ái

CE e Z1 Cv.xC Z1 / v 00 .x C Z1 /Z12 :

In particular

Á

' 0 .x; 0/ D exCv.x/ .1 C v 0 .x//E Z1 D 0 and ' 002 .x; 0/ D exCv.x/ .1 C v 0 .x//2 C v 00 .x/



so that ' 002 .x0 ; 0/ < 0 which implies that there exists a small enough

that ' 0 .x0 ; / < 0 on .0; 0  so that

7 ! '.x0 ; / is decreasing on .0;



0



> 0 such



0 :



This clearly exhibits a counter-example to Proposition 1 when the convexity

assumption is fulfilled neither by the functions . k /kD0Wn nor the functions .Äk /kD0Wn

(here with n D 1).



2.5 Lévy Driven Diffusions

Let Z D .Zt /t2Œ0;T be a Lévy process with Lévy measure

satisfying

Z

p

1

jzj .dz/ < C1, p 2 Œ1; C1/. Then Zt 2 L .P/ for every t 2 Œ0; T.

jzj 1



Assume furthermore that E Z1 D 0 so that .Zt /t2Œ0;T is an F Z -martingale.

Theorem 2 Let Z D .Zt /t2Œ0;T be a martingale Lévy process with Lévy measure

satisfying .jzjp / < C1 for a p 2 .1; C1/ if Z has no Brownian component and

.z2 / < C1 if Z has a Brownian component. Let Äi W Œ0; T R ! R, i D 1; 2, be

continuous functions with linear growth in x uniformly in t 2 Œ0; T. For i D 1; 2, let

.Ä /

X .Äi/ D .Xt i /t2Œ0;T be the weak solution, assumed to be unique, to

.Äi /



dXt



.Ä /



.Ä /



D Äi .t; Xt i /dZt i ;



.Äi /



X0



D x 2 R;



(10)



where Z .Äi / , i D 1; 2 have the same distribution as Z. Let F W D.Œ0; T; R/ ! R

be a Borel convex functional, PX .Äi / -a:s: Sk-continuous, i D 1; 2, with .r; k:ksup /polynomial growth for some r 2 Œ1; p/ i.e.

8 ˛ 2 D.Œ0; T; R/;



jF.˛/j Ä C.1 C k˛krsup /:



50



G. Pagès



.a/ Convex Partitioning function: If there exists a function Ä W Œ0; T R ! RC

such that Ä.t; :/ is convex for every t 2 Œ0; T and 0 Ä Ä1 Ä Ä Ä Ä2 , then

E F.X .Ä1 / / Ä E F.X .Ä2/ /:

.a0 / An equivalent form for claim .a/ is: if 0 Ä Ä1 Ä Ä2 and, either Ä1 .t; :/ is convex

for every t 2 Œ0; T, or Ä2 .t; :/ is convex for every t 2 Œ0; T, then the conclusion

of .a/ still holds true.

.b/ Convex Domination property: If Z has a symmetric distribution, jÄ1 j Ä Ä2 and

Ä2 is convex, then

E F.X .Ä1 / / Ä E F.X .Ä2/ /:

Remark 3

• The PX .Äi / -a:s: Sk-continuity of the functional F, i D 1; 2, is now requested:

Sk-continuity no longer follows from the convexity since D.Œ0; T; R/; Sk is a

Polish space but not a topological vector space. Thus, the convex function ˛ 7!

j˛.t0 /j for a fixed t0 2 .0; T/ is continuous at a given ˇ 2 D.Œ0; T; R/ if and only

if ˇ is continuous at t0 (see [4, Chap. 3]).

• The result remains true under the less stringent moment assumption on the Lévy

measure : .jzjp 1fjzj 1g / < C1 but would require much more technicalities

since one has to carry out the reasoning of the proof below between two large

jumps of Z and “paste” these inter-jump results.

The following lemma is the key that solves the approximation part of the proof

in this càdlàg setting.

Lemma 3 Let ˛ 2 D.Œ0; T; R/. The sequence of stepwise constant approximations

defined by

˛n .t/ D ˛.tn /; t 2 Œ0; T;

converges toward ˛ for the Skorokhod topology.

Proof See [17, Proposition VI.6.37, p. 387] (second edition).

Proof (Proof of Theorem 2)

Step 1.



Let .XN tn /t2Œ0;T be the genuine Euler scheme defined by

Z

XN tn D x C



.0;t



Ä.sn ; XN snn /dZs



where Ä D Ä1 or Ä2 . Owing to the linear growth of Ä, we derive (see

e.g. Proposition 12 in Appendix 2) that

sup jXt j

t2Œ0;T



p



C sup

n 1



sup jXN tn j



t2Œ0;T



p



< C1:



Convex Order for Path-Dependent Derivatives: A Dynamic Programming Approach



51



We know, e.g. from Proposition 11 in Appendix 2, that .XN n /n 1 functionally

weakly converges for the Skorokhod topology toward the unique weak solution X

of the SDE dXk D Ä.t; Xt /dZt , X0 D x. In turn, Lemma 3 implies that .XN tnn /t2Œ0;T

Sk-weakly converges toward X.

Step 2. Let F W D.Œ0; T; R/ ! R be a PX -Sk-continuous convex functional. For

every integer n

1, we still define the sequence of convex functionals Fn W

n 1

Á

X

nC1

Nn

n

! R by Fn .x0Wn / D F

xk 1Œtkn ;tkC1

R

/ C xn 1fTg so that Fn .Xtn /0Wn D

k



kD0



.XN tnn /t2Œ0;T



F

.

Now, for every n 1, the discrete time Euler schemes XN .Äi/;n , i D 1; 2, related to

the jump diffusions with diffusion coefficients Ä1 and Ä2 are of the form (1) and

jFn .x0Wn /j Ä C.1 C kx0Wn kr /, r 2 Œ1; p/.

.a/ Assume 0 Ä Ä1 Ä Ä2 . Then, taking advantage of the partitioning function Ä,

.Ä /;n

it follows from Proposition 1.a/ that, for every n 1, E Fn .XN tn 1 /0Wn Ä

k

.Ä /;n

.Ä /;n

.Ä /;n

E Fn .XN n 2 /0Wn i.e. E F .XN t 1 /t2Œ0;T Ä E F .XN t 2 /t2Œ0;T . Letting

tk



n



n



n ! C1 completes the proof like for Theorem 1 since F is PX -a:s: Skcontinuous.

.b/ is an easy consequence of Proposition 1.b/.



3 Convex Order for Non-Markovian Itô and Doléans

Martingales

The results of this section illustrate another aspects of our paradigm in order to

establish functional convex order for various classes of continuous time stochastic

processes. Here we deal with (couples of) Itô integrals with the restriction that one

of the two integrands needs to be deterministic.



3.1 Itô Martingales

Proposition 4 Let .Wt /t2Œ0;T be a standard Brownian motion on a filtered probability space .˝; A ; .Ft /t2Œ0;T ; P/ where .Ft /t2Œ0;T satisfies the usual conditions

and let .Ht /t2Œ0;T be an .Ft /-progressively measurable process defined on the same

probability space. Let h D .ht /t2Œ0;T 2 L2T . Let F W C .Œ0; T; R/ ! R be a convex

functional with .r; k:ksup /-polynomial growth, r 1.

.a/ If jHt j Ä ht P-a:s: for every t 2 Œ0; T, then

ÂZ

EF



Ã



:

0



Hs dWs



ÂZ

Ä EF



:

0



Ã

hs dWs :



52



G. Pagès



.b/ If Ht



0 P-a:s: for every t 2 Œ0; T and jHjL2 2 Lr .P/, then



ht



T



ÂZ

EF



Ã



:

0



ÂZ

EF



Hs dWs



:

0



Ã

hs dWs :



Remark 4

• In the “marginal” case where F is of the form F.˛/ D f .˛.T//, it has been shown

in [12] that the above assumptions on H and h in .a/ and .b/ are too stringent and

can be relaxed into

Z



T

0



E Ht2 dt Ä



Z



T

0



h2t dt



Z



T



and

0



E Ht2 dt



Z



T



0



h2t dt



respectively. The main ingredient of the proof is the Dambis-Dubins-Schwartz

representation theorem for one-dimensional Brownian martingales (see e.g. Theorem 1.6 in [31, p. 181]).

• The first step of the proof below is a variant of Proposition 1 in a non-Markov

framework. It can be considered as an autonomous proposition devoted to

discrete time dynamics.

Proof Step 1 (Discrete Time). Let .Zk /1ÄkÄn be an n-tuple of independent symmetric (hence centered) R-valued random variables satisfying Zk 2 Lr .˝; A ; P/,

r

1, and let F0Z D f;; ˝g, FkZ D

Z1 ; : : : ; Zk , k D 1; : : : ; n be its

natural filtration. Let .Hk /kD0;:::;n be an .FkZ /kD0;:::;n -adapted sequence such that

Hk 2 Lr .P/, k D 1; : : : ; n.

Let X D .Xk /kD0Wn and Y D .Yk /kD0Wn be two sequences of random variables

recursively defined by

XkC1 D Xk CHk ZkC1 ;



YkC1 D Yk Chk ZkC1 ;



0 Ä k Ä n 1;



X0 D Y0 D x0 :



These are the discrete time stochastic integrals of .Hk / and .hk / with respect to

the sequence of increments .Zk /kD1Wn . It is clear by induction that Xk , Yk 2 Lr .P/

for every k D 0; : : : ; n since Hk is FkZ -measurable and ZkC1 is independent of

FkZ .

Let ˚ W RnC1 ! R be a convex function with r-polynomial growth. Let us focus

on the first inequality, discrete time counterpart of claim .a/. We proceed like in

the proof Proposition 1 to prove by three backward inductions that if jHk j Ä hk ,

for every k D 0; : : : ; n, then

E ˚.X/ Ä E ˚.Y/:

To be more precise, let us introduce by analogy with this proposition the sequence

.«k /kD0;:::;n of functions recursively defined by

«n D ˚; «k .x0Wk / D .QkC1 «kC1 .x0Wk ; xk C ://.hk /; x0Wk 2 RkC1 ; k D 0; : : : ; n 1:



Convex Order for Path-Dependent Derivatives: A Dynamic Programming Approach



53



First note that the functions «k satisfy the following linear dynamic programming

principle:

«k .Y0Wk / D E «kC1 .Y0WkC1 / j FkZ ; k D 0; : : : ; n



1;



so that, by the chaining rule for conditional expectations, we have

˚k .Y0Wk / D E ˚.Y0Wn / j FkZ ; k D 0; : : : ; n:

Furthermore, owing to the properties of the operator QkC1 , we already proved

that for any convex function G W RkC2 ! R with r-polynomial growth, the

function

.x0Wk ; u/ 7! .QkC1 G.x0Wk ; xk C ://.u/ D E G.x0Wk ; xk C uZkC1 /

is convex and even as a function of u for every fixed x0Wk . As a consequence, it

also satisfies the maximum principle established in Lemma 1.c/ since the random

variable Zk have symmetric distributions.

Now, let us introduce the martingale induced by ˚.X0Wn /, namely

Mk D E ˚.X0Wn / j FkZ ; k D 0; : : : ; n:

We show now by a backward induction that Mk Ä «k .X0Wk / for every k D

0; : : : ; n. If k D n, this is trivial. Assume now that MkC1 Ä «kC1 .X0WkC1 / for

a k 2 f0; : : : ; n 1g. Then we get the following string of inequalities

Mk D E.MkC1 j FkZ / Ä E.«kC1 .X0WkC1 / j FkZ /

D E.«kC1 .X0Wk ; Xk C Hk ZkC1 / j FkZ /

Á

D E.«kC1 .x0Wk ; xk C uZkC1 / j FkZ

D QkC1 «kC1 .x0Wk ; xk C :/.Hk /

Ä QkC1 «kC1 .x0Wk ; xk C :/.hk /



Á



Á



jx0Ik DX0Wk ;uDHk



jx0Ik DX0Wk



jx0Ik DX0Wk



D «k .X0Wk /

(11)



where we used in the fourth line that ZkC1 is independent of FkZ and, in the

penultimate line, the assumption jHk j Ä hk and the maximum principle. Finally,

at k D 0, we get E ˚.X0Wn / D M0 Ä ˚0 .x0 / D E˚.Y0Wn / which is the announced

conclusion.

Step 2 (Approximation-Regularization). We temporarily assume that the function

h has a modification which is bounded by a real constant so that P.d!/-a:s:

kH.!/ksup _ khksup Ä K. We first need a technical lemma adapted from



54



G. Pagès



Lemma 2.4 in [20, p. 132] about approximation of progressively measurable

processes by simple processes, with in mind the preservation of the domination

property requested in our framework. The details of the proof of this lemma are

left to the reader.

Lemma 4

.a/ For every " 2 .0; T/ and every g 2 L2 .Œ0; T; dt/ we define

Z



1

"



" g.t/ Á t 7 !



t

.t "/C



g.s/ds 2 C .Œ0; T; R/:



The operator " W L2T ! C .Œ0; T; R/ is non-negative. In particular, if g; 2 L2T

with jgj Ä

.

1 -a:e:, then j " gj Ä

" and k " gksup Ä jgjL1

T

.b/ If g 2 C .Œ0; T; R/, define for every integer m 1, the stepwise constant càglàd

(for left continuous right limited) approximation gQ m of g by

gQ m .t/ D g.0/1f0g .t/ C



m

X



g tkm



1.tkm



1



m

1 ;tk 



:



kD1

k : ksup



Then gQ m ! g as m ! C1. Furthermore, if g;

then jQgm j Ä Q m for every m 1.



2 C .Œ0; T; R/ and jgj Ä ,



By the Lebesgue fundamental Theorem of Calculus we know that

ˇ

ˇ

Since j

that



1

n



H



1

n



ˇ

H ˇL2



H



! 0 P-a:s:



T



HjL2 Ä 2K, the Lebesgue dominated convergence Theorem implies

T



Z

E



T



j

0



1

n



Ht



Ht j2 dt ! 0 as n ! C1:



(12)



By construction, 1 H is an .Ft /-adapted pathwise continuous process satisfying

n

the domination property j 1 Hj Ä 1 h so that, in turn, using this time claim .b/ of

n

n

the above lemma, for every n; m 1,

j

On the other hand, for every n

implies

Z

0



T



ˇ

ˇ



AH

1

n



t



m



1

n



AH

1

n



t



m







Ah :

1

n



t



m



1, the a:s: uniform continuity of



ˇ2

ˇ

Ht ˇ dt Ä T sup ˇ

t2Œ0;T



AH

1

n



t



m



1

n



1

n



H over Œ0; T



ˇ2

Ht ˇ ! 0 as m ! C1 P-a:s:



Convex Order for Path-Dependent Derivatives: A Dynamic Programming Approach



55



One concludes again by the Lebesgue dominated convergence Theorem that, for

every n 1,

Z

E



ˇ

ˇ

ˇ



T

0



AH

1

n



t



m



ˇ2

ˇ

Ht ˇ dt ! 0 as m ! C1:



1

n



One shows likewise for the function h itself that

ˇ

ˇ



1

n



ˇ

hˇL2 ! 0 as n ! C1



h



T



1,



and, for every n



ˇ

ˇ



eh



m



1

n



1

n



ˇ

hˇL2 ! 0 as m ! C1:

T



Consequently, there exists an increasing subsequence m.n/ " C1 such that

Z

E



T

0



ˇ

ˇ

ˇ



AH

1

n



t



m.n/



1

n



Z

ˇ2

ˇ

Ht ˇ dt C



T

0



ˇ

ˇ

ˇ



Ah

1

n



t



m.n/



1

n



ˇ2

ˇ

ht ˇ dt ! 0



n ! C1



as



which in turn implies, combined with (12) and its deterministic counterpart for h,

Z

E



T

0



ˇ

ˇ

ˇ



AH

1

n



t



Z

ˇ2

ˇ

Ht ˇ dt C



m.n/



T

0



ˇ

ˇ

ˇ



Ah

1

n



t



.n/



D



AH

1

n



t



m.n/



n ! C1:



as



1,



At this stage, we set for every integer n

Ht



ˇ2

ˇ

ht ˇ dt ! 0



m.n/



.n/



and ht D



Ah

1

n



t



m.n/



(13)



which satisfy

H .n/ j2L2 C jh



EjH



T



h.n/ jL2



T



! 0 as n ! C1:



(14)



It should be noted that these processes H .n/ , H and these functions h.n/ , h are all

bounded by 2K.

We consider now the continuous modifications of the four (square integrable)

Brownian martingales associated to the integrands H .n/ , H, h.n/ and h, the last two

being of Wiener type. It is clear by Doob’s Inequality that

Z t

Z t

ˇZ t

ˇ

ˇZ t

ˇ L2 .P/

ˇ

ˇ

ˇ

ˇ

sup ˇ

Hs.n/ dWs

Hs dWs ˇC sup ˇ

h.n/

dW

h

dW

ˇ ! 0 as n ! C1:

s

s

s

s



t2Œ0;T



0



0



Z



t



In particular

0



Hs.n/ dWs



t2Œ0;T



0



0



Z



Á

t2Œ0;T



t



functionally weakly converges to

0



Hs dWs



Á

t2Œ0;T



for the k : ksup -norm topology. We also have, owing to the B.D.G. Inequality, that



56



G. Pagès



for every p 2 .0; C1/,

ˇZ t

ˇp

ˇ

ˇ

p

Hs.n/ dWs ˇ Ä cpp EjH .n/ jL2 Ä cpp K p

E sup ˇ

t2Œ0;T



0



(15)



T



where cp is the universal constant involved in the B.D.G. Inequality. The same holds

true for the three other integrals related to h.n/ , H, and h.

.n/

.n/

Let n 1. Set Hkn D H m.n/ ; hnk D h m.n/ ; k D 0; : : : ; m.n/ 1 and Zkn D Wtm.n/

tk

tk

k

Z tkm.n/

k

X

Hs.n/ dWs D

H`n 1 Z`n , k D

Wtm.n/ , k D 1; : : : ; m.n/. One easily checks that

k 1



0



`D1



0; : : : ; m.n/, so that

ÂZ

Im.n/



:

0



Hs.n/ dWs



Ã

D im.n/



k

X



H`n 1 Z`n



`D1



!



Á

kD0Wm.n/



:



Let Fm.n/ be defined by (5) from the convex functional F (with .r; k : ksup /polynomial growth). It is clearly convex. One derives from Step 1 applied with

horizon m.n/ and discrete time random sequences .Zkn /kD1Wm.n/ , .Hkn /kD0Wm.n/ 1 ,

.hk /kD0Wm.n/ 1 that

Z

E FıIm.n/



:

0



Hs.n/ dWs



Á



D E Fm.n/



k

X



H`n 1 Z`n



kD0Wm.n/



`D1



Ä E Fm.n/



k

X

`D1



Z



D E FıIm.n/



:



0



hn` 1 Z`n



!



Á



!



Á

kD0Wm.n/



Á

h.n/

dW

s :

s



Combining the above functional weak convergence, Lemma 2 and the uniform

integrability derived form (15) (with any p > r) yields the expected inequality by

letting n go to infinity.

Step 3 (Second Approximation). Let K 2 N and K W R ! R be the thresholding

function defined by K .u/ D .u ^ K/ _ . K/. It follows from the B.D.G.

Inequality that, for every p 2 .0; C1/,

ˇZ t

ˇ

Hs dWs

E sup ˇ

t2Œ0;T



0



Z



t

0



ˇp

ˇ

p

.H

/dW

s

s ˇ Ä cp EjH

K



p



K



D cpp Ej jHj

Ä cpp j jhj



.H/jL2



T



K

K



p

j

C L2T



p

j

C L2T



(16)

(17)



Convex Order for Path-Dependent Derivatives: A Dynamic Programming Approach



57



where uC D max.u; 0/, u 2 R. The same bound obviously holds when

replacing H by h. This shows that the convergence holds in every Lp .P/ space,

p 2 .0; C1/, as K ! C1. Hence, one gets the expected inequality by letting K

go to infinity in the inequality

ÂZ

EF



:

0



Ã

ÂZ

Ä

EF

.H

/dW

s

s

K



Ã



:

0



ÂZ

D EF



K .hs /dWs



:

0



Ã

hs ^ KdWs :

(18)



.b/ We consider the same steps as for the upper-bound established in .a/ with the

same notations.

Step 1. First, in a discrete time setting, we assume that 0 Ä hk Ä Hk 2 Lr .P/

and we aim at showing by a backward induction that Mk «k .X0Wk / where

Mk D E ˚.X0Wn / j FkZ .

If k D n, the inequality holds as an equality since «n D ˚. Now assume

MkC1 «kC1 .X0WkC1 /. Then, like in .a/, we have

Mk D E MkC1 j FkZ

E « .X0WkC1 / j FkZ

D E « .X0Wk ; Xk CHk ZkC1 / jFkZ D Qk «kC1 .x0Wk ; xk C : /.Hk /

Qk «kC1 .x0Wk ; xk C : /.hk /



Á

jx0Wk DX0Wk



Á

jx0Wk DX0Wk



D «k .X0Wk /:



Step 2. This step is devoted to the approximation in a bounded setting where 0 Ä

ht Ä Ht Ä K. It follows the lines of its counterpart in claim .a/, taking

advantage of the global boundedness by K.

Step 3. This last step is devoted to the approximation procedure in the general

setting. It differs from .a/ since there is no longer a deterministic upper2

bound provided

R : by the function h 2 LT . Then, the key is to show rthat

the process 0 K .HRs /dWs converges for the sup norm over Œ0; T in L .P/

:

toward the process 0 Hs dWs as K ! C1. In fact, it follows from (16)

applied with p D r that

E



ˇZ t

ˇ

sup ˇ

Hs dWs

t2Œ0;T



0



Z



t

0



ˇr

ˇ

K .Hs /dWs ˇ



!

Ä crr E j jHj



K



jr

C L2T



Á

:



As jHjL2 2 Lr .P/, one concludes by the Lebesgue convergence Theorem by

T

letting K ! C1.

Remark 5

• Step 1 can be extended to non-symmetric, centered independent random variables

.Zk /1ÄkÄn if the sequences .Hk /0ÄkÄn 1 and .hk /0ÄkÄn 1 under consideration

satisfy 0 Ä Hk Ä hk , k D 0; : : : ; n 1.



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