3 Applications to (Brownian) Functional Peacocks and Option Pricing
Tải bản đầy đủ - 0trang
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
.Â /
model .St /t2Œ0;T related to the bounded volatility function Â 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 Â 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.Â / 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.Â /
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.Â / :
(8)
Proof We focus on the setting .i/. The second one can be treated likewise. First
note that Ä is bounded since Â 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.Â / 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
0Ä
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
jÄ
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.