arXiv:math/0610178v1 [math.PR] 5 Oct 2006
T he A nnals of A pplied P robability
2006, V ol. 16, N o. 3, 1124{1154
D O I: 10.1214/105051606000000060
c Institute of M athem atical Statistics, 2006
A D U A LITY A PPR O A C H FO R TH E W EA K A PPR O X IM ATIO N O F
STO C H A STIC D IFFER EN TIA L EQ U ATIO N S
B y E m m anuelle C lem ent,A rturo K ohatsu-H iga1
and D am ien Lam berton
U niversite de M arne-la-Vallee,O saka U niversity
and U niversite de M arne-la-Vallee
In this article w e develop a new m ethodology to prove w eak approxim ation results for general stochastic di erentialequations.Instead of using a partial di erentialequation approach as is usually
done for di usions,the approach considered here uses the properties
ofthe linear equation satis ed by the error process.T his m ethodology seem s to apply to a large class ofprocesses and w e present as an
exam ple the w eak approxim ation ofstochastic delay equations.
1. Introduction. T he Euler schem e for stochastic di erentialequations
is w idely used in applications as it is easy to com pute.T he Euler schem e
can be easily generalized to a variety of stochastic equations beyond the
fram ework ofdi usion equations,in particular Volterra SD Es,delay SD Es,
anticipating SD Es and nonlinear SD Es.
O n the other hand, the theoretical properties of the Euler schem e are
m ostly studied for the di usion case as m ost ofthe results available so far
are in this fram ework.In som e cases,extensions to other sim ilar equations
arestraightforward butin othercases,additionalnontrivialwork isrequired.
For exam ple,see [8]for extensions to sem im artingales,and [1,11]for approxim ations of an irregular functionalof a di usion w hich is approached
using a Euler type schem e.It is also wellknow n that the de nition of an
extension ofthe Euler schem e for delay type system s is straightforward but
thetechnicalresultson theweak rateofconvergence arelim ited.See[4,6,9].
In this article we propose a generalization ofthe theory ofweak approxim ationsw hich studiestherateofconvergence oftheEulerschem econsidered
R eceived D ecem ber 2004;revised O ctober 2005.
Supported in part by G rants B FM 2003-03324 and B FM 2003-04294.
A M S 2000 subject classi cations.60H 07,60H 10,60H 35,65C 30.
K ey words and phrases.Stochastic di erential equation, w eak approxim ation, Euler
schem e,M alliavin calculus.
1
T his is an electronic reprint ofthe originalarticle published by the
Institute ofM athem aticalStatistics in T he A nnals ofA pplied Probability,
2006,Vol.16,N o.3,1124{1154.T his reprint di ers from the originalin
pagination and typographic detail.
1
2
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
in law .T his generalization nds as an application the weak rate ofconvergence ofsm ooth functionals ofgeneraldelay type system s and also covers,
w ith a furtherstudy ofthe M alliavin covariance m atrix,the case ofirregular
functions ofthe solution ofthe stochastic equation.
T he m ain idea is to change com pletely the approach used untilnow to
prove weak approxim ation rate results.T hisnew idea,w hich usesthe w hole
path ofthe processunderstudy ratherthan the partialdi erentialequation
associated to the problem ,should allow to obtain various other straightforward generalizations ofresults ofthe weak rate ofconvergence.
In order to describe our approach roughly,let X denote the solution ofa
stochastic equation and X the Euler schem e associated to it.T he problem
ofweak rate ofconvergence consists in nding the rate at w hich E (f(X )
f(X )) converges to zero for various classes of functionals f.T he optim al
rate is the step size of the schem e even though the equations considered
m ay di er.
T he classicalproofofthis result for di usions is based on the associated
partialdi erentialequation,that is,E f(X ) has through the Feynm an{K ac
form ula an interpretation using PD Es.T his is the im portant point in the
classical approach w hich is not used in our approach.In the case of som e
stochastic equations,if f is regular enough,the proof is sim ilar if the associated PD E exists. If f is an irregular function, then the issue of the
nondegeneracy ofthe M alliavin covariance m atrix ofthe Euler schem e becom es an im portantissue as has been show n in [1,11],butthis extension is
nontrivial.
In this article we propose a com pletely di erent m ethod to prove weak
approxim ation results based on a pathw ise appr
R oach. T hat is,we use the
m ean value theorem to rew rite f(X ) f(X )= 01 f0(aX + (1 a)X )da(X
X ).T hen,we derive a linear equation satis ed by Y = X
X .W hen this
equation can be explicitly solved,w hich seem sto be true only fordi usions,
one can obtain the rate of convergence by using the duality property of
stochastic integrals.T hism ethodology was rstintroduced by K ohatsu-H iga
and Pettersson [7]and used in G obet and M unos [5].It seem s to be quite
generalexceptfortheexplicitexpression forY w hich can bedoneonly in the
case ofdi usions.T hisarticle presentsa generalfram ework to analyze weak
approxim ations in stochastic equations.In particular we solve the problem
w ithout having an explicit expression for the solution of stochastic linear
equations,by using a duality argum ent.T his duality form ula (see Section
3) show s explicitly the weak error as a by-product ofthe expectation ofan
error process (called G in Section 3). To nish the proof one has to use
the duality form ula for stochastic integrals.T herefore our approach works
m ostly for stochastic equations w ith regular coe cients.For this reason we
have to study the stochastic derivatives ofthe solution process.Itshould be
3
W EA K A PPR O X IM AT IO N O F SD ES
em phasized that this approach applies to regular functionals ofthe process
X and not only to functions ofthe value ofX at a xed tim e t.
Furtherm ore, the fram ework introduced here also extends naturally to
the case ofirregular functions f.T hat is,one uses the integration by parts
form ula ofM alliavin calculus to regularize the function f.W e believe that
this approach for the irregular case follow s naturally from the regular case.
Italso explains clearly thatto obtain the resultforthe irregular case is just
a m atter of studying the nondegeneracy of the lim iting stochastic process
and not the approxim ating process.
In order to show w hat are the elem ents in each theorem we also consider
as an exam ple a delay type equation w hich does not have a clear associated
PD E in orderto extend theclassicalproof.T hegeneralfram ework isdirectly
applied,w hich gives the weak rate ofconvergence.
T his article is organized as follow s.In Section 2 we present the exam ple
ofa di usion process to introduce the m ethodology.Section 3 contains the
m ain results in the generalfram ework and an application to the weak rate
of convergence of approxim ations of delay equations for regular functions.
T hese results are then extended to irregular functions in Section 4.
2. The case of one-dim ensionaldi usions. To clarify the m ethodology,
weconsidera sm ooth function and a W ienerprocessW and a realdi usion
process
Z
t
(X s)dW s;
X t= x +
t2 [0;T ]:
0
Its Euler approxim ation is given by
Z
t
(X
X t= x +
(s))dW s;
t2 [0;T ];
0
w here (s)= kT=n forkT=n
satis es
Z
s< (k+ 1)T=n.T heerrorprocessY = X
X
t
( (X s)
Yt =
(X
(s)))dW s
0
Z tZ
1
0
=
(aX s + (1
a)X
(s))da(X s
X
(s))dW s;
0 0
this can be w ritten as
Z
(1)
t
1(s)Ys dW s +
Yt =
G t;
0
t T;
(X
(s))(W s
0
w ith
Z
1
0
(aX s + (1
1(s)=
a)X
(s))da;
0
Zt
Z
1(s)(X s
G t=
0
X
t
1(s)
(s))dW s =
0
W
(s))dW s:
4
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
In this sim ple case we have an explicit expression for Yt:
Z
t
Yt = Et
0
Es 1(dG s
1(s)dhG ;W
is);
w here E is the unique solution of
Z
t
1(s)Es dW s:
Et = 1+
0
Finally we obtain
Z
t
Yt = Et
0
Es 1
Z
t
Et
0
1(s)
Es 1
(X
(s))(W s
2
1(s)
(X
W
(s))(W s
(s))dW s
W
(s))ds:
N ow let f be a regular function.W e are interested in the determ ination of
the rate ofconvergence ofEf(X T ) to Ef(X T ).W e rst w rite the di erence
Z
1
f0(aX T + (1
Ef(X T ) Ef(X T )= E
a)X T )daYT :
0
R eplacing YT by itsexpression,we obtain w ith the additionalnotation F h =
R
1 0
0
f (aX T + (1
a)X T )da,
Z
EF h YT = EF h ET
T
Es 1 1(s) (X
0
(2)
Z
T
h
EF ET
0
Es 1
2
1(s)
(s))(W s
(X
W
(s))dW s
(s))(W s
W
(s))ds:
A pplying the duality for stochastic integrals,this gives
Z
EF h YT = E
T
0
D s(F h ET )Es 1
Z
EF h ET
T
0
Es 1
1(s)
2
1(s)
(X
(X
(s))(W s
(s))(W s
W
W
(s))ds
(s))ds;
w hereD denotesthestochasticderivative.C onsequently,thedi erenceEf(X T )
Ef(X T ) can be w ritten as
Z
Ef(X T ) Ef(X T )= E
T
0
U sh (W
s
W
(s))ds;
w ith
U sh = (D s(F h ET ) F h ET
1(s))(Es
1
1(s)
(X
(s))):
5
W EA K A PPR O X IM AT IO N O F SD ES
W e nally obtain the rate ofconvergence by applying once m ore the duality
for stochastic integrals
Z
T
Z
s
Ef(X T ) Ef(X T )= E
0
(s)
D u U sh du ds:
T hislastform ula m akesclearthatjEf(X T ) Ef(X T )j T=n and leadsto an
expansion ofEf(X T ) Ef(X T ) w ith som e additionalwork (see Section 3).
In otherstochastic equations,the errorprocessalso satis esa linearequation [sim ilar to (1)]but in a m ore generalform ;see Section 3.4.T he aim
of the next section is to establish a form ula (called the duality form ula)
w hich w illbe a substitute for (2) w hen the error process Y does not have
an explicit expression.
3. D uality for the error process and application to delay equations.
3.1. G eneralform ofthe error process equation. T hroughoutthe paper,
we consider a com plete probability space ( ;F ;P), w hich is the canonical space of a d-dim ensional B row nian m otion w ith nite horizon W =
f(W t1;:::;W td),0 t T g.W e denote by F = (F t)0 t T the usualaugm entation ofthe natural ltration ofW .IfH is a separable H ilbert space and
p 2 [1;+ 1 ),we denote by L pa([0;T ];H ) the space ofallm easurable adapted
R
processes X = (X t)0 t T , w ith values in H such that E 0T jX tjp dt< 1 ,
w here j jdenotes the norm on H .
R ecall that according to the It^
o representation theorem , any H -valued
random variable X such that EjX j2 < 1 can be w ritten in the follow ing
form :
Xd Z
T
X = E(X )+
i= 1 0
Jsi(X )dW si;
w here J 1(X ),...,J d(X )2 L 2a([0;T ];H ).N ote that this de nes J 1;:::;J d
as linear operators m apping L 2(F T ;H ) into L 2a([0;T ];H ). W e w ill often
use the notation J(X ) for the vector (J 1(X );:::;J d(X )) and Z dW for
P
d
i
i
i= 1 Z dW .
W e consider 1;:::; d, ,d+ 1 linearcontinuousoperatorson theH ilbert
space L 2a([0;T ];H ).T he aim ofSection 3.1 is to study the follow ing generalization of(1):
Xd Z
(3)
Z
t
Yt =
i(Y )(s)dW
i= 1 0
i
s+
t
(Y )(s)ds+ G t;
0
t T;
0
w here G 2 L 2a([0;T ];H ).
In fact,the study of (3) in the space L 2a w illnot be su cient (see Section 3.4) and we w illneed L p-estim ates for p large enough.For sim plicity
we state the assum ption as follow s.
6
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
A ssum ption A 1. For every p 2,there exists a positive constant C p
such that,for allt2 [0;T ],Y 2 L pa([0;T ];H ),
Z
t
! p=2
Xd
2
j i(Y )(s)j
(4) E
0
Z
Z
t
ds+ E
p
j (Y )(s)j ds
0
i= 1
t
C pE
jYsjp ds:
0
P roposition 1. Let A ssum ption A 1 hold and let p 2. G iven G 2
L pa([0;T ];H ),(3) has a unique solution Y 2 L pa([0;T ];H ) and we have
Z
Z
T
T
p
jYtj dt C pE
E
0
jG tjp dt;
0
for som e constant C p depending on C p (and T ) only.
k
P roof. Let k be a positive realnum ber.W e de ne an equivalent norm
k on the space L pa([0;T ];H ) by setting
k
Z
kY kpk
T
e
=E
kt
jY (t)jp dt:
0
Let A and B be the operators de ned by
Xd Z
t
A (Y )(t)=
(5)
i= 1 0
Z
(6)
i
i(Y )(s)dW s;
t
(Y )(s)ds:
B (Y )(t)=
0
W e have, using the B urkholder{D avis{G undy inequality (B D G inequality
in the sequel) and p 2,
Z
kA (Y )kpk
= E
T
dte
kt
Xd Z
p
t
i(Y )(s)dW
i= 1 0
0
Z
T
dte
K pE
0 i= 1
Z
T
K pE
2
j i(Y )(s)j ds
0
Z
! p=2
Z t Xd
kt
i
s
dte
kt p=2 1
0
Z
K pC pT p=2 1E
dte
kt
0
ds
i= 1
Z
T
2
j i(Y )(s)j
t
0
! p=2
Xd
t
t
dsjYsjp;
0
w here the constant K p com es from the B D G inequality and C p from A ssum ption A 1.U sing Fubini’s theorem ,we derive
(7)
kA (Y )kpk
K pC pT p=2
k
1
kY kpk :
7
W EA K A PPR O X IM AT IO N O F SD ES
A sim ilar argum ent leads to
Cp
kY kpk :
k
Itfollow sthat,fork largeenough,theoperatorA + B isa contraction forthe
norm k kk.H ence,ifI denotesthe identity,the operatorI (A + B )(acting
on L pa([0;T ];H )) is invertible and this im plies existence and uniqueness for
the solution of(3).
kB (Y )kpk
(8)
R em ark 2. N ote that ifthe process G has right-continuous (resp.continuous) paths,the solution of(3) has a right-continuous (resp.continuous)
m odi cation.T his m odi cation w illstillbe denoted by Y .
3.2. T he duality form ulae. T hepurposeofthissection isto establish two
duality form ulae relating EhF;YT i to G T for F in L 2( ).W e rstintroduce
som e notation.W e denote by h ; i the inner product on H .T he operators
A :L 2a([0;T ];H )! L 2a([0;T ];H ) and B :L 2a([0;T ];H )! L 2a([0;T ];H ) are
the adjoints of the operators A and B de ned by (5) and (6). If we set
(Y )= ( 1(Y );:::; d(Y )), we can view
as a linear operator m apping
the H ilbert space L 2a([0;T ];H ) into L 2a([0;T ];H d).T he adjoint operator
m aps L 2a([0;T ];H d) into L 2a([0;T ];H ).T he follow ing proposition relates the
operators A and B to
and .
P roposition 3.
T he operators A and B
Z
Xd
A (Z )(t)=
Ji
i
T
Z s ds
(t)
0
i= 1
Z
=
are given by
T
J
Z s ds
(t);
0
Z
B (Z )(t)=
T
Z s ds F
E
(t):
P roof. For allY ,Z 2 L 2a([0;T ];H ),we have
Z
E
Xd Z
T
0
hZ t;A (Y )(t)idt= E
t
Z t;
i= 1 0
Xd Z
i= 1 0
Z
T
0
0
Z
T
=
= E
Z
T
E
Z
s
dt
t
0
T
i
i(Y )(s)dW s
hJsi(Z t); i(Y )(s)idsdt
Xd
!
hJsi(Z t); i(Y )(s)idt
i= 1
ds:
8
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
N ote that,since Z t is F t-m easurable,we have Jsi(Z t)= 0 for t< s.H ence,
using the linearity ofthe operators J i,
Z
Z
T
Z
T
hJsi(Z t); i(Y )(s)idt
i= 1
hZ t;A (Y )(t)idt= E
E
0
0
Z
0
Z
Xd
T
Jsi
= E
0
Z
!
Xd
T
*
Z t dt ; i(Y )(s) ds
Z
Xd
= E
J
i
0
T
0
i= 1
T
ds
i
+
T
Z tdt
(s);Y (s) ds;
0
i= 1
w hich proves the form ula for A .W e proceed sim ilarly w ith B :
Z
Z
T
hZ t;B (Y )(t)idt= E
E
0
Z
T
t
Z t;
0
Z
T
(Y )(s)ds dt
0
Z
T
Z t dt; (Y )(s) ds
=E
0
Z
s
Z
T
=E
T
Z tdt F
E
(s);Y (s) ds:
0
T he nexttheorem statesthe two basic dual
R ity form ulae.In orderto relate
Eh ;Y T i to G T , we need a form ula for E 0T hF t;Ytidt w hen (F t)0 t T 2
L 2a([0;T ];H ).
T heorem 4.
1. IfF = (F t)0
Let G 2 L 2a([0;T ];H ) and let Y be the solution of(3).
t T
2 L 2a([0;T ];H ),we have
Z
Z
T
T
hF t;Ytidt= E
E
0
h t;G tidt;
0
with = (I A
B ) 1(F ).
2. If G has a continuous m odi cation, then Y has a continuous m odi cation (which we stilldenote by Y ),and if is an F T -m easurable square
integrable random variable with values in H ,we have
Z
Eh ;Y T i= Eh ;G
T i+
T
E
h^t;G tidt;
0
with
^ = (I
(I
A
B ) 1[ (J( ))+
(E( jF ))]:
P roof. T he rst part of the theorem com es from the equality Y =
A B ) 1(G ) and standard duality theory. For the second part,it is
9
W EA K A PPR O X IM AT IO N O F SD ES
clear from (3) that ift7
! G t is continuous,the process Y has a continuous
m odi cation.W e also have
Z
Eh ;Y T i= Eh ;G
T i+ E
Z
T
;
(Y )(s)
T
dWs +
0
(Y )(s)ds :
0
N ow
Z
Z
T
dWs = E
(Y )(s)
;
E
0
T
0
Xd
hJsi( ); i(Y )(s)ids
i= 1
and
Z
Z
T
;
E
T
(Y )(s)ds = E
0
hE( jF s); (Y )(s)ids:
0
T herefore
Z
Eh ;Y T i= Eh ;G
T
hF s;Ysids;
T i+ E
0
w ith F t =
part.
(J( ))(t)+
(E( jF ))(t).A nd the resultfollow sfrom the rst
R em ark 5. T he processes and ^ given by T heorem 4 can also be
characterized in connection w ith backward stochastic di erentialequations
(B SD Es).First note that satis es the dualequation
(9)
= F + A ( )+ B ( );
w hich,using Proposition 3,can be w ritten
Z
(10)
=F+
Z
T
J
s ds
+
T
s ds F
E
:
0
N ow ,observe that,given 2 L2a([0;T ];H ) and
processes (Y~ ;Z~),de ned by
Z
Y~t = E
Z
T
s ds F t
+
2 L 2(F T ;H ),the pair of
and Z~t = Jt
s ds
0
t
is the unique solution ofthe follow ing B SD E:
~
dY~t =
t dt+ Z t
Y~T = :
It follow s that,if
T
+
dWt;
satis es (10),we have
t=
Ft +
(Z )(t)+
(Y )(t);
;
10
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
w here the pair (Y;Z ) solves the follow ing B SD E:
dYt =
(F t +
(Z )(t)+
(Y )(t))dt+ Z t
dWt;
YT = 0:
Sim ilarly,the process ^ in T heorem 4 is given by
^t =
(Z^)(t)+
(Y^ )(t);
w here the pair (Y^ ;Z^) solves the follow ing B SD E:
dY^t =
( (Z^)(t)+
(Y^ )(t))dt+ Z^t
dWt;
Y^T = :
3.3. Estim ates for the stochastic derivatives ofthe dualequation. In this
section we study the dualequation (9).W e w illestablish estim ates for the
derivatives of the process involved in the duality form ulae. T hese estim ates w illbe usefulfor the study of the weak rate of convergence of the
Euler schem e for the com putation ofexpectations ofregular functions (see
Section 3.4).U nder A ssum ption A 1,we know from Proposition 1 that the
operator I A B is invertible w hen considered on the space L pa([0;T ];H )
for p 2.T he operators A and B can be viewed as bounded linear operatorson L pa([0;T ];H )forp 2,and itfollow sthatthe operator(I A
B )
is invertible on the space L pa([0;T ];H ),for p 2.
In other words,we m ay assertthat,given F 2 L pa([0;T ];H ) (w ith 1 < p
2),there exists a unique 2 Lpa([0;T ];H ),satisfying
= F + A ( )+ B ( ):
U sing Proposition 3,we have
Z
(11)
=F+
Z
T
s ds
J
+
E
T
s
F
:
0
W e want to di erentiate this equation,in order to estim ate the M alliavin
derivatives of .W e w illneed som e regularity assum ptions on the operators
and .For the basic theory of Sobolev spaces on W iener space and for
standard notation,we refer the reader to [13].
W e denote by D the derivative operator.IfX is a sim ple functionalw ith
values in H , D X is a random variable w ith values in L 2([0;T ];H d) and
can be viewed as a nonadapted process (D tX )0 t T . W e w ill say that a
functional (or random variable) X w ith values in H is sm ooth, if it can
be w ritten as a nite sum of m ultiple W iener integrals w ith continuous
determ inistic integrands.N ote that ifX is sm ooth,the process (D tX ) has
a right-continuous m odi cation.W e denote by S a([0;T ];H ) the space ofall
adapted processes (Yt)0 t T w ith values in H ,w ith continuous paths such
11
W EA K A PPR O X IM AT IO N O F SD ES
that,for each t2 [0;T ],the random variable Yt is sm ooth in the previous
sense.T he space Sa([0;T ];H ) is dense in L pa([0;T ];H ) for allp 2 [1;+ 1 ).
O ur regularity assum ptions on and can now be form ulated as follow s.
A ssum ption A 2. For =
and =
and for all Z 2 Sa([0;T ];H ),
the process (Z ) is in D 2;2 and,for all u;v 2 [0;T ],there exist operators
2
,such that,for Z 2 Sa([0;T ];H ),
denoted by D u ,D uv
D u ( (Z ))= (D u
)(Z )+
2
2
D uv
( (Z ))= (D uv
(D u Z );
)(Z )+ (D u
+ (D v
)(D u Z )+
)(D vZ )
2
(D uv
Z ):
M oreover,these operators satisfy the follow ing estim ate,for allq1;q2 w ith
1 q1 < q2 2:
Z
1=q1
T
jD u
E
0
(12)
2
(Z )(t)jq1 + jD uv
Z
C q1 ;q2 E
T
(Z )(t)jq1 dt
1=q2
jZ tjq2 dt
;
0
w here the constants C q1 ;q2 do not depend on u;v (or Z ).
T he estim ate (12) for and allow s us to extend the operators D u ,
2
2
D uv
, D u , D uv
as continuous operators from L qa2 ([0;T ];H ) into
L qa1 ([0;T ];H ), for 1 q1 < q2 2.N ote that in typical exam ples [see Section 3.4, (21), (22)], the operators D u , D u
are not bounded from
L pa([0;T ];H ) into itself.In the sequel,it w illbe convenient to use the follow ing notation.For a random variable Z in D 2;2,and u;v 2 [0;T ],let
2
n(Z;u;v)= jZ j+ jD u Z j+ jD vZ j+ jD uv
Z j:
P roposition 6. LetA ssum ptions A 1 and A 2 hold.LetF 2 L 2a([0;T ];H ).
W e assum e that F has a m odi cation F t such that, for each t2 [0;T ],
R
F t 2 D 2;2 and sup0 u;v T E 0T n(F t;u;v)2 dt< 1 .T hen,the solution of(11)
has a m odi cation t satisfying,for 1 p < q 2,
Z
T
n ( t;u;v)dt
E
0
Z
1=p
p
T
K p;q E
1=q
nq(F t;u;v)dt
;
0
where the constants K p;q depend on T and the constants C p and C q1 ;q2 in
A ssum ptions A 1 and A 2 (and noton u;v or F ).
For the proofof Proposition 6,we w illneed the follow ing com m utation
relations between the operators J and the derivative and conditionalexpectation operators.W e om itthe proofw hich involves only classicalargum ents
ofanalysis on W iener space.
12
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
Lem m a 7.
For allX 2 D 1;2(H ),we have
D u (J(X )(v))= J(D u (X ))(v)1fu
vg
D u (E(X jF v))= E(D u (X )jF v )1fu
vg;
and
du dv a.e.
P roof of P roposition 6. T he form aldi erentiation of(11) gives
Z
Du
t = D u F t + (D u
Z
T
s ds
) J
(t)+ (D u
T
s ds F
) E
(t)
0
Z
+
Z
T
s ds
Du J
(t)+
T
s ds F
Du E
(t):
0
U sing Lem m a 7 and the linearity ofD u ,we get
Z
Du
t = D u F t + (D u
T
) J
s ds
(t)
0
Z
+ (D u
T
s ds F
) E
(13)
Z
1[u;T ]J
+
T
Du
s ds
Du
s ds F
(t)
0
Z
1[u;T ]E
+
(t)
T
(t):
N ow let Iu;T be the operator on L 2a([0;T ];H ) de ned by
Iu;T (Y )= 1[u;T ]Y:
C learly,Iu;T is a self-adjoint operator and de nes an operator w ith norm 1
on L pa([0;T ];H ) for every p 2 [1;+ 1 ).W e have
Iu;T = (Iu;T
) and
Iu;T = (Iu;T
);
and the operatorsIu;T
and Iu;T
satisfy A ssum ptionA 1 w ith the sam e
constants C p as and .N ow let A u and B u be the operators de ned on
L pa([0;T ];H ) (p 2) by
Xd Z
t
A u (Y )(t)=
i= 1 0
Z
t
B u (Y )(t)=
0
1[u;T ](s) i(Y )(s)dW si;
1[u;T ](s) (Y )(s)ds:
U sing Proposition 3,we can rew rite (13) as
Du =
u
+ (A u + B u )(D u );
13
W EA K A PPR O X IM AT IO N O F SD ES
w here
is the adapted process de ned by
u
Z
u (t)=
D u F t + (D u
T
s ds
) J
(t)
0
Z
+ (D u
T
(t):
s ds F
) E
It follow s from Proposition 1 and the properties of the operators Iu;T
and Iu;T
that,for p 2 [1;2],
Z
E
Z
T
jD u tj dt C pE
0
T
p
j u (t)jp dt;
0
w here C p does not depend on u.O n the other hand,we deduce from A ssum ption A 2 and the assum ptions on F that,for 1 p < q 2,
Z
Z
1=p
T
p
T
Cp E
j u (t)j dt
E
1=p
jD u F tjp dt
0
0
Z
+ C p;q
Z
T
+
0
Z
T
E
dt
s ds
0
Z
1=q
q
T
Jt
E
dt
s ds F t
0
1=q
q
T
E
;
t
w herehereagain theconstantsC p and C p;q do notdepend on u.H erewehave
used the notation Jt(Z ) for J(Z )(t).W e have,using Jensen’s and H older’s
inequalities,
Z
Z
T
E
dt
s ds F t
E
0
Z
1=q
q
T
Z
T
s ds
0
t
1=q
q
T
E
dt
t
Z
T
T E
1=q
j tjq dt
:
0
O n the other hand,using 1 < q
Z
Z
T
0
1=q
q
T
dt
s ds
Jt
E
2 and the B D G inequality,we have
0
Z
T 1=q
1=2
0
1=2
Cq E
Z
T 1=2C q E
T
0
q 1=q
T
s ds
0
1=q
j sjq ds
q=2 1=q
2
T
s ds
0
Z
T 1=q
Z
T
Jt
E
:
dt
14
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
R ecallthat = (I
B ) 1(F ),so that
A
Z
Z
1=q
T
q
Z
p< q
p
0
Z
1=p
T
p
+ C p;q E
jD u F tj dt
Cp E
jD u tj dt
E
:
2,
Z
1=p
T
1=q
jF tjq dt
0
0
H ence,we have,for 1
T
Cq E
j tj dt
E
0
T
1=q
jF tjq dt
;
0
w here the constants C p and C p;q depend on T but not on u.W e can now
di erentiate (13) w ith respect to v and derive in a sim ilar m anner the estim ate
Z
jD
E
0
Z
1=p
T
2
p
uv tj dt
T
Cp E
0
1=p
2
jD uv
F tjp dt
Z
T
+ C p;q E
1=q
(jF tj+ jD u F tj+ jD vF tj)q dt
:
0
N ote that in order to justify the form aldi erentiations,it su ces to use an
iterating procedure ofthe form 0 = F and j+ 1 = F + (A + B )( j).T he
regularity of j carries over to j+ 1 and we have convergence of j toward
,together w ith the derivatives,w ith the suitable norm s.
For the application of our m ethodology to num erical schem es, we w ill
need to consider fam ilies ofoperators ( h ; h )h 0 and introduce som e additionalnotation.Let A 1;2 denote the space ofalloperators ( ; ) satisfying
A ssum ptionsA 1 and A 2.For( ; )2 A 1;2,letC p( ; )[resp.C q1 ;q2 ( ; )]be
the sm allest constant for w hich (4) [resp.(12)]holds.T he follow ing proposition follow s easily from Proposition 6.
P roposition 8. A ssum e ( h ; h )h 0 is a fam ily ofoperators in A 1;2,
satisfying,for allp 2 [2;+ 1 ) and for allq1;q2 with 1 q1 < q2 2,
lim C p(
h
;
h
)= 0 and
lim C q1 ;q2 (
h! 0
h
;
h
)= 0:
h! 0
If(F h )h 0 is a fam ily ofadapted processes satisfying sup0
u;v)2 dt< 1 and
Z
lim
h! 0 0 u;v T
then the processes
h
T
sup E
0
[de ned by
n(F th
h
lim
= (I
sup E
h! 0 0 u;v T
E
RT
F t0;u;v)2 dt= 0;
Z
8 p 2 [1;2)
u;v T
Ah
T
0
n( th
B h ) 1(F h )]satisfy
0
p
t ;u;v) dt=
0:
0
n(F th ;
15
W EA K A PPR O X IM AT IO N O F SD ES
3.4. A pplication to the Eulerapproxim ation ofdelay equations:the regular
case. In this section we assum e to sim plify the notation that allprocesses
take values in R .W e are interested in the expansion ofEf(X T ) Ef(X T )
w here the process (X t)t2 [ r;T ] solves the stochastic delay equation
Z
Z
0
0
X t+ s d (s) dt;
X t+ s d (s) dW t + b
dX t =
Xs=
s;
t 0;
r
r
s2 [ r;0];
w here r> 0, 2 C1([ r;0];R ) and is a nite m easure.
W e denote by X the follow ing Euler approxim ation of (X t)t2 [
step h = r=n:
Z
Z
0
dX t =
X
(t)+ (s) d
Xs=
s;
0
(s) dW t + b
r
r;T ] w ith
X
(t)+ (s) d
(s) dt;
t 0;
r
s2 [ r;0];
w ith (s)= [ns=r]
n=r ,w here [t]stands for the entire part oft.W e assum e that
the functions,f, and b are in Cb3.N ote that if is the D irac m easure at
0,we have a standard di usion.
For existence,uniqueness and m om ent estim ates ofsolutions ofstochastic delay equations in the above form ,we refer to [12].C onsistency of the
Euler schem e for delay equations in the case w here is a D irac m ass has
been studied in [9]through an extension ofthe PD E m ethod.A n in nitedim ensionalextension ofthe PD E m ethod is used in [4]but their result is
lim ited to drift coe cients linearly dependent on the past and nondelayed
di usion coe cient.
O urexpansion resultisderived from theduality form ula and thefollow ing
lem m a.
Lem m a 9. Let(U sh )be a fam ily ofF T -m easurable real-valued processes,
such that8 s2 [0;T ],8 h 2 [0;1],U sh 2 D 1;2.W e assum e that,for som e p > 1,
we have
Z
lim E
h! 0
T
0
jU sh
U s0jp ds= 0
and
Z
0
Z
T
lim
sup
h! 0
r 0 v2 [ (s)+ (u);s+ u]
kD vU sh
D s+ u U s0kp1
(s)+ (u) 0
dsd (u)= 0:
T hen
Z
T
E
0
Z
U sh
0
(X s+ u
r
X
(s)+ (u) )d
(u)ds= hC (U 0)+ Ih (U 0)+ o(h);
16
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
where
Z
C (U 0)=
Z
0
1
2
T
E
0
r
[U s0(~
b0s+ u 1s+ u
0
s+ u 1s+ u< 0)
0+
0
0
+ ~s+
u D s+ u U s 1s+ u
Z
Z
0
Ih (U 0)=
T
[U s0(~
b0s+ u 1s+ u
E
r
0
Z
0
~
b0t = b
r
0]ds(u
(u))d (u);
0
and ~t0 =
X t+ v d (v)
(u);
0
s+ u 1s+ u< 0)
0+
0
0
+ ~s+
u D s+ u U s 1s+ u
Z
0]dsd
X t+ v d (v) :
r
R em ark. O bserve that jIh (U 0)j hC . M oreover, w hen
is a D irac
m assat r a good discretization ofthe tim eintervalgivesIh (U 0)= 0.W hen
is an absolutely continuous m easure,we obtain Ih (U 0)= hC (U 0)+ o(h).
P roof of Lem m a 9. W e have
Z
X s+ u
X
s+ u
=
(s)+ (u)
(s)+ (u)
(s)+ (u)
~
bh(t) dt 1
(s)+ (u) 0
(s)+ (u) )1 (s)+ (u) 0< s+ u ;
+ (0
w here
Z
0
~th =
X t+
(v) d
(v)
r
and
Z
~
bht = b
(15)
s+ u
(s)+ (u) )1s+ u< 0
+ ( s+ u
(14)
Z
~h(t) dW t +
0
X t+
(v) d
(v) :
r
T his gives
Z
T
E
0
Z
U sh
Z
0
(X s+ u
X
(s)+ (u) )d
(u)ds
r
Z
0
=
Z
T
U sh 1
E
r
Z
0
+
r
Z
Z
0
E
+
0
Z
0
T
T
E
r
Z
0
Z
0
+
T
E
r
0
s+ u
(s)+ (u) 0
(s)+ (u)
Z
U sh 1 (s)+ (u) 0
U sh ( s+ u
U sh ( 0
~h(t) dW t dsd (u)
s+ u
(s)+ (u)
~
bh(t) dtdsd (u)
(s)+ (u) )1s+ u< 0 dsd
(u)
(s)+ (u) )1 (s)+ (u) 0< s+ u
dsd (u):
17
W EA K A PPR O X IM AT IO N O F SD ES
B y duality,we obtain for the rst term
Z
Z
0
Z
T
U sh 1 (s)+ (u) 0
E
r
0
Z
Z
0
=
(s)+ (u)
Z
T
1
E
0
r
s+ u
~h(t) dW t dsd (u)
s+ u
(s)+ (u) 0
(s)+ (u)
Since (s)+ (u) 0 < s+ u im plies u < s<
that
Z
Z
0
T
E
r
0
D tU sh ~h(t) dtdsd (u):
u + 2h,one can easily verify
(s)+ (u) )1 (s)+ (u) 0< s+ u
U sh ( 0
dsd (u)= o(h):
R
N ow recallthat ifg is an integrable function on [0;T ],we have 0T g(s)(s
R
(s))ds= h2 0T g(s)ds+ o(h).Itfollow sthatin orderto derive the expansion
stated in the lem m a,it is enough to prove that w hen h tends to 0,
Z
1
h
Z
0
Z
T
r
0
(16)
Z
Z
Z
0
T
0
Z
T
s+ u
Z
0
T
E
r
Z
0
Z
T
U sh
E
r
0
(18)
Z
0
Z
0
r
E
0
~s+
u D s+ u U
s+ u
0
0
s
s+ u dsd
0
t dt1s+ u< 0 dsd
0
s+ u 1s+ u< 0
U s0
s+ u dsd
(u) ! 0;
D tU hs ~h(t) dtdsd (u)
(s)+ (u)
T
~
bh(t) dtdsd (u)
U 0s~
b0s+ u
0
(s)+ (u)
0
r
Z
Z
E
E
(17)
1
h
(s)+ (u)
Z
r
1
h
s+ u
h
Us
E
s+ u dsd
(u) ! 0;
(u)
(u) ! 0;
w here s+ u = s+ u
(s) (u),Us = U sh 1 (s)+ (u) 0 and U 0s = U s01s+ u
U sing H older’s inequality,the left-hand side of(16) is bounded by
h
Z
T
0
U s0kp dssup k~
bht kq
kU sh
t
Z
T
+
0
kU s0kp dssup
Z
T
+ sup
u
sup
k~
bh(t)
u;s t2 [ (s)+ (u);s+ u]
0
kU sh kp1
(s)+ (u) 0 s+ u
~
b0s+ u kq
dssup k~
bht kq;
t
0.
18
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
R
bht kq =
w ith 1p + 1q = 1.O bserve that supu 0T kU sh kp1 (s)+ (u) 0 s+ u dssuptk~
o(h).In the sam e way the left-hand side of(17) is bounded by
Z
0
Z
T
sup
r 0 t2 [ (s)+ (u);s+ u]
Z
0
Z
T
+
D s+ u U s0kp1
kD tU sh
kD s+ u U s0kp dsd (u)sup
(s)+ (u) 0
dsd (u)supk~th kq
t
k~h(t)
sup
u;s t2 [ (s)+ (u);s+ u]
r 0
0
~s+
u kq
+ o(h):
Sim ilar bounds hold for the left-hand side of(18) and we obtain the result
ofLem m a 9 as soon as
sup k~
bhs kq < 1 and
sup k~sh kq < 1 ;
h> 0;s2 [0;T ]
h> 0;s2 [0;T ]
sup
sup
k~h(t)
0
~s+
u kq ! 0
k~
bh(t)
~
b0s+ u kq ! 0;
u;s t2 [ (s)+ (u);s+ u]
and
sup
sup
u;s t2 [ (s)+ (u);s+ u]
for allq2 [1;+ 1 ).T his can be proved as in [6].
T he follow ing theorem can be viewed as an analogue ofthe classicalexpansion ofthe error for di usions (see [15]).
T heorem 10. W e have,iff 2 Cb3,
Ef(X T ) Ef(X T )= hC f + Ih (f)+ o(h);
where C f = C (U 0) and Ih (f)= Ih (U 0) are de ned as in Lem m a 9 with
Z
U s0 =
Z
0
Z
Z
0
0
t dt
Z
0
0
T
t dt
X s+ u d (u)
+b
s
r
0
is the unique solution of
Z
t=
Z
T
X s+ u d (u) D s
r
and
X s+ u d (u) f0(X T )
r
r
+
0
X s+ u d (u) D sf0(X T )+ b0
0
J f0(X T )+
Z
T
s ds
E f0(X T )+
(t)+
T
s ds F
0
with
Z
(X )(t)= E
0
0
Z
0
u+ v d
(v) X t
ud
(u)F t ;
Xt
u+ v d
(v) X t
ud
(u)F t :
0
0
b
m ax(t T ; r)
Xt
r
m ax(t T ; r)
Z
(X )(t)= E
Z
0
r
(t)
19
W EA K A PPR O X IM AT IO N O F SD ES
P roof. W e have
Z
1
f0(aX T + (1
Ef(X T ) Ef(X T )= E
a)X T )da(X T
X T ):
0
Let F h =
R1
0
f0(aX T + (1
Z
a)X T )da and Yt = X t
Z
0
X s+ u d (u)
b
0
X
b
(s)+ (u) d
(u)
r
r
Z
=
X t.W e have
0
bh1 (s)
(X s+ u
X
(s)+ (u) )d
Z
0
(u)
r
and
Z
0
X
X s+ u d (u)
(s)+ (u) d
(u)
r
r
Z
0
h
1 (s)
=
(X s+ u
X
(s)+ (u) )d
(u);
r
w here
Z
(19)
Z
1
0
h
1 (s)=
X s+ u d (u)+ (1
(20)
Z
1
X
(s)+ (u) d
(u) da;
X
(s)+ (u) d
(u) da:
(Y )(t)dt+ dG ht ;
t 0;
r
Z
0
0
bh1 (s)=
X s+ u d (u)+ (1
b a
0
a)
r
r
0
0
a)
r
0
Z
Z
0
a
W e deduce that Yt is a solution of
dYt =
h
h
(Y )(t)dW t +
Ys = 0;
s2 [ r;0];
w ith
Z
(21)
h
0
h
1 (s)
(Y )(s)=
Ys+ u d (u);
r
Z
(22)
h
(Y )(s)=
0
bh1 (s)
Ys+ u d (u);
r
Z
G ht
Z
t
0
h
1 (s)
=
(X s+ u
0
X
(s)+ (u) )d
(u)dW s
r
Z
t
+
Z
bh1 (s)
0
(X s+ u
X
(s)+ (u) )d
(u)ds:
r
0
T he operators h and h satisfy A ssum ption A 1 uniform ly w ith respect to
h.T he adjoint operators h and h are given by
Z
h
0
h
1 (s
(X )(s)= E
m ax(s T ; r)
u)X s
ud
(u)F s ;
20
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
Z
h
0
bh1 (s
(X )(s)= E
u)X s
ud
(u)F s :
m ax(s T ; r)
T hey satisfy A ssum ption A 2 uniform ly in h. N ote that the estim ates on
D u h ,D u h ,and so on,follow from the boundednessofthe derivatives of
b and and the follow ing estim ates (see [6]):
8 p 2 [1;+ 1 )
Enp(X t;u;v)< 1
sup
and
0 u;v;t T
sup
Enp(X t;u;v)< 1 :
sup
h> 0 0 u;v;t T
U sing T heorem 4,we obtain
Z
Ef(X T ) Ef(X T )= EF h YT = EF h G hT + E
T
0
h h
s G s ds;
w ith
h
(B h ) ) 1[
(A h )
= (I
h
(J(F h ))+
h
(E(F h jF ))]
B ut
Z
EF h G hT = E
Z
T
D sF h
0
h
1 (s)
(X s+ u
0
X
(s)+ (u) )d
(u)ds
r
Z
T
+E
Z
0
F h bh1 (s)
(X s+ u
0
X
(s)+ (u) )d
(u)ds
r
and
Z
E
T
0
Z
h h
s G s ds=
E
Z
T
T
Ds
0
0
Z
T
Z
T
+E
0
s
Z
h
t dt
(X s+ u
X
(s)+ (u) )d
(u)ds
r
Z
h
t dt
0
h
1 (s)
bh1 (s)
0
(X s+ u
X
(s)+ (u) )d
(u)ds:
r
W e end the proofapplying Lem m a 9 w ith
Z
U sh
=D
h h
sF
1 (s)+
F h bh1 (s)+
T
Ds
0
Z
h
t dt
h
1 (s)+
Since lim h sup0 u;v;t T Enp(X t X t;u;v)= 0 for p
U sh and its derivatives follow s from Proposition 8.
4. The irregular case.
bh1 (s)
T
s
h
t dt:
1,the convergence of
21
W EA K A PPR O X IM AT IO N O F SD ES
4.1. D uality for the derivatives ofY . In thissection we wantto establish
duality form ulae sim ilar to those in T heorem 4,for the derivatives of the
solution of(3).T his is m otivated by the treatm ent ofirregular functions of
the Euler schem e,w hich w illinvolve integrations by parts (see Section 4.2).
W e w illneed regularity assum ptions on the operators and .W e w ill
assum e that for Z 2 Sa([0;T ];H ), (Z ) and (Z ) are in D 1 and that we
can de ne recursively the operators D sk1 k s and D sk1 k s so that,for k > 1
and s1;:::;sk 2 [0;T ],
D s1 ((D sk2
1
D s1 ((D sk2
1
s)(Z ))=
(D sk1
k
s)(Z )+
(D sk2
1
k
s)(Z ))=
(D sk1
k
s)(Z )+
(D sk2
1
k
k
s)(D s1 Z );
k
s)(D s1 Z ):
k
N ote that D sk1 k Zst can be viewed as an elem ent ofH d ,w ith coordinates
l here the l superscriptsreferto the coordinatesofW w ith respect
D sl11 kk ,w
i
s
to w hich di erentiation occurs(li = 1;:::;d).W enow introducethefollow ing
assum ption.
A ssum ption A 3. For = and = ,the operators D sj1 j s de ned
j
above are bounded from L qa([0;T ];H ) into L pa([0;T ];H d ) for 2 p < q <
1 .M oreover,their adjoints satisfy the follow ing estim ates.For allpositive
integers j and for 1 p < q 2,there exists a positive constant C p;q;j such
that for allu;v;s1;:::;sj 2 [0;T ],we have
k(D sj1
j
s) kq! p +
kD u (D sj1
j
s) kq! p +
2
kD uv
(D sj1
j
s) kq! p
C p;q;j;
w here k qk! p stands for the operator norm from L qa into L pa,and the op2 (D j
erators D u (D sj1 j s) ,D uv
s1 j s) are de ned in the sam e way as D u
in A ssum ption A 2.
R ecallthe notation
2
n(Z;u;v)= jZ j+ jD u Z j+ jD vZ j+ jD uv
Z j:
T heorem 11. LetA ssum ptions A 1,A 2 and A 3 hold.LetG = (G t)0 t T
be a continuous adapted process with values in H , satisfying G t 2 D 1 , for
allt2 [0;T ].T hen the solution of(3) satis es Y t 2 D 1 ,for t2 [0;T ].
M oreover, given an adapted process F = (F t)0 t T with values in
k
L 2([0;T ]k ;H d ) such that F t 2 D 2;2 for t2 [0;T ],and
Z
sup
0 u;v;s1 ;:::;sk T
T
E
0
1=2
n2(F t(s1;:::;sk);u;v)dt
<1 ;
22
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
there exist adapted processes (0);:::; (j);:::; (k), with values in H ;:::;
k
j
L p([0;T ]j;H d );:::;L p([0;T ]k ;H d ) respectively for allp 2 [1;2),such that
Z
(23)
T
E
hF t;D k Ytidt=
0
and,for 1
Z
Xk
0
j= 0
p< q
T
E
(j)
h t ;D jG tidt;
2,j= 0;:::;k,
Z
sup
T
E
0
u;v;s1 ;:::;sj 2 [0;T ]
1=p
(j)
np( t (s1;:::;sj);u;v)dt
(24)
Z
C p;q;k
sup
E
u;v;s1 ;:::;sk 2 [0;T ]
T
1=q
nq(F t(s1;:::;sk);u;v)dt
;
0
where the constants C p;q;k do notdepend on F .
C orollary 12. In addition to the assum ptions of T heorem 11, we
assum e that the process (D kG t) is right-continuous with respect to t for
k 1. T hen, if Y is the solution of (3), the process (D k Yt)0 t T has a
right-continuous m odi cation. M oreover, given a random variable with
k
values in L 2([0;T ]k ;H d ),such that 2 D 2;2 and
En2( (s 1;:::;sk);u;v)< 1 ;
sup
u;v;s1 ;:::;sk 2 [0;T ]
there exist adapted processes ^(0);:::;^(j);:::;^(k), with values in H ;:::;
k
j
L p([0;T ]j;H d );:::;L p([0;T ]k ;H d ) respectively for allp 2 [1;2),such that
k
Eh ;D YT i= Eh ;D G T i+
(25)
Z
Xk
k
0
j= 0
and,for 1
p< q
T
E
(j)
h^t ;D jG tidt;
2,j= 0;:::;k,
Z
sup
T
E
u;v;s1 ;:::;sj 2 [0;T ]
0
(j)
np(^t (s1;:::;sj);u;v)dt
1=p
(26)
C p;q;k
sup
(Enq( (s 1;:::;sk );u;v))1=q;
u;v;s1 ;:::;sk 2 [0;T ]
where the constants C p;q;k do notdepend on .
P roof. B y di erentiating (3) k tim es,we get
Z
D k Yt = D kG t +
t
0
Z
D k( (Y )(s))
t
dWs +
0
(k)
D k( (Y )(s))ds+ It ;
23
W EA K A PPR O X IM AT IO N O F SD ES
w ith
Xk
(k)
It (s1;:::;sk)=
D sk1
s^k (s (Y )(si))1fsi tg;
1
i
i= 1
w here the notation s^i m eans that the variable si is om itted.M ore precisely,
k
recallthat D sk1 k Yst can be viewed as an elem ent ofH d ,w ith coordinates
l here the l superscriptsreferto the coordinatesofW w ith respect
D sl11 kk ,w
i
s
to w hich di erentiation occurs.W ith this m ore precise notation,we have
Itl1
k
l
(s
1;:::;sk )=
Xk
^i
l
D sl11
i
k l
s^k (s li(Y )(si))1fsi tg:
Z
t
i= 1
W e can w rite
D
k
s1
(k)
G t (s1;:::;sk)+
Y =
k st
(27)
Z
t
(D sk1
+
0
k
(D sk1
0
k
Ys )(s)
dWs
Ys )(s)ds;
w ith
(k)
G t (s1;:::;sk)=
D
Z
Xk X
k
s1
G +
k st
0
j= 1 2 A k
j
(28)
Z
t
+
0
(D sj
t
(D sj )(D sk jY )(s)
dWs
(k)
)(D sk jY )(s)ds+ It (s1;:::;sk);
w here A kj
s = si1
that D sk1
is the set of j-tuples = (i1;:::;ij), w ith 1 i1 <
j< ik
27)
s
is the ordered com plem ent of .W e deduce from (
ij ,and
p
Y
s
ol
ves
an
equat
i
on
s
i
m
i
l
ar
t
o
(
3)
and
we
can
der
i
ve
L
Lq
k st
estim ates for the derivatives ofY using A ssum ption A 3 and the regularity
of G .H ere again the form aldi erentiation can be justi ed by a standard
approxim ation argum ent.
W e now prove (23)by induction.For k = 0,the resultreducesto the rst
part ofT heorem 4 and Proposition 6.N ow assum e that (23) and (24) hold
up to the order k 1.W e rst deduce from (27) that
Z
T
E
Z
hF t;D kYtidt= E
0
T
0
(k)
t (s1;:::;sk )=
(k)
(k)
hG t ; t idt;
1(F (s ;:::;s )).T he estim
1
k
(I A
B )
w ith
follow from Proposition 6.It follow s from (28) that
Z
E
T
0
Z
(k)
(k)
hG t ; t idt= E
T
0
(k)
hD kG t; t idt+ R T ;
ates for
(k)
24
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
w here R T is the sum of term s w hich are of three types, w hich we study
successively.
T ype 1.
Z
Z
Z
T
dt
E
[0;T ]k
0
w ith j
Z
(D tj1
0
k j
t)(D tj+ 1
j
Yt)(s) dWs
k
1.W e have
Z
Z
T
dt
E
t
(k)
k dtt (t1;:::;tk );
dt1
(k)
k dtt (t1;:::;tk );
dt1
[0;T ]k
0
t
0
(D tj1
j
k j
t)(D tj+ 1
k
Yt)(s)
dWs
Z
=
dt1
k dt
[0;T ]k
Z
Z
T
(k)
t (t1;:::;tk );
dt
E
0
Z
Z
=
dt1
T
dt
kE
[0;T ]k
0
t
0
(D tj1
h~s(t1;:::;tk );D tkj+ j1
k j
t)(D tj+ 1
j
k
k
Yt)(s)
dWs
Ytsids;
w ith
Z
~t(t1;:::;tk )= (D j
t1
k
t)
T
(k)
s (t1;:::;tk )ds
J
0
(t):
H ere, we have used Proposition 3, w ith D tj1 k t instead of . U sing A ssum ption A 3,we have,as in the proofofProposition 6,
Z
T
E
1=p
np(~t(t1;:::;tk);u;v)dt
0
Z
T
C p;q E
0
1=q
(k)
nq( t (t1;:::;tk );u;v)dt
;
for 1 p < q 2.W e can now apply the induction hypothesis,for a xed
t1;:::;tj, to the process ~t(t1;:::;tj) considered as a process w ith values
k j
in L p([0;T ]k j;H d ).N ote that these processes are not in L 2 but,by a
suitable density argum ent,the induction hypothesis can be applied.
T ype 2.
Z
Z
Z
T
dt
E
dt1
[0;T ]k
0
w ith j
Z
T
0
t
(D tj1
0
k j
t)(D tj+ 1
j
k
Yt)(s)ds
1.W e have
Z
Z
dt
E
(k)
k dtt (t1;:::;tk );
dt1
[0;T ]k
(k)
k dtt (t1;:::;tk );
t
0
(D tj1
j
k j
t)(D tj+ 1
k
Yt)(s)ds
;
;
25
W EA K A PPR O X IM AT IO N O F SD ES
Z
=
dt1
k dt
[0;T ]k
Z
Z
T
(k)
t (t1;:::;tk );
dt
E
0
Z
Z
=
dt1
T
0
(D tj1
0
k j
t)(D tj+ 1
j
dth~t(t1;:::;tk);D tkj+ j1
dt
kE
[0;T ]k
t
k
k
Yt)(s)ds
Ytti;
w ith
Z
~t(t1;:::;tk)= (D j
t1
T
(k)
s (t1;:::;tk )ds F
t) E
k
(t):
H ere again,we have used a variant ofProposition 3,w ith D tj1 k t instead
of ,and the L p L q estim ate follow s from A ssum ption A 3 as in the proof
ofProposition 6.
T ype 3. T he term s oftype 3 com e from I(k).T hey are ofthe follow ing
form (we let = (k)):
Z
Z
T
dt
E
ds1
k ds
[0;T ]k
0
h(D sj1
1
j
s1
)(D skj+ j1
k
Ys )(sj)1fsj
and can be treated as follow s (the notation
tion w ith respect to sj is om itted):
Z
R
[0;T ]k
1
ds^j m eans that integra-
Z
T
dt
E
tg t(s1;:::;sj;:::;sk )i
ds1
k ds
[0;T ]k
0
h(D sj1
1
j
s1
)(D skj+ j1
k
Ys )(sj)1fsj
tg; t(s1;:::;sj;:::;sk )i
Z
=
1
[0;T ]k
Z
ds^j
Z
T
0
sj
Z
Z
=
[0;T ]k
T
dsj
E
1
dth t(s1;:::;sj;:::;sk);(D sj1
Z
T
ds^j E
Z
=
[0;T ]k
1
T
ds^j E
0
s1
)(D skj+ j1
k
u (s1;
;s; k;s
)du F s ;
1
)(D skj+ j1
s
(D sj1
Z
j
T
ds E
0
1
j
s1
k
Ys)(s)
h~t(s1;:::;sj 1;sj+ 1;:::;sk);D skj+ j1
k
Ystidt;
Ys )(sj)i
26
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
w ith
~t(s1;:::;sj 1;sj+ 1;:::;sk)
Z
= (D sj1
1
s1
j
T
u (s1;:::;sj 1;
) E
;s
j+ 1;:::;sk )du F
(t):
W e have,using A ssum ption A 3,
Z
E
T
1=p
j~t(s1;:::;sj 1;sj+ 1;:::;sk)jp dt
0
Z
C p;q E
Z
T
0
Z
Z
T
1=q
q
T
u (s1;:::;sj 1;t;sj+ 1;:::;sk )du
E
dt
Ft
t
1=q
q
T
s(s1;:::;sj 1;t;sj+ 1;:::;sk )ds
C p;q E
dt
t
0
Z
C p;qT sup
T
E
1=q
j s(s1;:::;sj 1;t;sj+ 1;:::;sk )jq ds
:
0
0 t T
R
R
(j)
Finally,we have E 0T hF t;D kYtidtasa sum ofterm slike E 0T h t ;D jG tidt,
R
(i)
w ith j k,orE 0T hF t ;D iYtidt,w ith i k 1,w ith appropriate estim ates
for the processes F (i).W e can now apply the induction hypothesis to term s
R
(i)
like E 0T hF t ;D iYtidt.T he various ’s given by the induction hypothesis
com bine to produce the nalform stated in T heorem 11.
P roof of C orollary 12. T hecontinuity ofthederivativesofY follow
easily from the assum ptions on G and (27).U sing the notation ofthe proof
ofT heorem 11,we have
Z
Eh ;D YT i= Eh ;G
(k)
T i+
= Eh ;G
(k)
T i+
k
Z
+E
T
0
T
T
dWs +
(D k Y )(s)ds
0
0
Z
E
k
(D Y )(s)
;
E
Z
T
h (J( ))(s);D k Ysids
0
h (E( jF ))(s);D k Ysids:
For the last two term s, we can apply T heorem 11, and the fact that, for
1 p < q 2,
Z
T
E
1=p
np(Jt( (s 1
C p;q(Enq( (s 1
s;u;v)dt
k ))
s
)1=q
k );u;v)
0
and
Z
E
T
0
1=p
np(E( (s 1
s
F t);u;v)dt
k )j
C p;q(Enq( (s 1
s
)1=q:
k );u;v)
27
W EA K A PPR O X IM AT IO N O F SD ES
For the rst term ,we have,using (28),
(k)
T i=
Eh ;G
Eh ;D kYT i+ R T ;
w here R T is a sum ofterm s ofthree di erent types,w hich can be treated
in the sam e way as in the proofofT heorem 11.
4.2. A pplication to the Euler approxim ation ofdelay equations:the irregular case. In thissection we considerthe processesX and X ofSection 3.4.
W e assum e thatf isa m easurable bounded function,b and are in Cb1 and
that the variable X T is nondegenerate:
(29)
E((
XT
) p)< 1
8 p > 1;
w here X T denotesthe M alliavin covariance m atrix ofX T .T hiscondition is
satis ed in the uniform ly elliptic case,thatis, (x) a > 0 forallx 2 R (see
[10]) and under weaker assum ptions (see [2]) w hen is a D irac m easure.
in Cb1 and X T satisfying (29),we have for
T heorem 13. For b and
f m easurable bounded:
jEf(X T ) Ef(X T )j C f h:
Sketch of proof. W e consider the follow ing truncation function.Let
:[0;+ 1 )7
! R bea C 1 function w ith bounded derivativessuch that1[0;1=8]
1 [0;1=4] and let X T be the M alliavin covariance m atrix ofX T be w hich
R
in our one-dim ensionalsetting reduces to X T = 0T (D u X T )2 du.W e de ne
h by
T
RT
h
T
(30)
(D u X T
0
=
D u X T )2 du
:
XT
O bserve that
Z
P(
h
T
= 1)
6
1
P
XT
T
(D u X T
Z
p
8pE
D u X T )2 du > 1=8
0
p
T
(D u X T
XT
D u X T )2 du
forallp > 0.U sing H older’sinequality and supu EjD u X T
one can easily prove that for allp 1
(31)
P(
;
0
h
T
D u X T jp
= 1) C hp:
6
M oreover,we have
Z
f
h
T
T
(D u X T
= 0g
6
0
D u X T )2 du
XT
=4 :
C hp=2
28
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
Since
Z
XT
aX T + (1 a)X T
we obtain for 0
a
(32)
=2
a)2
(1
T
(D u X T
D u X T )2 du;
0
1
h
T
f
= 0g
6
f
aX T + (1 a)X T
1
4 XT
g:
N ow we have
Ef(X T ) Ef(X T )
h
T )g +
= Ef(f(X T ) f(X T ))(1
Ef(f(X T ) f(X T ))
h
T g:
B y construction of hT ,the rstterm is oforder h p for allp 1 and we just
have to prove that the second one is oforder h.
Let YT = X T X T and let (fm ) be a sequence ofC1 functions such that
kfm k1
kfk1 and (fm ) converges dx a.e.to f.W e have
Z
h
T
E(fm (X T ) fm (X T ))
1
=
0
Efm0 (aX T + (1
a)X T )YT
h
T
da:
Since X T adm its a density Efm (X T ) hT converges to Ef(X T ) hT .N ow on
the set f hT 6
= 0g,det X T > 0 and from [13],C orollary 2.2.1,page 88 (see
[3]in higher dim ension) X T has an absolutely continuous law conditioned
by f hT 6
= 0g and Efm (X T ) hT converges to Ef(X T ) hT .Itrem ainsto prove
that E(fm (X T ) fm (X T )) hT is bounded by C f h w here the constant C f
only depends on f through kfk1 .U sing the M alliavin integration by parts
form ula (see [14]) we obtain
E(fm (X T ) fm (X T ))
Z
h
T
1
Egm (aX T + (1
=
a)X T )H 3(aX T + (1
a)X T ;YT
0
R
R
h
T )da;
w here gm (x)= 0x dy 0y dzfm (z) so that gm is in C3 and gm00 = fm and H 3 is
de ned recursively by
H 1(F;G )= G
1
F
(D F ) hD (G
H k(F;G )= H 1(F;H k
1
F
1(F;G ));
);D F i;
k
2;
w ith
Z
T
D u H D u F du:
hD H ;D F i=
0
H ence for any m easurable set A ,we have (see [1])
(33)
kH k(F;G )1A kp
Ck
1A kkp11 kF kkp22 ;q2 kG kp3 ;q3
1
F
29
W EA K A PPR O X IM AT IO N O F SD ES
for som e constants C ,k1,p1,k2,p2,q2,p3,q3,depending on k and p.
O bserve that
H 3(aX T + (1
X3
h
T )=
a)X T ;YT
h
h
i
i ;D YT i;
i= 0
h ,and
i
for sm ooth variables
h
T
E(fm (X T ) fm (X T ))
nally
Z
X3
1
= E
gm (aX T + (1
i= 0
X3
a)X T )
0
h
i
i da;D YT
hF mi;h ;D iYT i:
= E
i= 0
M oreover,the variables F mi;h 2 D 2;2 and from (33) and (39) we deduce the
follow ing estim ate:
sup En2(F mi;h ;u;v) C k
u;v
1 k1
X T kp1 kaX T
a)X T kkp22 ;q2 ;
+ (1
forsom e k1,p1,k2,p2,q2,w ith C independentofm .A pplying C orollary 12,
this gives
h
T
E(fm (X T ) fm (X T ))
X3
EhF mi;h ;D iG T i+
=
i= 0
Z
Xi
T
E
0
j= 0
!
(i;j)
h^t ;D iG tidt ;
w here the process G ht is de ned in Section 3.4 by
Z
G ht
Z
t
0
h
1 (s)
=
(X s+ u
0
Z
t
+
Z
(s)+ (u) )d
(u)dW s
(X s+ u
0
Z
h
1 (s)
=
t
+
0
Z
s+ u
(s)+ (u)
r
0
Z
X
(s)+ (u) )d
(u)ds
r
Z
t
0
bh1 (s)
0
Z
X
r
0
bh1 (s)
r
Z
~h(v) dW v d (u)dW s
s+ u
(s)+ (u)
~
bh(v) dvd (u)ds:
and ~
bhv are,respectively,de ned in (19),(20),(14) and (15).W e
end the proofusing the duality relationship as in Lem m a 9.
h
h
h
1 ,b1 ,~v
R em ark 14. T heaboveproofcarriesoverto a m ultidim ensionalsetting.
T he only technicaldi culty isto extend the localization argum ent.Suppose
X T takes itsvaluesin R r w ith r> 1,and the M alliavin covariance m atrix of
X T is de ned by ( X T )i;j = hD X Ti ;D X Tj i,1 i j r.W e want to de ne
30
E.C L EM EN T ,A .K O H AT SU -H IG A A N D D .LA M B ERT O N
a sm ooth functional hT such that outside a set w here all the M alliavin
derivatives D k hT vanish,we have a uniform controlofthe determ inant of
the M alliavin m atrix ofaX T + (1 a)X T ,0 a 1.Let :[0;+ 1 )! R be
1 [0;1=4].W e
a C1 function w ith bounded derivatives such that 1[0;1=8]
de ne hT by
h
T
(34)
=
X T )j2(1+ k
det X T
jD (X T
k22)(r
1)=2
;
R
P
P
XT
w herejD X T j2 = ijD X Ti j2 = i;k 0T (D ukX Ti )2 du and k X T k2 istheH ilbert{
Schm idtnorm ofthe m atrix X T ,thatis,the l2 norm ofthe coe cients.W e
denote by k X T k the operator norm .
N ote that hT 2 D 1 [the sum of squares of the coe cients ( X T )i;j is
sm ooth]and that
f
h
T
= 1g
6
so that for allp
X T )j2
jD (X T
det X T
8(1 + k X T k22)(r
1)=2
;
1,we have as in the one-dim ensionalcase
(35)
9C > 0
h
T
P(
= 1) C hp;
6
p
provided kD (X T X T )kp C p h for allp 2 [1;+ 1 ).
N ow observethatwehavethefollow ing inequality forany positive-de nite
r-dim ensionalm atrix A :
p
(36)
rkA k:
kA k kA k2
M oreover,if
1(A ) is the sm
allest eigenvalue ofA ,we have
r
1(A )
(37)
r 1
:
1(A )kA k
detA
O bserve that 1(A )= infj j= 1 tA ,w here j jis the Euclidean norm of in
R r. N ow for a 2 [0;1], we derive a uniform lower bound for the sm allest
eigenvalue of X T + (1 a)X T :
q
q
1(
X T + (a 1)(X T
t
X T ))= inf
j j= 1
X T + (a 1)(X T
X
iD
= inf
j j= 1
(38)
(X Ti + (a
1)(X Ti
X Ti ))
i
q
inf
XT)
t
j j= 1
X
XT
iD
sup
j j= 1
(X Ti
i
q
1( X T
) jD (X T
X T )j:
X Ti )
31
W EA K A PPR O X IM AT IO N O F SD ES
M oreover,we have
[
fD k
h
T
= 0g
6
det X T
4(1 + k X T k22)(r
X T )j2
jD (X T
k
:
1)=2
B ut from (36) and (37) we have
det X T
4(1+ k X T k22)(r
1( X T
1)=2
4
)
:
W e deduce that
[
fD k
h
T
= 0g
6
jD (X T
1( X T
X T )j2
4
k
N ow ifjD (X T
X T )j2
1( X T
)
:
)=4,it follow s from (39) that
q
q
1(
X T + (1 a)(X T
X T ))
1( X T
)=2
and consequently,using (37),
det
X T + (1 a)(X T
XT)
Finally we obtain for 0
[
(39)
fD k
h
T
= 0g
6
a
1( X T
)r=4r
(det
XT
)r=(k
XT
kr(r
1) r
4 ):
1
det
aX T + (1 a)X T
k
(det X T )r
:
k X T kr(r 1)4r
S
= 0g,we have a controlofthe determ inant
T herefore on the set k fD k hT 6
of the inverse of the M alliavin m atrix of aX T + (1 a)X T by the random
variable
k X T kr(r 1)
(det X T )r .
A know ledgm ents. T his research originated from discussions w ith V lad
B ally.T he authors are gratefulfor his specialcontribution and encouragem ent.A .K ohatsu-H iga thanks the support ofthe M ath project (IN R IA )
and the D epartm ent ofM athem atics ofthe U niversity M arne-la-Vallee for
their hospitality during various visits.
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E . C lem ent
D . Lam berton
Laboratoire d’A nalyse et de
M athem atiques A ppliquees
et P rojet M athfi
U niversite de M arne-la-Vallee
5 B ld D escartes, C ham ps-sur-m arne
77454 M arne-la-Vallee C edex 2
F rance
E -m ail:em m anuelle.clem ent@ univ-m lv.fr
A . K ohatsu-H iga
G raduate School
of E ngineering Sciences
O saka U niversity
M achikaneyam a cho 1-3
O saka 560-8531
Japan