Good Rough Path Sequences and Applications to Anticipating & Fractional Stochastic Calculus

0.35 MB·English

by  Laure Coutin

GOOD ROUGH PATH SEQUENCES AND APPLICATIONS TO
ANTICIPATING & FRACTIONAL STOCHASTIC CALCULUS
5
0
0
LAURECOUTIN,PETERFRIZ,NICOLASVICTOIR
2

n
Abstract. WeconsideranticipativeStratonovichstochasticdifferentialequa-
a
J tions driven by some stochastic process (not necessarily a semi-martingale).
No adaptedness of initial point or vector fields is assumed. Under a simple
3
condition on the stochastic process, we show that the unique solution of the
1
above SDE understoodinthe roughpath senseisactuallyaStratonovich so-

lution. This condition issatisfied bythe Brownianmotion andthe fractional
]
R BrownianmotionwithHurstparameter greaterthan 1/4. Asapplication, we
obtainratherflexibleresultssuchassupporttheorems,largedeviationprinci-
P
plesandWong-Zakai approximationsforSDEsdrivenbyfractionalBrownian
h. Motionalonganticipatingvectorfields. Inparticular,thisunifiesmanyresults
t onanticipative SDEs.
a
m
[

1 1. Introduction
v
7 We fix a filtered probabilityspace Ω,F,P,(Ft)0 t 1 satisfying the usualcon-
9 ditions. Itoˆ’s theory tells us that ther(cid:16)e exists a uniqu≤e≤so(cid:17)lutionto the Stratonovich
1
stochastic differential equation
1
0 dY =V (Y )dt+ d V (Y ) dBi
5 (1.1) t 0 t i=1 i t ◦ t
Y =y ,
0 (cid:26) 0 0 P
/ where B = B1, ,Bd is a standard d-dimensional ( )-Brownian motion,
h t ··· t 0 t 1 Ft
t V a C1 vector field on R≤d,≤V , ,V some C2 vector fields on Rd, and y a -
a 0 (cid:0) (cid:1) 1 ··· d 0 F0
m measurable random variable.
We remind the reader that, by definition (see [25]), a process z is Stratonovich
:
v integrable with respect to a process x if for all t there exists a random variable
i t
X denoted 0 zu◦dxu (theStratonovichintegralofz withrespecttoxbetweentime0
r andt)suRchthat for allsequences Dn =(tni)i n 0ofsubdivisions of[0,t]suchthat
a Dn 0, the following convergence holds i≥n probability
| |→n→∞ (cid:0) (cid:1)
1 tni+1 t
Xi tni+1−tni Ztni zudu!(cid:16)xtni+1 −xtni(cid:17)→n→∞ Z0 zu◦dxu.
Another wayto expressthis is by introducing xD the D-linear approximationof x,
where D =(t ) is a subdivision of [0,1]:
i
t t
(1.2) xD(t)=x + − i x x if t t t .
ti t t ti+1 − ti i ≤ ≤ i+1
i+1 i
−
(cid:0) (cid:1)
Then,z isStratonovichintegrablewith respecttox ifandonly iffor allsequences
of subdivisions Dn which mesh size tends to 0 and for all t [0,1], tz dxDn(u)
∈ 0 u
t
converges in probability to z dx .
0 u◦ u R
1
R
2 LAURECOUTIN,PETERFRIZ,NICOLASVICTOIR
Ocone and Pardoux [26] showed that there exists a unique solution to equation
(1.1)evenifthevectorfieldV andtheinitialconditiony wereallowedtobe( )-
0 0 1
F
randomvariables. TheydidsorelatingtheSkorokhodintegralandtheStratonovich
one, and using Malliavin Calculus techniques. This solution has been studied in
various directions: existence and study of the density of Y [2, 28, 30], a Freidlin-
Wentzell’s type theorem [23], results on the support of the law of Y ,0 t 1
t
{ ≤ ≤ }
[3, 22], approximation of Y by some Euler’s type schemes [1]... The case where
t
the vectorfields V , ,V areallowedto be ( )-measurablewasdealtin[12,13],
1 d 1
··· F
under the strong condition that the (V ) commute. This is an application of
i 1 i d
the Doss-Sussman theorem. The latter sa≤ys≤that if V = (V , ,V ) are d vector
1 d
···
fields smooth enough such that [V ,V ] = 0, for all i,j 1, and if V is another
i j 0
vectorfields,thenthemapϕy0 whichatasmoothpath≥x:[0,1] Rdassociates
(V0,V) →
the path y which is the solution of the differential equation
dy =V (y )dt+ d V (y )dxi
(1.3) t 0 t i=1 i t t
y =y ,
(cid:26) 0 0 P
is continuous when one equips the space of continuous functions with the uniform
topology. One can then define ϕy0 on the whole space of continuous function,
(V0,V)
in particular ϕy0 (B) is then well defined, and is almost-surely the solution of
(V0,V)
the Stratonovich differential equation (1.1). This remains true even if the vector
fields and the initial condition are allowed to be random.
RoughpaththeorycanbeseenasamajorextensionoftheDoss-Sussmanresult.
Oneofthemainthingtorememberfromthistheoryisthatitisnotxwhichcontrols
the differential equation (1.3), but the lift of x to a path in a Lie group lying over
Rd. The choice of the Lie group depends on the roughness of x.
If x is a Rd-valued path of finite p-variation,p 1, one needs to lift x to a path
x with values in G[p] Rd , the free nilpotent Lie≥group of step [p] over Rd. When
x is smooth, there exists a canonical lift of x to a path denoted S(x) with values
(cid:0) (cid:1)
in G[p] Rd (S(x) is obtain from x by computing the ”first [p]” iterated integrals
of x). If x is a smooth path, then there exists a solution y to the differential
equatio(cid:0)n (1(cid:1).3). We denote by Iy0 the map which at S(x) associates S(x y).
(V0,V) ⊕
Denoting C G[p] Rd the set of continuous paths from [0,1] into G[p] Rd , we
see that Iy0 is a map from a subset of C G[p] Rd to C G[p] Rd Rn .
(V0(cid:0),V) (cid:0) (cid:1)(cid:1) (cid:0)⊕ (cid:1)
Lyons showed that this map is (locally uniformly) continuous when one equips
(cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:0) (cid:1)(cid:1)
C G[p] Rd and C G[p] Rd Rn with a ”p-variation distance”. Hence one
can define Iy0 on the closur⊕e (in this p-variation topology) of the canonical lift
(cid:0) (cid:0) (cid:1)((cid:1)V0,V) (cid:0) (cid:0) (cid:1)(cid:1)
of smooth paths. This latter set is the set of geometric p-rough paths.
Inthecasex=B,theBrownianmotionbeingalmostsurely1/p-H¨older,2<p<
3, one needs to lift B to a process with values in G2(Rd) to obtain the solution in
theroughpathsenseofequation(1.1). Thisisequivalenttodefineitsareaprocess.
The standard choice for the area process of the Brownian motion is the L´evy area
[18,20],althoughonecouldchooseverydifferentareaprocesses[16]. Choosingthis
area,we lift B to a geometric p-roughpath B, Iy0 (B) is then the Stratonovich
(V0,V)
solutionof the stochastic differential equation (1.1), together with its lift (i.e. here
its area process).
GOOD ROUGH PATH SEQUENCES 3
Justasbefore,whenthe vectorfields V arealmostsurely”smoothenough”and
i
y is almost surely finite, the Itoˆ map Iy0 is still well defined and continuous
0 (V0,V)
almostsurely,andthere is no problematall ofdefinition of Iy0 (B). Therefore,
(V0,V)
thetheoryofroughpathprovidesameaningandauniquesolutiontothestochastic
differential equation (1.1), even when the vector fields and the initial condition
depend on the whole Brownian path. Moreover, the continuity of the Itoˆ map
provides for free a Wong-Zakai theorem, and is very well adapted to obtaining
large deviation principles and support theorems.
The only work not completely given for free by the theory of rough path is to
provethatthesolutionyofequation(1.1)usingtheroughpathapproachisactually
solution of the Stratonovich differential equation, i.e. that for all t,
t d t
y =y + V (y )du+ V (y ) dBi.
t 0 0 u i u ◦ u
Z0 i=1Z0
X
When the vector fields and the initial condition are deterministic, this is usually
proved using the standard Wong-Zakai theorem.
We provide in this general case here a solution typically in the spirit of rough
path, by separating neatly probability theory and differential equation theory. We
will show via a deterministic argument that to obtain our result we only need to
checkthat,ifDn isasequenceofsubdivisionswhichstepstendsto0, .B dBDn
and .BDn dBDn converges in an appropriate topology to .B 0dBu⊗. u
0 u ⊗ u 0 u⊗R◦ u
The paper is organized as follow: in the first section, we present quickly the
R R
theory of rough path (for a more complete presentation, see [19, 20] or [15]). The
secondsectionintroducethenotionofgoodroughpathsequenceanditsproperties.
We will then show that Bn defines a good rough path sequence associated to B,
and this will imply that the solution via rough path of equation (1.3) with signal
B is indeed solution of the Stratonovich stochastic differential equation (1.1).
We conclude with a few applications: an approximation/Wong-Zakai result, a
large deviation principle, and some remarks on the support theorem.
2. Rough Paths
Bypathwewillalwaysmeanacontinuousfunctionfrom[0,1]intoa(Lie)group.
If x is such a path, x is a notation for x 1.x .
s,t −s t
2.1. Algebraicpreliminaries. Wepresentthetheoryofroughpaths,inthefinite
dimensionalcasepurelyforsimplicity. Allargumentsarevalidininfinitedimension.
We equip Rd with the Euclideanscalarproduct .,. andthe Euclideannorm x =
h i | |
x,x 1/2. We denote by Gn(Rd), the free nilpotent of step n over Rd, which
h i ⊗
is imbedded in the tensor(cid:0)algebra Tn(cid:1)(Rd) = nk=0 Rd ⊗k. We define a family of
dilations on the group Gn(Rd) by the formula
L (cid:0) (cid:1)
δ (1,x , ,x )=(1,λx , ,λnx ),
λ 1 n 1 n
··· ···
where xi Rd ⊗i, (1,x1, ,xn) Gn(Rd) and λ R. Inverse, exponential and
∈ ··· ∈ ∈
logarithm functions on Tn(Rd) are defined by mean of their power series [19, 27].
(cid:0) (cid:1)
4 LAURECOUTIN,PETERFRIZ,NICOLASVICTOIR
We define on Rd ⊗k the Hilbert tensor scalar product and its norm
n (cid:0) (cid:1) n
x1 xk, y1 yk = x1,y1 xk,yk ,
* i ⊗···⊗ i j ⊗···⊗ i+ i j ··· i j
i=1 j=1 i,j
X X X(cid:10) (cid:11) (cid:10) (cid:11)
x = x,x 1/2.
| | h i
ThisyieldsafamilyofcomptibletensornormsonRd anditstensorproductspaces.
Since all finite dimensional norms are equivalent the Hilbert structure of Rd was
only used for convenience. In fact, one can replace Rd by a Banach-spaceand deal
with suited tensor norms but this can be rather subtle, see [20] and the references
therein.
For (1,x1, ,xn) Gn(Rd), with xi Rd ⊗i, we define
··· ∈ ∈
(cid:0) (cid:1)
(1,x , ,x ) = max (i! x )1/i .
1 n i
k ··· k i=1, ,n | |
··· n o
. is then a symmetric sub-additive homogeneous norm on Gn(Rd) ( g = 0 iff
kgk= 1, and δ g = λ . g for all (λ,g) R Gn(Rd), for all g,hk kGn(Rd),
λ
k k | | k k ∈ × ∈
g h g + h , and g 1 = g ). This homogeneous norm allows us to
−
k ⊗ k ≤ k k k k k k
define on Gn(Rd) a left-invariant distance with the formula
(cid:13) (cid:13)
(cid:13) (cid:13)
d(g,h)= g−1 h .
⊗
Fromthisdistancewecandefinesome(cid:13)distances(cid:13)onthespaceofcontinuouspaths
(cid:13) (cid:13)
from [0,1] into G[p](Rd) (p is a fixed real greater than or equal to 1):
(1) the p-variation distance
1/p
p
d (x,y)=d(x ,y )+sup d x ,y
p−var 0 0 ti,ti+1 ti,ti+1 !
i
X (cid:0) (cid:1)
where the supremum is over all subdivision (t ) of [0,1]. We also define
i i
x = d (1,x), and x < means that x has finite p-
vkarkipa−tivoanr. p−var k kp−var ∞
(2) modulus distances: We say that ω : (s,t),0 s t 1 R+ is a
{ ≤ ≤ ≤ } →
control if
ω is continuous.
ω is super-additive, i.e. s<t<u, ω(s,t)+ω(t,u) ω(t,u).
 ∀ ≤
ω is zero on the diagonal, i.e. ω(t,t)=0 for all t [0,1]

∈
Whenω is non-zero off the diagonal we introduce the distance dω,p by
d(x ,y )
s,t s,t
d (x,y)=d(x ,y )+ sup .
ω,p 0 0 ω(s,t)1/p
0 s<t 1
≤ ≤
Thesimplestexampleofsuchacontrolis givenbyω(s,t)=t s. Inthis case,d
ω,p
defines then a notion of 1/p-H¨older distance on Gn(Rd)-valu−ed paths. In general,
if x =d (1,x)< , we say that x has finite p-variation controlled by ω.
k kω,p ω,p ∞
GOOD ROUGH PATH SEQUENCES 5
2.2. Gn(Rd)-Valued Paths. Let x : [0,1] Rd be a path of bounded variation
→
and define S(x) to be the solution of the ordinary differential equation
dS(x) = S(x) dx
t t t
⊗
S(x) = exp(x ),
0 0
where isthemultiplicationofTn(Rd)andexpistheexponentialonTn(Rd). S(x)
actuall⊗y takes its values in Gn(Rd). Moreover, if the 1-variation of x is controlled
by ω, then so is the 1-variationof S(x) [19, Theorem 1].
Example 1. If n = 2, S(x) = exp x(t)+ 1 tx dx 1 tdx x . The
t 2 0 u⊗ u− 2 0 u⊗ u
term 12 0txu⊗dxu− 12 0tdxu⊗xu i(cid:16)s the areaRbetween the lineRjoining x0(cid:17)and xt
and the path (x ) .
R u 0 u t R
≤ ≤
Definition 1. A path x : [0,1] G[p](Rd) is a geometric p-rough path if there
→
exists a sequence of paths of bounded variation (x ) such that
n n N
∈
lim d (S(x ),x)=0.
p var n
n −
→∞
Notation 1. Let C0,p var G[p](Rd) denote the space of geometric p-rough paths,
−
and C0,ω,p G[p](Rd) the set of paths x:[0,1] G[p](Rd) for which there exists a
(cid:0) (cid:1) →
sequence of paths (x ) of bounded variation such that
(cid:0) n(cid:1) n N
∈
lim d (S(x ),x)=0.
ω,p n
n
→∞
In particular, any G[p](Rd)-valued path with finite q-variation, q < p, is a geo-
metric p-roughpath, and a geometricp-roughpath has finite p-variation. We refer
to [11] for a ”Polish” study of the space of geometric rough paths.
If x is a geometric p-rough path, we denote by xi the projection of x onto
s,t
Rd ⊗i. We also say that x lies above x1.
(cid:0) (cid:1)
2.3. The Itˆo map. We fix p 1, and a control ω.
When A is vector field on≥Rd, we denote by dkA its kth derivative (with the
convention d0A=A), and its γ-Lipschitz norm by
A =max max dkA , d[γ]A ,
wh,ewreeks.aky∞thisattkhAeksiLusipap(nγLo)irpm(γa)n-vde(cid:26)kc.tkko=βr0,tfi·h·e·e,l[dγβs]-(cid:13)(cid:13)Hon¨olRd(cid:13)(cid:13)edr.∞no(cid:13)(cid:13)(cid:13)rm,0≤(cid:13)(cid:13)(cid:13)γβ−[γ<](cid:27)1. IfkAkLip(γ) <
∞We consider V =(V , ,V ), where the V are Lip(p+ε)-vector fields on Rn,
1 d i
ε>0. V can be identified··w· ith a linear map from Rd into Lip(p+ε)-vector fields
on Rn,
d
V(y)(dx1, ,dxd)= V (y)dxi.
i
···
i=1
X
For a Rd-valued path x of bounded variation, we define y to be the solution of the
ordinary differential equation
dy =V(y )dx
(2.1) t t t
y =y .
0 0
(cid:26)
6 LAURECOUTIN,PETERFRIZ,NICOLASVICTOIR
Liftingxand(x y)toS(x)andS(x y)(theircanonicallifttopathswithvalues
⊕ ⊕
inthefreenilpotentgroupofstepp),weconsiderthemapwhichatS(x)associates
S(x y). We denote it I . We refer to [19, 20] for the following theorem:
⊕ y0,V
Theorem 1 (Universal Limit Theorem (Lyons)). The map I is continuous
y0,V
from C0,ω,p G[p](Rd) ,d into C0,ω,p G[p](Rd Rn) ,d .
ω,p ω,p
⊕
Let(cid:0)x be(cid:0)asequenc(cid:1)e ofpa(cid:1)thofb(cid:0)ounded(cid:0)variationsuch(cid:1)thatS(cid:1)(x )convergesin
n n
the d -topology to a geometric p-rough path x, and define y to be the solution
ω,p n
of the differential equation (2.1), where x is replaced by x . Then, the Universal
n
Limit Theorem says that S(x y ) convergesin the d -topology to a geometric
n n ω,p
⊕
p-roughpath z. We say that y, the projectionof z onto G[p](Rn) is the solution of
the rough differential equation
dy=V(y)dx
with initial condition y . It is interesting to observethat Lyons’estimates actually
0
give that for all R > 0 and sequence x C0,ω,p G[p](Rd Rn) converging to
n
∈ ⊕
x C0,ω,p G[p](Rd Rn) in the d -topology,
∈ ⊕ ω,p (cid:0) (cid:1)
(2.2) (cid:0) |ys0u|≤pR(cid:1) dω,p(Iy0,V (xn),Iy0,V (x))→n→∞ 0.
V R
k kLip(p+ε)≤
Toseethis,one firstobservesthatthe continuityofintegrationalongofa one-form
is uniform over the set of one-forms with Lipschitz norm bounded by a given R.
Then, the path I (x) over small time is the fixed point of a map (integrating
y0,V
along a one-form) which is a contraction. Reading the estimate in [20], one sees
thatthismapisuniformly,overtheset y R; V R ,acontraction
| 0|≤ k kLip(p+ε) ≤
with parameter strictly less than 1. n o
The next theorem was also proved in [20], and deals with the continuity of the
flow.
Theorem 2. If (yn) is a Rd-valued sequence converging to y , then for all R>0,
0 n 0
sup dω,p Iy0n,V (x),Iy0,V (x) →n→∞ 0.
x R
k kω,p≤ (cid:0) (cid:1)
V R
k kLip(p+ε)≤
The next theorem shows that the Itoˆ map it continuous when one varies the
vector fields defining the differential equation. It does not seem to have appeared
anywhere, despite that its proof does not involve any new ideas.
Theorem 3. Let Vn = (Vn, ,Vn) be a sequence of d Lip(p+ε)-vector fields
on Rn such that 1 ··· d
lim max Vn V =0.
n i k i − ikLip(p+ε)
→∞
Then, for all R>0,
lim sup d (I (x),I (x))=0.
n→∞kxkω,p≤R ω,p y0,Vn y0,V
|yo|≤R
Proof. We use the notations of [19]. First consider the Lip(p+ε 1)-one-forms
θ :Rd Hom Rd,Rn , i N . We assume that θ converge−s to θ in the
i n
→ ∈ ∪{∞} ∞
(cid:0) (cid:1)
GOOD ROUGH PATH SEQUENCES 7
(p+ε 1)-Lipschitz topology when n . For n N , θ (x)dx is the
n
− → ∞ ∈ ∪{∞}
unique rough path associated to the almost multiplicative functional
R
[p] 1
i −
Z(x)ns,t = dl1θn x1s ⊗···⊗dliθn x1s  π xis+,tl1+···+li 
(cid:16) (cid:17) l1,·X··,li=0 (cid:0) (cid:1) (cid:0) (cid:1)π∈Π(Xl1,···,li) (cid:16) (cid:17)
 
(see [19] for the definition of Π). It is obvious that
i i
Z(x)n Z(x)∞
s,t − s,t
xsup Rmiax(cid:12)(cid:12)(cid:12)(cid:16) ω(cid:17)(s,t)(cid:16)i/p (cid:17) (cid:12)(cid:12)(cid:12) →n→∞ 0,
k kω,p≤ (cid:12) (cid:12)
which implies by theorem 3.1.2 in [19] that
sup d θ (x)dx, θ (x)dx 0.
ω,p n n
∞ → →∞
kxkω,p≤R (cid:18)Z Z (cid:19)
Now consider the Picard iteration sequence (zn) introduced in [19, formula
m m 0
(4.10)] to construct to I (x): ≥
y0,Vn
zn = (0,y )
0 0
zn = h (zn)dzn,
m+1 n m m
Z
whereh istheone-formdefinedbytheformulah (x,y)(dX,dY)=(dX,V (y)dX)
n n
(by V(y)dX we mean V (y)dXi). Lyons proved that
i
lim sup supd (zn,zn )=0;
P ω,p m
m→∞kxkω,p≤R n ∞
|yo|≤R
we have the supremum over all n here because the p +ε-Lipschitz norm of the
Vn are uniformly bounded in n. Moreover, we have just seen that for all fixed
m, lim sup d (zn,z ) = 0. Therefore, with a 3ε-type argument, we
obtainno→ur∞theo|rye0m|≤.R ω,p m ∞m (cid:3)
The three previous theorems actually gives that the map
(y ,V,x) I (x)
0 → y0,V
iscontinuousintheproducttopologyRn (Lip(p+ǫ) on Rn)d C0,ω,p G[p](Rd) .
× ×
In the reminder of this section, p is a real in [2,3). Now consider a Lip(1+ε)-
vector field on Rn denoted V , and for a path x of bounded variation, w(cid:0)e consid(cid:1)er
0
y to be the solution of
dy =V (y )dt+V(y )dx
(2.3) t 0 t t t
y =y ,
t 0
(cid:26)
Lifting x and (x y) to S(x) and S(x y) we consider the map which at S(x)
⊕ ⊕
associatesS(x,y). WedenoteitI . ThefollowingextensionoftheUniversal
y0,(V0,V)
Limit Theorem was obtained in [17].
Theorem 4. The map I is continuous from C0,ω,p G[p](Rd) ,d into
y0,(V0,V) ω,p
C0,ω,p G[p](Rd Rn) ,d . More precisely, for all R>0,
⊕ ω,p (cid:0) (cid:0) (cid:1) (cid:1)
((cid:0)2.4) (cid:0) V |ys0u|≤pR (cid:1)Rdω,p(cid:1)(cid:0)Iy0,(V0,V)(xn),Iy0,(V0,V)(x)(cid:1)→n→∞ 0.
k kLip(p+ε)≤
8 LAURECOUTIN,PETERFRIZ,NICOLASVICTOIR
Letx beasequenceofpathsofboundedvariationsuchthatS(x )convergesin
n n
thed -topologytoageometricp-roughpathx,anddefiney tobethesolutionof
ω,p n
equation(2.3)replacingx by x . Then, the previoustheoremsaysthat S(x y )
n n n
⊕
converges in the d -topology to a geometric p-rough path z. We say that y, the
ω,p
projection of z onto G[p](Rn) is the solution of the rough differential equation
dy =V (y )dt+V(y )dx
t 0 t t t
with initial condition y .
0
We obtain, as before, the following two theorems:
Theorem 5. If (yn) is a Rd-valued sequence converging to y , then
0 n 0
sup d I (x),I (x) 0.
x R ω,p y0n,(V0,V) y0,(V0,V) →n→∞
Vk kω,p≤ R (cid:0) (cid:1)
k kLip(p+ε)≤
kV0kLip(1+ε)≤R
Theorem 6. Let (Vn =(Vn, ,Vn)) be a sequence of d Lip(p+ε)-vector
1 ··· d n 0
fields on Rn and (Vn) a sequence of L≥ip(1+ε)-vector fields on Rn, such that
0 n 0
≥
lim max Vn V , max Vn V =0.
n k 0 − 0kLip(1+ε) 1 i dk i − ikLip(p+ε)
→∞ (cid:26) ≤≤ (cid:27)
Then, if x C0,ω,p G[p](Rd) ,
∈
lim (cid:0) sup d(cid:1) I (x),I (x) =0.
n→∞kxkω,p≤R ω,p(cid:16) y0,(V0n,Vn) y0,(V0,V) (cid:17)
|yo|≤R
2.4. Solving Anticipative Stochastic Differential Equations Via Rough
Paths. We fix a p (2,3) and, for simplicity, the control ω(s,t) = t s (i.e. we
∈ −
deal with Ho¨lder topologies), although we could have been more general and have
considered a wide class of controls as in [10] (i.e. we could have consider modulus
type topologies). We define B the Stratonovichlift to a geometric p-roughpath of
the Brownian motion B with the formula
t
B = 1,B , B dB .
t t u u
⊗◦
(cid:18) Z0 (cid:19)
B is a G2 Rd -valued path, and almost surely, B < .
k kω,p ∞
Consider V a random vector field on Rn almost surely in Lip(1+ε), i.e. a
(cid:0) (cid:1)0
measurable map from
V :Ω Rn Rn
0
× →
suchthatV (ω,.) Lip(1+ε)forω inasetafullmeasure,andV =(V , ,V ),
0 1 d
where V , ,V a∈re random vector fields on Rn almost surely in Lip(2+··ε·), and
1 d
a random·v·a·riable y Rn finite almost surely.
0
I (B) is the∈n almost surely well defined, and its projection onto G2(Rn)
y0,(V0,V)
is the solution, in the rough path sense, of the anticipative stochastic differential
equation
(2.5) dy =V (y )dt+V(y )dB
t 0 t t t
with initial condition y .
0
The next section introduces the notion of good rough paths sequence, and its
properties. Showing that linear approximations of Brownian motion form good
rough path sequences (in some sense that will be precise later on) will prove that
GOOD ROUGH PATH SEQUENCES 9
y1 issolutionofthe anticipativeStratonovichstochasticdifferentialequation(1.1).
In particular, the solution that we construct coincides with the one constructed in
[26].
3. Good Rough Path Sequence
3.1. Definitions. We fix a parameter p > 2, and a control ω. Rd and Rd will
denote two identical copies of Rd.
Let p > 2, and q such that 1/p+1/q > 1. We consider x and y two Rd-fvalued
paths of bounded variation. We let y=S(y) to be the canonical lift of y to a
G[p] Rd -valued path. We let
(3.1)(cid:0) (cid:1) S′(x,S(y)):=S(x y)
⊕
be the canonical lift of x y to a G[p] Rd Rd -valued path and
⊕ ⊕
(cid:16) (cid:17)
(3.2) S (S(x)):=S (x,S(x))=S(x x)
′′ ′
⊕
be the canonical lift of y y to A G[p] Rd Rd -valued path.
⊕ ⊕
(cid:16) (cid:17)
Proposition 1. Let x be a Rd-valued path of finite q-variation, and y a G[p] Rd -
valued path of finite p-variation. Let (x ,y ) be a sequence of Rd Rd-valued path
n n ⊕ (cid:0) (cid:1)
such that d (x ,x) 0 and d (S(y ),y) 0. Then,
ω,p n n ω,p n n
→ →∞ → →∞
(i) S (x ,S(y )) converges in the d -topology, and the limit is independent of the
′ n n ω,p
choice of the sequence (x ,y ). We denote this limit element S (x,y).
n n ′
(ii) S′′(S(y ),S(y )) converges in d -topology, and the limit is independent of
n n ω,p
the choice of the sequence y . We denote this elememt S (y,y).
n ′′
Proof. This is simply obtained using theorem 3.1.2 in [19], which says that the
procedure which at an almost multiplicative functional associates a rough path is
continuous, and we leave the details to the reader. (cid:3)
Example 2. If 2 p<3,
≤
S (x,y)
′ t
t t t
= 1,x y1, x dx x dy1 y1 dx y2 .
t⊕ t u⊗ u⊕ u⊗ u⊕ u⊗ u⊕ t
(cid:18) Z0 Z0 Z0 (cid:19)
The three integrals are well defined Young’s integrals [33].
We introduce the notion of a good p-rough path sequence.
Definition2. Let(x ) beasequenceofRd-valuedpathsofboundedvariation, and
n n
x a geometric p-rough path. We say that (x ) is a good p-rough path sequence
n n N
(associated to x) (for the control ω) if and only∈if
(3.3) lim d (S (x ,x),S (x))=0.
ω,p ′ n ′′
n
→∞
In particular, if (x ) is a good p-rough path sequence associated to x, for the
n n
control ω, x converges to x in the topology induced by d .
n ω,p
Proposition 2. Assume 2 p < 3. The sequence (x(n)) of paths of bounded
≤ n
variation is good rough path sequence associated to x, for the control ω, if and only
10 LAURECOUTIN,PETERFRIZ,NICOLASVICTOIR
if
x(n) x1
lim sup s,t− s,t = 0,
n→∞0≤s<t≤1(cid:12)(cid:12) ω(s,t)1/p (cid:12)(cid:12)
tx(n) dx(n) x2
s s,u⊗ u− s,t
lim sup = 0,
n→∞0≤s<t≤1(cid:12)(cid:12)(cid:12)R ω(s,t)2/p (cid:12)(cid:12)(cid:12)
tx1 dx(n) x2
s s,u⊗ u− s,t
lim sup = 0.
n→∞0≤s<t≤1(cid:12)(cid:12)(cid:12)R ω(s,t)2/p (cid:12)(cid:12)(cid:12)
Proof. d (S (x(n),x),S (x,x)) is bounded by a constant times
ω,p ′ ′′
max An, An, An, An ,
1 2 3 4
where n p p p o
x1 x(n)
An = sup s,t− s,t ,
1 ω(s,t)1/p
0 s<t 1(cid:12) (cid:12)
≤ ≤ (cid:12) (cid:12)
tx(n) dx(n) x2
An = sup s s,u⊗ u− s,t ,
2 0 s<t 1(cid:12)(cid:12)R ω(s,t)2/p (cid:12)(cid:12)
≤ ≤ (cid:12) (cid:12)
tx(n) dx1 x2
An = sup s s,u⊗ u− s,t ,
3 0 s<t 1(cid:12)(cid:12)R ω(s,t)2/p (cid:12)(cid:12)
≤ ≤ (cid:12) (cid:12)
tx1 dx(n) x2
An = sup s s,u⊗ u− s,t .
4 0 s<t 1(cid:12)(cid:12)R ω(s,t)2/p (cid:12)(cid:12)
≤ ≤ (cid:12) (cid:12)
Let π be the linear operator from Rd Rd onto itself defined by π(x y) =
⊗ ⊗
π(y x). Observethatπ x2 = x1 ⊗2 x2 . Also,wehaveforallz Rd Rd,
⊗ s,t s,t − s,t ∈ ⊗
z = π(z) . Using this property and an integration by part, we see that
| | | | (cid:0) (cid:1) (cid:0) (cid:1)
t
x1 dx(n) x2
s,u⊗ u− s,t
(cid:12)Zs (cid:12)
(cid:12) t (cid:12)
= (cid:12)(cid:12)x1s,t⊗x(n)s,t− dx1s,u(cid:12)(cid:12) ⊗x(n)u−x2s,t
(cid:12) Zs (cid:12)
(cid:12) t (cid:12)
= (cid:12)(cid:12)x1s,t⊗ x(n)s,t−x1s,t − dx1s,u⊗x(n(cid:12)(cid:12))u+ x1s,t ⊗2−x2s,t
(cid:12)(cid:12) (cid:0) (cid:1) Zs t (cid:16)(cid:0) (cid:1) (cid:17)(cid:12)(cid:12)
≤ (cid:12)(cid:12)x1s,t . x(n)s,t−x1s,t + x2s,t− x(n)s,u⊗dx1u . (cid:12)(cid:12)
(cid:12) Zs (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Hence, An (cid:12) x (cid:12) (cid:12).An+An, wh(cid:12)ich(cid:12)proves the proposition.(cid:12) (cid:3)
4 ≤k kω,p 1 3 (cid:12) (cid:12)
By definition of a geometric p-rough path, there always exists a sequence of
smooth x such that S(x ) converges to x in the topology induced by d . How-
n n ω,p
ever, this does not imply that equation (3.3) holds (or equivalently for [p] = 2 :
that the conditions in proposition2 hold). We now give anexample ofa geometric
rough path x for which there exists no good sequence associated to it.