Errata - Friz, Victoir: Multidimensional stochastic processes as rough paths
Last modi…ed: 16. February 2011
Errata Chapter 5.2
p 81. line 18. There is a small gap in the estimate. The left hand side [of line 18] must include
(since t 2
= D in general!) an additional term ( );
X
p
d xti ; xti+1 + ( ) + ! x;p t; t + h) > ! x;p (s; t + h) "
ti 2D1 =(s=t0 <t1 <
where ( ) = d xtk ; xtk+1
p
<tk
1)
p
= d (xtk ; xt ) + d xtk ; xtk+1
p
p
p
d (xtk ; xt ) . Hence,
p
! x;p (s; t) + d xtk ; xtk+1
d (xtk ; xt ) + ! x;p (t; t + h) > ! x;p (s; t + h)
{z
}
|
"
!0 as h!0
(since tk+1 2 [t; t + h] ! t as h ! 0, and x is (uniformly) continuous on [0; T ]). The rest of the
argument is unchanged. (We also note that the proof that ! (t; t+) = 0 can be a bit simpli…ed; e.g.
along the "outer continuity" argument in [2, page 12].)
p 91. line 1, include the word "with "after conclude
Errata Chapter 5.5
p 105. De…nition 5.50. We need to modify the de…nition of jf jp-var to1
jf jp-var;[s;t]
[u;v]
:=
X
sup
2P([s;t] [u;v])
A2
p
jf (A)j
!1=p
;
2
a partition
of a rectangle R [0; T ] is a …nite set of (closed) rectangles, essentially disjoint,
whose union is R; the family of all such partitions is denoted by P (R). We then maintain the
2
de…nition that C p-var [0; T ] ; Rd denotes the space of continuous f with jf jp-var;[0;T ]2 < 1 and
say that any such f has …nite controlled p-variation. Indeed, lemma 5.52 (which is correct with
1 Recall
that f (A) is the rectangular increment of A = ((a; b) ; (c; d)) 2
f
a; b
c; d
b
d
:= f
and we regard A as (closed) rectangle A
a
c
+f
f
a
d
T,
T
f
b
c
;
[0; T ]2 ,
A :=
a; b
c; d
:= [a; b]
[c; d] :
If a = b or c = d we call A degenerate; recall also that two rectangles are called essentially disjoint if their
intersection is empty or degenerate.
1
p
this modi…ed de…nition, see below) asserts that ! (R) := jf jp-var;R is a 2D control function such
that (obviously)
2
p
8R [0; T ] : jf (R)j
! (R) .
Any continuous f which satis…es the above estimate for some 2D control ! is said to have …nite
p-variation controlled (equivalently: dominated) by !; this is a quantitative way of saying that
2
f 2 C p-var [0; T ] ; Rd since super-addivity immediately gives
8R
2
p
[0; T ] : jf (R)j
! (R) =) jf jp-var;R
! (R) ;
cf. part (ii) of the corrected lemma 5.52 below. We remark that the di¤erence between this de…nition
of controlled p-variation and our original one is that, in our original de…nition, the supremum is
restricted to grid-like partitions,
ti ; ti+1
t0j ; t0j+1
:1
i
m; 1
j
where D = (ti : 1 i m) 2 D ([s; t]) and D0 = t0j : 1
continuous f such that2
Vp (f ; [s; t]
0
[u; v]) := @
sup
(ti )2D([s;t]);(t0j )2D([u;v])
j
X
i;j
2
n ;
f
n 2 D ([u; v]); i.e we consider
ti ; ti+1
t0j ; t0j+1
2
Clearly, not every partition is grid-like (consider e.g. [0; 2] = [0; 1] [ [1; 2]
hence
2
8R [0; T ] : Vp (f ; R) jf jp-var;R :
p
1 p1
A < 1:
[0; 1] [ [0; 2]
[1; 2])
p
The trouble is that Vp (f ; ) is not super-additive in 2D sense3 , hence not a 2D control, whereas
p
jf jp-var; based on all partitions does yield a 2D control; hence our modi…ed de…nition 5.50. Even
2
so, the class of such functions remains important and we say that any f with Vp f ; [0; T ] < 1
has …nite p-variation. It is worth noting that this distinction is not seen when p = 1 (the short
proof of this [2] is based on the idea that further re…ning of a partition to a grid-like partition can
only increase the 1-variation; this is false for p-variation, p > 1), nor in the 1D case of course, and
we are dealing with a phenomena speci…c to higher dimensional p-variation with p > 1. That said,
it is possible to show ([2] for full details) that
8R
2
[0; T ] : jf j(p+")-var;R
c (p; ") Vp (f ; R)
so that the two notions of p-variations (controlled vs. genuine) are ""-close".
p 105. De…nition 5.51 replace "which is super-additive in the sense that .... ! (R1 ) + ! (R2 )
! (R)" by the more convenient "which is super-additive in the sense that
n
X
i=1
2 The
3 We
! (Ri )
! (R) ; whenever fRi : 1
i
ng is a partition of R
notation Vp (f ) is consistent with the notation and terminology of [Towghi, 2002].
thank Bruce Driver for pointing this out by constructing an explicit counter-example.
2
p 105. Lemma 5.52, part (i) is correct as stated with the new de…nition of jf jp-var ; in part
(ii) we need to insert the word "controlled " (the statement should read "f is of …nite controlled
p
p-variation if and only if there exists a 2D control ! such that for all R : jf (R)j
! (R) :") We
include the proof.
Proof. Let us start with the remark that the super-addivity of property of 2D controls, cf. our
slightly modi…ed de…nition 5.51 above, immediately gives
8R
2
p
[0; T ] : jf (R)j
! (R) =) jf jp-var;R
! (R) :
p
This settles part (ii) and we focus on part (i). De…ne ! (R) := jf jp-var;R where R is a rectangle of
the form [s; t]
[u; v]
2
[0; T ] . By assumption ! (R)
2
! [0; T ]
< 1 and it is immediate from
the de…nition of jf jp-var;R that ! is zero on degenerate rectangles. We need to check super-addivity
and continuity.
Super-additivity: Assume fRi : 1 i ng constitutes a partition of R. Assume also that i is
a partition of Ri for every 1 i n. Clearly, := [ni=1 i is a partition of R and hence
n X
X
i=1 A2
i
p
jf (A)j =
Now taking the supremum over each of the
i
X
A2
p
jf (A)j
! (R)
gives the desired result.
At last, we note that for similar ideas as in the 1D case (cf. p.81) give continuity of ! is a map
from T
T ! [0; 1); details can be found in [2].
p 106. Lemma 5.54 concerning the reduction of partitions based on D D0 to partitions
based on D D is also incorrect (because we use control argument for objects which are not
controls), and should be removed. The lemma is used in Proposition 15.5 (variational regularity of
fBM covariance), the (solution) to Exercise 15.6 and lemma 15.8. In each case the conclusion can
obtained with an alternative argument; details are discussed at those places.
Errata Chapter 15.1
p 403. Lines 9,10, 18. Replace j j -var by V ( ). The comments and recalls on -variation in 2D
sense should also be extended such as to brie‡y mention controlled -varation.
2
p 406. Proposition 15.5: replace "controlled by ! H (:; :) = :::)" by "i.e. V1=(2H) RH ; [0; 1] <
1". Replace
2H
" RH 1 -var;[s;t]2 CH jt sj
so that ...."
2H
by
1
RH ; [s; t]
V 2H
2
CH jt
sj
2H
:
( )
1
; RH is of …nite (Hölder) controlled -variation."
Add: "This implies in particular that for > 2H
0
The original proof only considered D = D . The arguments for general D; D0 are similar;
nonetheless we include full details:
3
New proof: (By fractional scaling it would su¢ ce to consider [s; t] = [0; 1] in ( ) but this does
not simplify the argument which follows.) Consider D = (ti ) ; D0 = t0j 2 D [s; t]. Clearly,
31
1
2H
X
E
j
h
H
ti ;ti+1
i
H
t0j ;t0j+1
1
2H
1
2H
31
E
h
h
E
H
ti ;ti+1
H
ti ;ti+1
h
+ E
h
+ E
H
i
H
1
2H
1
2H
H
ti ;ti+1
H
H
ti ;ti+1
H
i
i
1
2H
1
2H
-var;[s;t]
(1)
-var;[s;ti ]
1
2H
i
1
2H
1
2H
-var;[ti ;ti+1 ]
1
2H
-var;[ti+1 ;t]
(2)
;
(3)
by super-additivity of (1D!) controls. The middle term (2) is estimated by
E
h
H
ti ;ti+1
H
i
1
2H
1
2H
=
-var;[ti ;ti+1 ]
X
sup
E
(sk )2D[ti ;ti+1 ] k
h
H
ti ;ti+1
H
sk ;sk+1
i
1
2H
cH jti+1 ti j ;
i
h
2H
H
where we used that [sk ; sk+1 ]
[ti ; ti+1 ] implies E H
cH jsk+1 sk j . The
ti ;ti+1 sk ;sk+1
…rst term (1) and the last term (3) are estimated by exploiting the fact that disjoint increments of
fractional Brownian motion have negative correlation when H < 1=2 (resp. zero correlation in the
Brownian case, H = 1=2); that is, E
(1) as follows;
E
h
H
ti ;ti+1
H
i
1
2H
1
2H
-var;[s;ti ]
=
H H
c;d a;b
E
2
h
1
2H
0 whenever a
H
ti ;ti+1
1
E
H
s;ti
h
i
b
h
H
ti ;ti+1
H
i
1
2H
1
2H
-var;[s;ti ]
d. We can thus estimate
1
2H
H
ti ;ti+1
H
s;ti
i
1
2H
The covariance of fractional Brownian motion gives immediately E
1
i 2H
h
H
On the other hand, [ti ; ti+1 ] [s; ti+1 ] implies E H
ti ;ti+1 s;ti
E
c
cH jti+1
+E
H
ti ;ti+1
H
ti ;ti+1
2
cH jti+1
2
1
2H
!
:
1
2H
= cH (ti+1
ti ).
ti j; hence
ti j .
As already remarked, the last term is estimated similarly. It only remains to sum up and to take
the supremum over all dissections D and D0 .
p 407. Exercise 15.6. It should be assumed that H 2 (0; 1=2] and the exercise should be
rephrased as: show that R X ,the covariance of X has …nite 1=2H-variation. More precisely, show
that there exists a constant CH such that for all s < t in [0; 1] ;
2
1
V 2H
RX ; [s; t]
CH jt
4
2H
sj
:
H
Proof. Let
leads to
be a fractional Brownian motion with Hurst parameter H. The argument that
2
E jXs;t j
and, for s
t
u
2
cH E
H
s;t
cH E
H H
s;t u;v
;
v;
jE (Xs;t Xu;v )j
(4)
1
is unchanged. To prove that the covariance of X has …nite 2H
-variation, we follow the (above)
proof of proposition 15.5 and see that we need (replace [ti ; ti+1 ] by some generic [u; v] [s; t])
1
jE [Xu;v X ]j 2H
1
-var;[s;u]
cH jv
uj
jE [Xu;v X ]j
cH jv
uj
cH jv
uj :
2H
1
2H
1
2H -var;[u;v]
1
2H
1
2H -var;[v;t]
jE [Xu;v X ]j
The …rst and third inequality follow from (4) and the corresponding fBM estimates contained in
the (above) proof of proposition 15.5. So it only remains to establish the "middle" estimate, after
renaming [u; v]
[s; t], we need, for any s < t in [0; 1]
1
cH jt
jE [Xs;t X ]j 2H
1
-var;[s;t]
2H
Let again [u; v]
sj :
[s; t]. The triangle inequality gives
2
jE (Xs;t Xu;v )j
+ jE (Xv;t Xu;v )j
jE (Xs;u Xu;v )j + E jXu;v j
cH
H
H
s;u u;v
E
2
H
u;v
+E
But using the structure of fractional Brownian motion (using H
=
cH
E
H
H
s;u u;v
+E
=
cH
E
H H
s;t u;v
+ 2E
cH E
H H
s;t u;v
H
u;v
H H
v;t u;v
E
2
H
u;v
+ 2cH E
1=2) we see that
2
H
u;v
H H
v;t u;v
+ E
2
:
Hence,for a suitable constant c~ = c~ (H) which may change from line to line,
jE (Xs;t Xu;v )j
1
2H
c~H E
H H
s;t u;v
= c~H E
H H
s;t u;v
and then
1
jE [Xs;t X ]j 2H
1
-var;[s;t]
2H
c~H E
h
5
H H
s;t
1
2H
1
2H
i
+ c~H E
+ c~H jv
1
2H
1
2H
H
u;v
-var;[s;t]
2
1
2H
uj
+ c~H jt
sj :
=:
Since E
h
H H
s;t
i
1
2H
1
2H
-var;[s;t]
= c~H jt
sj, this was seen in the (above) proof of proposition 15.5, the
exercise is now completed.
p 415. Lemma 15.8 The second claimed estimate (" RA
rephrased as4
2
2
V2 RA ; [s; t]
2
jRj2-var;[s;t]2 ") should be
2-var;[s;t]2
V2 R; [s; t]
:
2
The given proof (we may take [s; t] = [0; 1] without loss of generality) of lemma 15.8 actually
only shows this estimate when supD;D0 2D[0;1] in the de…nition of V2 is replaced by the sup over all
D = D0 2 D [0; 1]. This gap is closed by the following (new) lemma which may be interesting in its
own right.
Lemma 1 De…ne R (s; t) := E (Xs Xt ) for some stochastic process (Xt : t 2 [0; 1]). Then
sup
X
2
ti ; ti+1
t0j 0 ; t0j+1
R
D;D 0 2D[0;1] i;j
=
sup
X
D2D[0;1] i;j
R
ti ; ti+1
tj; tj+1
2
;
where we write D = (ti ) and D0 = t0j .
Proof. We only need to show "
". Set Xi = Xti ;ti+1 and Xj 0 = Xt0j0 ; t0j+1 so that
R
ti ; ti+1
t0j 0 ; t0j+1
= E (Xi Xj 0 ) :
~ so that
Consider an IID copy of X, say X,
R
ti ; ti+1
t0j; t0j+1
2
~iX
~ j 0 = E Xi Xj 0 X
~iX
~j0 :
= E (Xi Xj 0 ) E X
It follows that
X
i;j
R
ti ; ti+1
t0j; t0j+1
X
2
=
i;j
h
i
~iX
~j0
E Xi Xj 0 X
2
E4
=
X
~i
Xi X
i
!0
@
V2 R; [s; t]2 = supD;D0 2D[s;t]
P
j
13
~ j 0 A5
Xj 0 X
v 2
v 2
0
12 3
u
!2 3u
u
u
X
X
u
u 4
~ i 5uE 6
~j0 A 7
tE
Xj 0 X
Xi X
5
t 4@
i
4 Recall
X
i;j
R
ti ; ti+1
t0j 0 ; t0j+1
6
j
2
:
where we used Cauchy-Schwarz. Since
2
2
3
!2 3
h
i X
X
X
X
~i 5 = E 4
~ i Xk X
~k 5 =
~iX
~k =
E4
Xi X
R
Xi X
E [Xi Xk ] E X
i
20
12 3
6 X 0 ~ 0A 7
Xj Xj
E 4@
5
i;k
=
i;k
(as above)
=
j
X
ti ; ti+1
tk; tk+1
i;k
j;l
;
2
t0j 0 ; t0j+1
t0l; t0l+1
R
2
we see that
0
X
@
R
i;j
ti ; ti+1
t0j; t0j+1
2
12
X
A
R
i;k
X
2
ti ; ti+1
tk; tk+1
t0j 0 ; t0j+1
t0l; t0l+1
R
j;l
2
:
Since we managed to factorize the dependence on D = (ti ) 2 D [0; 1] and D0 = t0j 2 D [0; 1] on the
right-hand-side the conclusion follows immediately upon taking supD;D0 …rst on the right-hand-side,
then on the left-hand-side.
p 431-432. Exercise 15.36 is corrected as stated (in particular, under the assumption of
Theorem 15.33 concerning …nite controlled -variation of the covariance). However, if one wants
to apply this to fractional Brownian motion with H < 1=2, say, one has to work with ~-variation,
~ := 1= (2H) + ", any " > 0, rather than := 1= (2H). The conclusion in part (iii) of this exercise,
1 -variation. In
…nite 2~;~ -variation, then is not optimal: one wants (optimal) …nite 2 ; = H1 ; 2H
2
fact, one does get this result upon realizing that by fractional scaling !
~ [s; t]
In particular, (15.20) applied with !
~ then yields
p
2
C q!
~ [s; t]
jd (Xs ; Xt )jLq
…nite
2 ;
1
2~
p
= C q jt
1
~
= (const) jt
1
sj .
1
sj 2 ;
-variation of sample paths is then a standard consequence of the results in section A.4. (In
a similar spirit one can show that the assumption of …nite -variation, rather than …nite controlled
-variation, leads to 2 ; -variation of the sample paths.)
p 438. Replace " RA 2-var;[s;t]2 jRj2-var;[s;t]2 " (15.28) by
2
2
V2 RA ; [s; t]
V2 R; [s; t]
:
((15.28))
p 440, remove the text from line 7 until the end of the proof, and replace it with the following
argument.
u; v
s; u
u; v
For u < v in [s; t], de…ne Q1u;v =
; Q2u;v =
; Q3u;v =
; rewrite the
u; v
u; v
s; u
previous equation as
fu;v = RX Ac ;i Q1u;v + RX Ac ;i Q2u;v + RX Ac ;i Q3u;v :
It follows that, for " > 0 and c1 = 32+"
31
(2+")
2+"
jfu;v j
RX Ac ;i Q1u;v
1
,
2+"
+ RX Ac ;i Q2u;v
2+"
2+"
+ RX Ac ;i Q3u;v
2+"
jRX Ac ;i j(2+")-var;[u;v]2 + jRX Ac ;i j(2+")-var;[s;t]
7
2+"
2+"
[u;v]
+ jRX Ac ;i j(2+")-var;[u;v]
[s;t]
:
Since the last line is (1D) super-additive in [u; v] it follows that
2+"
jf j(2+")-var;[s;t]
2+"
32+" jRX Ac ;i j(2+")-var;[s;t]2
2
c1 V2 RX Ac ;i ; [s; t]
2
c1 V2 RX i ; [s; t]
use (5.28)
c1 jRX j2-var;[s;t]2 :
By taking " > 0 small enough, the proof is …nished with the same argument, namely the YoungWiener estimate of Proposition 15.39.
References
[1] Friz, P., Victoir, N.: Multidimensional stochastic processes as rough paths. Theory and applications. Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press,
Cambridge, 2010
[2] Friz, P., Victoir, N.: A note on higher dimensional p-variation; arXiv-preprint
8