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Quasi sure quadratic variation of local times of smooth semimartingales
Bull. Sci. math. 128 (2004) 91–104 www.elsevier.com/locate/bulsci
Quasi sure quadratic variation of local times of smooth semimartingales ✩ Kai He a , Jiagang Ren a,b,∗ a Department of Mathematics, Huazhong University of Science and Technology, Wuhan,
Hubei 430074, PR China b School of Mathematics and Computational Science, Zhongshan University, Guangzhou,
Guangdong 510275, PR China Received 10 November 2003; accepted 30 November 2003
Abstract In this paper, we prove that the process of the quadratic variation of local times of smooth an an semimartingales can be constructed as the quasi sure limit of the form ∆n (Lt i+1 − Lt i )2 , where n ) is a sequence of subdivisions of [a, b], a n = i(b − a)/2n + a, i = 0, 1, . . . , 2n . ∆n = (ain , ai+1 i 2003 Elsevier SAS. All rights reserved. MSC: 60H07; 60G44; 31C15; 31C25 Keywords: Local time; Quadratic variation; Capacity; Quasi sure convergence
1. Introduction The study of regularity of local times in the sense of Malliavin calculus traces back to the original work of Shikegawa [12] in which he proved the 1-quasi sure existence of the Brownian local time. Later Nualart and Vives in [7] changed the angle and proved the Brownian local time is in the fractional Sobolev spaces Dr2 for all r < 1/2. This result p was then improved by Watanabe in [14] by replacing Dr2 by Dr for all p > 1 and was further extended to semimartingales in [1]. In [4] a comparison theorem for capacities was established and it turned out then that Shigekawa’s quasi sure existence theorem can be deduced from Nualart–Watanabe’s result and extended to semimartingales. ✩
Work supported partially by Projects 973.
* Corresponding author.
E-mail address: [email protected], [email protected] (J. Ren). 0007-4497/$ – see front matter 2003 Elsevier SAS. All rights reserved. doi:10.1016/j.bulsci.2003.11.001
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K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
In this paper, we continue the study in this line by looking at the quasi sure properties of the quadratic variations of local times of smooth semimartingales. The quadratic variation of the local time of the 1-dimensional Brownian motion or, more generally, a continuous martingale, was studied in Bouleau and Yor [3], Perkins [8], see also [11]. Denoting by Lxt the local time of a continuous martingale, they showed that, for n any sequence of partition ∆n = (ain , ai+1 ) of [a, b] with mesh size converging to zero, lim
n→∞
an
a n 2
Lt i+1 − Lt i
∆n
b =4
Lxt dx a
in probability. In the present work we shall prove that for smooth semimartingales this convergence n ) where a n = i(b − a)/2n + a, i = can be enhanced to hold quasi surely for ∆n = (ain , ai+1 i n 0, 1, . . . , 2 . This can be considered as a refinement of the results of Bouleau, Yor and Perkins. It may happen that x → Lxt is a semimartingale. But it is utter impossible that it is a smooth semimartingale except in the trivial case (see [1, Theorem 2]). Hence the results in [10] is not applicable here. This paper is organized as follows. In Section 2, we give necessary notations and notions. In Section 3, we state and prove our main results.
2. Preliminaries Now let us recall and fix some notations and notions. We shall work on the classical Wiener space (X, F , µ; H ), where X = C0 ([0, T ] → Rd ) is the space of continuous maps from [0, T ] to Rd , null at zero; H is the usual Cameron–Martin subspace and µ the p standard Wiener measure. Denote by Dα the Sobolev space of order α and of power p p over X. The norm in Dα is defined by F p,α = (I − L)α/2 F p , where L is the Ornstein–Uhlenbeck operator. Given an open set O of X, its (p, α)-capacity is defined by Cp,α (O) = inf F p,α : F 0, F 1 µ-a.s. on O , and for any subset A ⊂ X, by Cp,α (A) = inf Cp,α (O): O open and O ⊃ A . Let {Ws = (Ws1 , . . . , Wsd ) , 0 s T } be the d-dimensional Brownian motion realized by the coordinate process on (X, F , µ; H ) and (Ft )t 0 the σ -algebra generated by {Ws , 0 s t}. Following Malliavin and Nualart [6], we give Definition 2.1. M is called a smooth semimartingale if it can be represented as: Mt = Nt + Vt
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
93
where Nt =
d
t
t j fs
j dWs
=
j =1 0
fs dWs ,
f = f 1, . . . , f d ,
0
t Vt =
gs ds, 0
with d j p fs ds + gs pp,α ds < ∞, p,α T
T
j =1 0
∀p > 1, α 0.
(1)
0
t j t By [6,10], M and its quadratic variation [M] = dj=1 0 (fs )2 ds = 0 |fs |2 ds admit ∞-modifications, and sup0t T Mt p,α < ∞, sup0t T [M]t p,α < ∞. For the smoothness in the sense of Malliavin calculus of local times of smooth semimartingales, the following is proved in [1]: Theorem 2.2. Let M be a smooth semimartingale, and suppose that there exists an (Ft )adapted process ξt with T
p
ξs p ds < ∞,
∀p > 1,
(2)
0
such that |gt | |ft | · |ξt | a.s. for Lebesgue almost all t ∈ [0, T ]. Let L be the local time of M. Then Lat ∈ Dαp for all p > 1, 0 < α < 1/2. The following lemma comes from [4, Theorem 2.23] and [11, Theorem 1.7 and Exercise 1.32, Chapter VI]: Lemma 2.3. In the same situation as in the above theorem, for every N > 0, p > 1, 0 < α < 1/2, 0 < γ < 1/2 − α, 0 r < 1, there exist positive constants C1 = C1 (N, p, α, γ ), C2 = C2 (N, p, r) such that x L − Lys C1 |t − s|γ + |x − y|γ , (3) t p,α p x y L − Ls C2 |t − s|p/2−1+r + |x − y|p/2 , (4) t p for all s, t ∈ [0, T ] and |x|, |y| N . In particular, x L < ∞ sup t p,α 0t T ,|x|N
(5)
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K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
and x L < ∞ t p
sup
(6)
0t T ,x∈R
for every p 2. We also need the following simple lemma: Lemma 2.4. (1) For every n ∈ N, p > 1, α > 0, there exists a positive constant C = C(n, p, α) such that n F C F n (7) np,α p,α np
for all F ∈ Dα . (2) For every p > 1, 0 < α < 1/2, 0 σ < 1/2 − α, there exists a constant C = C(p, α, σ ) such that σ
α
α+σ F p,α C F pα+σ F p,α+σ
(8)
p
for all F ∈ Dα+σ ; and for every n ∈ N, there exists a constant C = C(n, p, α, σ ) such that nσ nα n α+σ α+σ F C F np F np,α+σ . p,α
(9)
Proof. (7) comes from repeated uses of the inequality (see [5,14]) F · G r,α C F p,α G q,α ,
1 1 1 + = , p, q, r > 1. p q r
(8) is deduced from the Reiteration theorem (see [2,13]), and (9) is a combination of (7) and (8). ✷
3. Main results n Let ∆n = (ain , ai+1 ) be a sequence of subdivisions of [a, b], where ain = i(b − a)/2n + n a, i = 0, 1, . . . , 2 . We can now state our main results.
Theorem 3.1. The convergence lim
n→∞
n −1 2
i=0
an Lt i+1
a n 2 − Lt i
b =4
Lxt dx a p
holds uniformly in t ∈ [0, T ], (p, α)-q.s. and Dα for every p > 1, 0 < α < 1/6. Thus, in particular, it holds uniformly in t ∈ [0, T ], (2, ∞)-q.s.
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
95
We need to prepare a series of lemmas for the proof. Put
X 2
−n
,t =
n −1 2
a n 2
an
Lt i+1 − Lt i
,
i=0
and −n n+1 −n X(2 , t) + 2 (s − 2 ) −n −(n+1) ×(X(2 , t) − X(2 , t)) X(s, t) = b x 4 a Lt dx
if 2−(n+1) s 2−n , if s = 0.
Let t Ytx
=
I(x,∞) (Ms ) dMs , 0
t Ztx =
I(x,∞) (Ms ) dNs , 0
t Axt
=
I(x,∞) (Ms ) dVs , 0
x,y Yt
= Ytx
y − Yt
t =
I(x,y](Ms ) dMs
if x < y,
I(x,y](Ms ) dNs
if x < y,
I(x,y](Ms ) dVs
if x < y,
0 x,y Zt
= Ztx
y − Zt
t = 0
x,y At
= Axt
y − At
t = 0
φtx = (Mt − x)+ − (M0 − x)+ . Then we have Ytx = Ztx + Axt , x,y
Yt
x,y
= Zt
x,y
+ At
if x < y,
and Tanaka formula says Lxt = 2 φtx − Ytx = 2 φtx − Ztx − Axt . The proof of the following lemma is trivial and is therefore omitted.
(10)
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K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
Lemma 3.2. (1) There exists a constant C > 0 such that
y p E φtx − φt C|x − y|p
(11)
for all p 1. (2) For every p 2, 0 r < 1, there exists a constant C = C(p, r) > 0 such that
p E φtx − φsx C|t − s|p/2−1+r . (12) (3) For every p 2, there exists a constant C = C(p) > 0 such that
x,y p E Yt C|x − y|p/2,
x,y p E Zt C|x − y|p/2,
x,y p E At C|x − y|p/2 .
(13) (14) (15)
(4) For every p 2, there exists a constant C = C(p) > 0 such that p p E Y x,y t = E Z x,y t C|x − y|p .
(16)
(5) For every p 2, 0 r < 1, there exists a constant C = C(p, r) > 0 such that
x,y x,y p E Yt − Ys C|t − s|p/2−1+r , (17)
p
x,y x,y (18) E Zt − Zs C|t − s|p/2−1+r ,
x,y x,y
p p−1+r
E At − As C|t − s| . (19) (6) For every p 1, 0 r < 1, there exists a constant C = C(p, r) > 0 such that
p p E Y x,y t − Y x,y s = E Z x,y t − Z x,y s C|t − s|p−1+r . (20) In the following, we fix p 2, t1 < t2 , and C denotes a positive constant whose value may change from line to line. Lemma 3.3. There exists a constant C = C(p) > 0 such that p p−1 supX 2−n , t − X(0, t)p C2−n 2 .
(21)
t
Proof. By using Tanaka formula and Itô formula, we obtain X 2−n , t − X(0, t) 2n −1 b n n n a n ain ,ai+1 ai+1 ain 2 ain ain ,ai+1 2 x i+1 + Yt − 2 φt − φt Yt φt − φt − Lt dx =4 i=0
=4
n −1 2
i=0
a an φt i+1
a n 2 − φt i
t +2 0
a n ,a n Ys i i+1
a n ,a n dYs i i+1
an a n a n ,a n − 2 φt i+1 − φt i Yt i i+1
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
=4
n −1 2
an φt i+1
a n 2 − φt i
t +2
i=0
a n ,a n Ys i i+1
a n ,a n dZs i i+1
t +2
0
97
n a n ,ai+1
Ys i 0
n n ai+1 ain ain ,ai+1 . − 2 φt − φt Yt
Trivially, 2n −1 p n −1 an
an n p−1 2 ain 2 a n 2p i+1 φ − φ 2 E φt i+1 − φt i t t i=0 i=0 p n p−1 n −2np C 2 2 2 = C2−np (by (11)). Moreover, we have 2n −1 t p n n ain ,ai+1 ain ,ai+1 Ys dZs i=0 0
p
p
t 2n −1
a n ,a n
i i+1 n ] (Ms ) fs dWs = E Ys I(ain ,ai+1
i=0
0
t 2n −1 2 p/2 a n ,a n i i+1 2 n ] (Ms ) CE Ys I(ain ,ai+1 |fs | ds i=0 0
t 2n −1
p
a n ,a n
i i+1 p n ] (Ms ) |fs | ds Ys I(ain ,ai+1
CE 0
= CE
i=0
n −1 t 2
0
i=0
i=0
0
n
ain ,ai+1
Ys
p I(a n ,a n ] (Ms ) |fs |p ds i i+1
1/4 t 1/4 n −1 t 2 n
ain ,ai+1 4p 4p−4
ds E Ys E |fs | C ds t 1/2 2 n ] (Ms )|fs | ds × E I(ain ,ai+1 0
C
n −1 2
2
−np/2
i=0
0
n −np/2 −n/2
C2 2 and
0
t 1/2 n ] (Ms ) d[M]s E I(ain ,ai+1
2
= C2−n
p−1 2
(by (16))
n a n ,ai+1
dAs i
(by (13) and (1))
98
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
2n −1 t p n n ain ,ai+1 ain ,ai+1 Ys dAs i=0 0
p
t 2n −1
p
a n ,a n
i i+1 n ] (Ms ) gs ds = E Ys I(ain ,ai+1
i=0 0
t 2n −1
p
a n ,a n
p n ] (Ms ) |gs | ds Ys i i+1 I(ain ,ai+1
E
i=0
0
a n ,a n p i i+1
Ys
I(a n ,a n ] (Ms ) |fs |p |ξs |p ds i i+1
t 2n −1 E 0
n −1 2
i=0
t 1/4 t 1/8 t 1/8 n
ain ,ai+1 4p 8p−8 8p−8
E Ys E |fs | E |ξs | ds ds ds
i=0
0
0
0
t 1/2 2 n ] (Ms )|fs | ds × E I(ain ,ai+1 0 2n −1
C
2
−np/2
t 1/2 n ] (Ms ) d[M]s E I(ain ,ai+1
i=0
(by (13), (1) and (2))
0
n −np/2 −n/2
C2 2
2
= C2−n
p−1 2
(by (16)).
Finally, it is easily seen that 2n −1 p an n ain ain ,ai+1 i+1 φt − φt Yt i=0
p
p−1 2n
n −1 2
p−1 2n
n −1 2
a n a n a n ,a n p E φt i+1 − φt i Yt i i+1
i=0 n n ai+1 a n 2p 1/2 ain ,ai+1
2p 1/2 E φt − φt i E Yt
i=0
p−1 n −np −np/2 C 2n 2 2 2 = C2−np/2
(by (11) and (13)).
Summing these we obtain −n X 2 , t − X(0, t)p p p 2n −1 t p 2n −1 an n n ain ,ai+1 ain ,ai+1 ain 2 i+1 φt − φt C Ys dZs + i=0
p
i=0 0
p
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
99
p 2n −1 p 2n −1 t an n n n a n ,a n ain ,ai+1 ain ,ai+1 a φt i+1 − φt i Yt i i+1 + Ys dAs +
i=0 0
i=0
p
C 2−np + 2
−n p−1 2
+2
−n p−1 2
+2
− np 2
C2
p
−n p−1 2
as desired. ✷ Lemma 3.4. There exists a constant C = C(p) > 0 such that p−1 X(0, t1 ) − X(0, t2 )p C|t1 − t2 | 2 , p
(22)
and p p−1 supX 2−n , t1 − X 2−n , t2 p C|t1 − t2 | 4 .
(23)
n
Proof. By using (4) and (1), we have
p
b b
p
p x
x X(0, t1 ) − X(0, t2 ) = E 4 Lt1 − Lt2 dx C E Lxt1 − Lxt2 dx p
a
C|t1 − t2 |
a
p−1 2
.
This proves (22). Now we turn to (23). By (10), we obtain X 2−n , t1 − X 2−n , t2 =4
n −1 2
a n 2
an
φt1i+1 − φt1i
a n ,a n 2 a n ,a n 2 an a n 2 + Yt1i i+1 − Yt2i i+1 − φt2i+1 − φt2i
i=0
a n an a n a n ,a n a n a n ,a n − 2 φt1i+1 − φt1i Yt1i i+1 − φt2i+1 − φt2i Yt2i i+1 . We treat it term by term. First we have p 2n −1 a n n n n a a a 2 2 φt1i+1 − φt1i − φt2i+1 − φt2i i=0
p
p−1 2n
n −1 2
a n an an a n p E φt1i+1 − φt1i + φt2i+1 − φt2i
i=0
a n an an a n p × φt i+1 − φt i+1 − φt i − φt i 1
2
1
2
−1 n ai+1 an p−1 2 an a n 2p 1/2 E φt1 − φt1i + φt2i+1 − φt2i 2n n
i=0
an an an a n 2p 1/2 × E (φt1i+1 − φt2i+1 − φt1i − φt2i
100
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
p−1 n −np p−1 C 2n 2 2 |t1 − t2 | 2 = C|t1 − t2 |
p−1 2
(by (11) and (12))
.
For the second term, we have 2n −1 p a n ,a n n ain ,ai+1 2 i i+1 2 − Yt2 Yt1 i=0
p
2n−1 t2
p t2
n n ain ,ai+1 ain ,ai+1
= E 2 Ys dYs + I(a,b] (Ms ) d[M]s
i=0 t1
t1
p
p
t2 2n −1 t2
n n ain ,ai+1 ain ,ai+1
C E Ys dYs
+ E I(a,b] (Ms ) d[M]s
i=0 t1
t1
p t2 2n −1
a n ,a n
i i+1 n ] (Ms ) fs dWs Ys I(ain ,ai+1 C E
i=0
t1
t2 2n −1
p
a n ,a n 1
i i+1 p− 2 n ] (Ms ) gs ds + |t1 − t2 | + E Ys I(ain ,ai+1
(by (20))
i=0
t1
t22n −1 2 p/2 a n ,a n i i+1 2 n ] (Ms ) C E Ys I(ain ,ai+1 |fs | ds i=0
t1
t2 2n −1
p
a n ,a n 1
p− i i+1 2 n ] (Ms ) gs ds + |t1 − t2 | + E Ys I(ain ,ai+1
t1
i=0
C |t1 − t2 |
p/2−1
p t2 2n −1
a n ,a n
i i+1 p n ] (Ms ) |fs | ds E Ys I(ain ,ai+1
t1
+ |t1 − t2 |
p−1
p t2 2n −1
a n ,a n
1
i i+1 p p− n ] (Ms ) |gs | ds Ys I(ain ,ai+1 + |t1 − t2 | 2 E
i=0
t1
= C |t1 − t2 |
p/2−1
t2 2n −1 a n ,a n p i p i+1
Ys
I(a n ,a n ] (Ms ) |fs | ds E i i+1 t1
+ |t1 − t2 |
p−1
i=0
i=0
t2 2n −1 a n ,a n p i i+1 p p
Ys
I(a n ,a n ] (Ms ) |fs | |ξs | ds E i i+1 t1
i=0
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
101
+ |t1 − t2 |
p− 21
(by (2))
2n −1 t2 1/2 t2 1/2 n ,a n
a 2p E Ys i i+1 ds E |fs |2p ds C |t1 − t2 |p/2−1
i=0
t1
t1
2n −1 t2 1/2 t2 1/4 n
ain ,ai+1 2p p−1 4p
ds + |t1 − t2 | E Ys E |fs | ds i=0
t1
t1
t2 1/4 4p p− 12 + |t1 − t2 | × E |ξs | ds
t1
C |t1 − t2 |p/2−1 2n 2−np/2 |t1 − t2 |1/4
1 (by (13)) + |t1 − t2 |p−1 2n 2−np/2 |t1 − t2 |1/8 |t1 − t2 |1/8 + |t1 − t2 |p− 2 p 3 p 3 3 1 C |t1 − t2 | 2 − 4 + |t1 − t2 |p− 4 + |t1 − t2 |p− 2 C|t1 − t2 | 2 − 4 .
The last one goes as follows. p 2n −1 a n n n n ai+1 ain ain ,ai+1 ain ain ,ai+1 i+1 − φt2 − φt2 Yt2 φt1 − φt1 Yt1 i=0 p
2n −1
a n a n a n ,a n an a n a n ,a n
φt1i+1 − φt1i Yt1i i+1 − Yt2i i+1 + φt1i+1 − φt2i+1 = E
i=0
p n ain ain ain ,ai+1
− φt1 − φt2 Yt2
p 2n −1
an n n ain ,ai+1 ain ain ,ai+1
i+1 φt1 − φt1 Yt1 C E − Yt2
i=0
p
2n −1
a n n n n a n ,a n a a a
φt1i+1 − φt2i+1 − φt1i − φt2i Yt2i i+1 + E
i=0 n −1
an n p−1 2 a n ,a n p a n p a n ,a n E φt i+1 − φt i Yt i i+1 − Yt i i+1 C 2 1
1
1
2
i=0 −1
a n an n p−1 2 an a n p a n ,a n p + 2 E φt1i+1 − φt2i+1 − φt1i − φt2i Yt2i i+1 n
i=0
−1 n n p−1 2 ai+1 a n ,a n 2p 1/2 a n 2p 1/2 ain ,ai+1 C 2n E φt1 − φt1i E Yt1 − Yt2i i+1 n
i=0
102
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104 −1 n ai+1 an p−1 2 an a n 2p 1/4 E φt1 − φt2i+1 − φt1i − φt2i + 2n n
i=0 n a n an an a n 2p 1/4 ain ,ai+1
2p 1/2 × E φt i+1 − φt i − φt i+1 − φt i E Yt 1
1
2
2
2
p−1 n −np p−1 2 2 |t1 − t2 | 2 C 2n p−1 n p−1 np np (by (11)–(13) and (17)) + 2n 2 |t1 − t2 | 4 2− 2 2− 2 p−1 p−1 p−1 = C |t1 − t2 | 2 + |t1 − t2 | 4 C|t1 − t2 | 4 . So we get finally −n X 2 , t1 − X 2−n , t2 p p p 2n −1 a n n ai+1 ain 2 ain 2 i+1 φt 1 − φt 1 − φt 2 − φt 2 C i=0 p p 2n −1 a n ,a n a n ,a n 2 2 Yt1i i+1 − Yt2i i+1 + i=0 p p 2n −1 a n n n n ai+1 ain ain ,ai+1 ain ain ,ai+1 i+1 φt1 − φt1 Yt1 + − φt2 − φt2 Yt2 i=0
C |t1 − t2 |
p−1 2
+ |t1 − t2 |
p 3 2 −4
+ |t1 − t2 |
p−1 4
p
C|t1 − t2 |
p−1 4
as desired. ✷ Note that X is piecewise linear in s, by (21)–(23) we conclude that: Corollary 3.5. For every p 2, there exists a constant C = C(p) > 0 such that p−1 p−1 X(s1 , t1 ) − X(s2 , t2 )p C |s1 − s2 | 2 + |t1 − t2 | 4 . p
(24)
So (s, t) → X(s, t) admit a continuous modification, and the convergence: lim
n→∞
n −1 2
an Lt i+1
a n 2 − Lt i
i=0
b =4
Lxt dx a
holds uniformly in t ∈ [0, T ], a.s. and Lp for every p 1. Lemma 3.6. There exists a constant C = C(p, β) > 0 such that supX(s, t)p,β C s,t
for every p > 1, 0 < β < 1/6.
(25)
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
Proof. For 0 < ε < 1/2, b x X(0, t) = 4 L dx t p,ε
b C
p,ε
a
x L dx C t p,ε
i=0
C
p,ε
n n ai+1 Lt − Lat i 2 2p,ε
i=0 n −2nγ
C2 2
(by (5)),
a
and for 0 < γ < 1/2 − ε, 2n −1 an −n ain 2 i+1 X 2 , t = L − L t t p,ε n −1 2
= C2n(1−2γ )
(by (7)) (by (3)).
Then we have −n X 2 , t − X(0, t) C2n(1−2γ ) . p,ε For 0 < β < 1/6, choose q p ∨ 2 such that β < ξ :=
(6q − 2)β . q −1
δ :=
ξq 2qβ = > 0. 3q − 1 q − 1
(26) q−1 6q−2
and let ξ ∈ (0, 1) be defined by
Set
Then β +δ=
1 (3q − 1)β ξ = < . q −1 2 2
Let now 1 η 1 − β − δ − < − β − δ, 2 2 2 where 0 < η < 1 − ξ . Then we have 1 β β 1 δ − + (2γ − 1) = 2γ > 0. 2 2q β + δ β +δ β +δ γ :=
Thus by (8) we obtain −n X 2 , t − X(0, t) X 2−n , t − X(0, t) p,β q,β β δ β+δ C X 2−n , t − X(0, t)qβ+δ X 2−n , t − X(0, t)q,β+δ
C2 = C2
β 1 δ −n( 12 − 2q ) β+δ −n(2γ −1) β+δ
2
β 1 δ −n ( 21 − 2q ) β+δ +(2γ −1) β+δ
103
(by (21) and (26))
C.
104
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
So X(2−n , t) p,β C.
✷
At last, we can now give Proof of Theorem 3.1. By (9), (24) and (25), for any fixed n ∈ N, we have X(s1 , t1 ) − X(s2 , t2 ) n p,α nσ nα α+σ X(s , t ) − X(s , t ) α+σ C X(s1 , t1 ) − X(s2 , t2 )np 1 1 2 2 np,α+σ np−1 σ np−1 σ C |s1 − s2 | 2p α+σ + |t1 − t2 | 4p α+σ , where p 2, 0 < α < 1/6, 0 < σ < 1/6 − α. Choosing n sufficiently big such that np−1 σ 4p α+σ > 2, we conclude that (s, t) → X(s, t) admit a (p, α)-modification by the quasi sure version of Kolmogorov’s criterion (see [9, Theorem 3.1]) and the first part of the theorem is therefore proved. Combining the first part and [4, Theorem 2.19] gives the second part and the proof is thus finished. ✷ Acknowledgements We are very grateful to Guilan Cao, Xicheng Zhang, and Jicheng Liu for their helps and valuable discussions.
References [1] H. Airault, J. Ren, X. Zhang, Smoothness of local times of semimartingales, C. R. Acad. Sci. Paris Sér. I 330 (2000) 719–724. [2] J. Bergh, J. Lofstrom, Interpolation Spaces, in: Grund. Math. Wiss., vol. 223, Springer-Verlag, 1976. [3] N. Bouleau, M. Yor, Sur la variation quadratique des temps locaux de certaines semimartingales, C. R. Acad. Sci. Paris Sér. I 292 (1981) 491–494. [4] J. Hu, J. Ren, Infinite dimensional quasi continuity, path continuity and ray continuity of functions with fractional regularity, J. Math. Pures Appl. 80 (1) (2001) 131–152. [5] S. Kusuoka, D. Stroock, Application of the Malliavin calculus, Part I, in: K. Itô (Ed.), Proc. Taniguchi Symp. on Stoch. Anal., Katata/Kyoto, 1982, in: Math Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 271–306. [6] P. Malliavin, D. Nualart, Quasi sure analysis of stochastic flows and Banach space valued smooth functionals on the Wiener space, J. Funct. Anal. 112 (1993) 287–317. [7] D. Nualart, J. Vives, Smoothness of local time and related Wiener functionals, Potential Anal. 1 (1992) 257–263. [8] E. Perkins, Local time is a semimartingale, Z. Wahr. verw. Geb. 60 (1982) 79–117. [9] J. Ren, Analysis quasi sûre des équation différentielles stochastiques, Bull. Sci. Math. 114 (1990) 187–213. [10] J. Ren, Quasi sure quadratic variation of smooth martingales, J. Math. Kyoto Univ. 34 (1) (1994) 191–206. [11] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, in: Grund. Math. Wiss., vol. 293, Springer-Verlag, Berlin, 1991. [12] I. Shigekawa, On a quasi everywhere existence of the local time of the 1-dimensional Brownian motion, Osaka J. Math. 21 (1984) 621–627. [13] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. [14] S. Watanabe, Fractional order Sobolev spaces on Wiener space, Probab. Theory Related Fields 95 (1993) 175–198.
Quasi sure quadratic variation of local times of smooth semimartingales ✩ Kai He a , Jiagang Ren a,b,∗ a Department of Mathematics, Huazhong University of Science and Technology, Wuhan,
Hubei 430074, PR China b School of Mathematics and Computational Science, Zhongshan University, Guangzhou,
Guangdong 510275, PR China Received 10 November 2003; accepted 30 November 2003
Abstract In this paper, we prove that the process of the quadratic variation of local times of smooth an an semimartingales can be constructed as the quasi sure limit of the form ∆n (Lt i+1 − Lt i )2 , where n ) is a sequence of subdivisions of [a, b], a n = i(b − a)/2n + a, i = 0, 1, . . . , 2n . ∆n = (ain , ai+1 i 2003 Elsevier SAS. All rights reserved. MSC: 60H07; 60G44; 31C15; 31C25 Keywords: Local time; Quadratic variation; Capacity; Quasi sure convergence
1. Introduction The study of regularity of local times in the sense of Malliavin calculus traces back to the original work of Shikegawa [12] in which he proved the 1-quasi sure existence of the Brownian local time. Later Nualart and Vives in [7] changed the angle and proved the Brownian local time is in the fractional Sobolev spaces Dr2 for all r < 1/2. This result p was then improved by Watanabe in [14] by replacing Dr2 by Dr for all p > 1 and was further extended to semimartingales in [1]. In [4] a comparison theorem for capacities was established and it turned out then that Shigekawa’s quasi sure existence theorem can be deduced from Nualart–Watanabe’s result and extended to semimartingales. ✩
Work supported partially by Projects 973.
* Corresponding author.
E-mail address: [email protected], [email protected] (J. Ren). 0007-4497/$ – see front matter 2003 Elsevier SAS. All rights reserved. doi:10.1016/j.bulsci.2003.11.001
92
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
In this paper, we continue the study in this line by looking at the quasi sure properties of the quadratic variations of local times of smooth semimartingales. The quadratic variation of the local time of the 1-dimensional Brownian motion or, more generally, a continuous martingale, was studied in Bouleau and Yor [3], Perkins [8], see also [11]. Denoting by Lxt the local time of a continuous martingale, they showed that, for n any sequence of partition ∆n = (ain , ai+1 ) of [a, b] with mesh size converging to zero, lim
n→∞
an
a n 2
Lt i+1 − Lt i
∆n
b =4
Lxt dx a
in probability. In the present work we shall prove that for smooth semimartingales this convergence n ) where a n = i(b − a)/2n + a, i = can be enhanced to hold quasi surely for ∆n = (ain , ai+1 i n 0, 1, . . . , 2 . This can be considered as a refinement of the results of Bouleau, Yor and Perkins. It may happen that x → Lxt is a semimartingale. But it is utter impossible that it is a smooth semimartingale except in the trivial case (see [1, Theorem 2]). Hence the results in [10] is not applicable here. This paper is organized as follows. In Section 2, we give necessary notations and notions. In Section 3, we state and prove our main results.
2. Preliminaries Now let us recall and fix some notations and notions. We shall work on the classical Wiener space (X, F , µ; H ), where X = C0 ([0, T ] → Rd ) is the space of continuous maps from [0, T ] to Rd , null at zero; H is the usual Cameron–Martin subspace and µ the p standard Wiener measure. Denote by Dα the Sobolev space of order α and of power p p over X. The norm in Dα is defined by F p,α = (I − L)α/2 F p , where L is the Ornstein–Uhlenbeck operator. Given an open set O of X, its (p, α)-capacity is defined by Cp,α (O) = inf F p,α : F 0, F 1 µ-a.s. on O , and for any subset A ⊂ X, by Cp,α (A) = inf Cp,α (O): O open and O ⊃ A . Let {Ws = (Ws1 , . . . , Wsd ) , 0 s T } be the d-dimensional Brownian motion realized by the coordinate process on (X, F , µ; H ) and (Ft )t 0 the σ -algebra generated by {Ws , 0 s t}. Following Malliavin and Nualart [6], we give Definition 2.1. M is called a smooth semimartingale if it can be represented as: Mt = Nt + Vt
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
93
where Nt =
d
t
t j fs
j dWs
=
j =1 0
fs dWs ,
f = f 1, . . . , f d ,
0
t Vt =
gs ds, 0
with d j p fs ds + gs pp,α ds < ∞, p,α T
T
j =1 0
∀p > 1, α 0.
(1)
0
t j t By [6,10], M and its quadratic variation [M] = dj=1 0 (fs )2 ds = 0 |fs |2 ds admit ∞-modifications, and sup0t T Mt p,α < ∞, sup0t T [M]t p,α < ∞. For the smoothness in the sense of Malliavin calculus of local times of smooth semimartingales, the following is proved in [1]: Theorem 2.2. Let M be a smooth semimartingale, and suppose that there exists an (Ft )adapted process ξt with T
p
ξs p ds < ∞,
∀p > 1,
(2)
0
such that |gt | |ft | · |ξt | a.s. for Lebesgue almost all t ∈ [0, T ]. Let L be the local time of M. Then Lat ∈ Dαp for all p > 1, 0 < α < 1/2. The following lemma comes from [4, Theorem 2.23] and [11, Theorem 1.7 and Exercise 1.32, Chapter VI]: Lemma 2.3. In the same situation as in the above theorem, for every N > 0, p > 1, 0 < α < 1/2, 0 < γ < 1/2 − α, 0 r < 1, there exist positive constants C1 = C1 (N, p, α, γ ), C2 = C2 (N, p, r) such that x L − Lys C1 |t − s|γ + |x − y|γ , (3) t p,α p x y L − Ls C2 |t − s|p/2−1+r + |x − y|p/2 , (4) t p for all s, t ∈ [0, T ] and |x|, |y| N . In particular, x L < ∞ sup t p,α 0t T ,|x|N
(5)
94
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
and x L < ∞ t p
sup
(6)
0t T ,x∈R
for every p 2. We also need the following simple lemma: Lemma 2.4. (1) For every n ∈ N, p > 1, α > 0, there exists a positive constant C = C(n, p, α) such that n F C F n (7) np,α p,α np
for all F ∈ Dα . (2) For every p > 1, 0 < α < 1/2, 0 σ < 1/2 − α, there exists a constant C = C(p, α, σ ) such that σ
α
α+σ F p,α C F pα+σ F p,α+σ
(8)
p
for all F ∈ Dα+σ ; and for every n ∈ N, there exists a constant C = C(n, p, α, σ ) such that nσ nα n α+σ α+σ F C F np F np,α+σ . p,α
(9)
Proof. (7) comes from repeated uses of the inequality (see [5,14]) F · G r,α C F p,α G q,α ,
1 1 1 + = , p, q, r > 1. p q r
(8) is deduced from the Reiteration theorem (see [2,13]), and (9) is a combination of (7) and (8). ✷
3. Main results n Let ∆n = (ain , ai+1 ) be a sequence of subdivisions of [a, b], where ain = i(b − a)/2n + n a, i = 0, 1, . . . , 2 . We can now state our main results.
Theorem 3.1. The convergence lim
n→∞
n −1 2
i=0
an Lt i+1
a n 2 − Lt i
b =4
Lxt dx a p
holds uniformly in t ∈ [0, T ], (p, α)-q.s. and Dα for every p > 1, 0 < α < 1/6. Thus, in particular, it holds uniformly in t ∈ [0, T ], (2, ∞)-q.s.
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
95
We need to prepare a series of lemmas for the proof. Put
X 2
−n
,t =
n −1 2
a n 2
an
Lt i+1 − Lt i
,
i=0
and −n n+1 −n X(2 , t) + 2 (s − 2 ) −n −(n+1) ×(X(2 , t) − X(2 , t)) X(s, t) = b x 4 a Lt dx
if 2−(n+1) s 2−n , if s = 0.
Let t Ytx
=
I(x,∞) (Ms ) dMs , 0
t Ztx =
I(x,∞) (Ms ) dNs , 0
t Axt
=
I(x,∞) (Ms ) dVs , 0
x,y Yt
= Ytx
y − Yt
t =
I(x,y](Ms ) dMs
if x < y,
I(x,y](Ms ) dNs
if x < y,
I(x,y](Ms ) dVs
if x < y,
0 x,y Zt
= Ztx
y − Zt
t = 0
x,y At
= Axt
y − At
t = 0
φtx = (Mt − x)+ − (M0 − x)+ . Then we have Ytx = Ztx + Axt , x,y
Yt
x,y
= Zt
x,y
+ At
if x < y,
and Tanaka formula says Lxt = 2 φtx − Ytx = 2 φtx − Ztx − Axt . The proof of the following lemma is trivial and is therefore omitted.
(10)
96
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
Lemma 3.2. (1) There exists a constant C > 0 such that
y p E φtx − φt C|x − y|p
(11)
for all p 1. (2) For every p 2, 0 r < 1, there exists a constant C = C(p, r) > 0 such that
p E φtx − φsx C|t − s|p/2−1+r . (12) (3) For every p 2, there exists a constant C = C(p) > 0 such that
x,y p E Yt C|x − y|p/2,
x,y p E Zt C|x − y|p/2,
x,y p E At C|x − y|p/2 .
(13) (14) (15)
(4) For every p 2, there exists a constant C = C(p) > 0 such that p p E Y x,y t = E Z x,y t C|x − y|p .
(16)
(5) For every p 2, 0 r < 1, there exists a constant C = C(p, r) > 0 such that
x,y x,y p E Yt − Ys C|t − s|p/2−1+r , (17)
p
x,y x,y (18) E Zt − Zs C|t − s|p/2−1+r ,
x,y x,y
p p−1+r
E At − As C|t − s| . (19) (6) For every p 1, 0 r < 1, there exists a constant C = C(p, r) > 0 such that
p p E Y x,y t − Y x,y s = E Z x,y t − Z x,y s C|t − s|p−1+r . (20) In the following, we fix p 2, t1 < t2 , and C denotes a positive constant whose value may change from line to line. Lemma 3.3. There exists a constant C = C(p) > 0 such that p p−1 supX 2−n , t − X(0, t)p C2−n 2 .
(21)
t
Proof. By using Tanaka formula and Itô formula, we obtain X 2−n , t − X(0, t) 2n −1 b n n n a n ain ,ai+1 ai+1 ain 2 ain ain ,ai+1 2 x i+1 + Yt − 2 φt − φt Yt φt − φt − Lt dx =4 i=0
=4
n −1 2
i=0
a an φt i+1
a n 2 − φt i
t +2 0
a n ,a n Ys i i+1
a n ,a n dYs i i+1
an a n a n ,a n − 2 φt i+1 − φt i Yt i i+1
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
=4
n −1 2
an φt i+1
a n 2 − φt i
t +2
i=0
a n ,a n Ys i i+1
a n ,a n dZs i i+1
t +2
0
97
n a n ,ai+1
Ys i 0
n n ai+1 ain ain ,ai+1 . − 2 φt − φt Yt
Trivially, 2n −1 p n −1 an
an n p−1 2 ain 2 a n 2p i+1 φ − φ 2 E φt i+1 − φt i t t i=0 i=0 p n p−1 n −2np C 2 2 2 = C2−np (by (11)). Moreover, we have 2n −1 t p n n ain ,ai+1 ain ,ai+1 Ys dZs i=0 0
p
p
t 2n −1
a n ,a n
i i+1 n ] (Ms ) fs dWs = E Ys I(ain ,ai+1
i=0
0
t 2n −1 2 p/2 a n ,a n i i+1 2 n ] (Ms ) CE Ys I(ain ,ai+1 |fs | ds i=0 0
t 2n −1
p
a n ,a n
i i+1 p n ] (Ms ) |fs | ds Ys I(ain ,ai+1
CE 0
= CE
i=0
n −1 t 2
0
i=0
i=0
0
n
ain ,ai+1
Ys
p I(a n ,a n ] (Ms ) |fs |p ds i i+1
1/4 t 1/4 n −1 t 2 n
ain ,ai+1 4p 4p−4
ds E Ys E |fs | C ds t 1/2 2 n ] (Ms )|fs | ds × E I(ain ,ai+1 0
C
n −1 2
2
−np/2
i=0
0
n −np/2 −n/2
C2 2 and
0
t 1/2 n ] (Ms ) d[M]s E I(ain ,ai+1
2
= C2−n
p−1 2
(by (16))
n a n ,ai+1
dAs i
(by (13) and (1))
98
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
2n −1 t p n n ain ,ai+1 ain ,ai+1 Ys dAs i=0 0
p
t 2n −1
p
a n ,a n
i i+1 n ] (Ms ) gs ds = E Ys I(ain ,ai+1
i=0 0
t 2n −1
p
a n ,a n
p n ] (Ms ) |gs | ds Ys i i+1 I(ain ,ai+1
E
i=0
0
a n ,a n p i i+1
Ys
I(a n ,a n ] (Ms ) |fs |p |ξs |p ds i i+1
t 2n −1 E 0
n −1 2
i=0
t 1/4 t 1/8 t 1/8 n
ain ,ai+1 4p 8p−8 8p−8
E Ys E |fs | E |ξs | ds ds ds
i=0
0
0
0
t 1/2 2 n ] (Ms )|fs | ds × E I(ain ,ai+1 0 2n −1
C
2
−np/2
t 1/2 n ] (Ms ) d[M]s E I(ain ,ai+1
i=0
(by (13), (1) and (2))
0
n −np/2 −n/2
C2 2
2
= C2−n
p−1 2
(by (16)).
Finally, it is easily seen that 2n −1 p an n ain ain ,ai+1 i+1 φt − φt Yt i=0
p
p−1 2n
n −1 2
p−1 2n
n −1 2
a n a n a n ,a n p E φt i+1 − φt i Yt i i+1
i=0 n n ai+1 a n 2p 1/2 ain ,ai+1
2p 1/2 E φt − φt i E Yt
i=0
p−1 n −np −np/2 C 2n 2 2 2 = C2−np/2
(by (11) and (13)).
Summing these we obtain −n X 2 , t − X(0, t)p p p 2n −1 t p 2n −1 an n n ain ,ai+1 ain ,ai+1 ain 2 i+1 φt − φt C Ys dZs + i=0
p
i=0 0
p
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
99
p 2n −1 p 2n −1 t an n n n a n ,a n ain ,ai+1 ain ,ai+1 a φt i+1 − φt i Yt i i+1 + Ys dAs +
i=0 0
i=0
p
C 2−np + 2
−n p−1 2
+2
−n p−1 2
+2
− np 2
C2
p
−n p−1 2
as desired. ✷ Lemma 3.4. There exists a constant C = C(p) > 0 such that p−1 X(0, t1 ) − X(0, t2 )p C|t1 − t2 | 2 , p
(22)
and p p−1 supX 2−n , t1 − X 2−n , t2 p C|t1 − t2 | 4 .
(23)
n
Proof. By using (4) and (1), we have
p
b b
p
p x
x X(0, t1 ) − X(0, t2 ) = E 4 Lt1 − Lt2 dx C E Lxt1 − Lxt2 dx p
a
C|t1 − t2 |
a
p−1 2
.
This proves (22). Now we turn to (23). By (10), we obtain X 2−n , t1 − X 2−n , t2 =4
n −1 2
a n 2
an
φt1i+1 − φt1i
a n ,a n 2 a n ,a n 2 an a n 2 + Yt1i i+1 − Yt2i i+1 − φt2i+1 − φt2i
i=0
a n an a n a n ,a n a n a n ,a n − 2 φt1i+1 − φt1i Yt1i i+1 − φt2i+1 − φt2i Yt2i i+1 . We treat it term by term. First we have p 2n −1 a n n n n a a a 2 2 φt1i+1 − φt1i − φt2i+1 − φt2i i=0
p
p−1 2n
n −1 2
a n an an a n p E φt1i+1 − φt1i + φt2i+1 − φt2i
i=0
a n an an a n p × φt i+1 − φt i+1 − φt i − φt i 1
2
1
2
−1 n ai+1 an p−1 2 an a n 2p 1/2 E φt1 − φt1i + φt2i+1 − φt2i 2n n
i=0
an an an a n 2p 1/2 × E (φt1i+1 − φt2i+1 − φt1i − φt2i
100
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
p−1 n −np p−1 C 2n 2 2 |t1 − t2 | 2 = C|t1 − t2 |
p−1 2
(by (11) and (12))
.
For the second term, we have 2n −1 p a n ,a n n ain ,ai+1 2 i i+1 2 − Yt2 Yt1 i=0
p
2n−1 t2
p t2
n n ain ,ai+1 ain ,ai+1
= E 2 Ys dYs + I(a,b] (Ms ) d[M]s
i=0 t1
t1
p
p
t2 2n −1 t2
n n ain ,ai+1 ain ,ai+1
C E Ys dYs
+ E I(a,b] (Ms ) d[M]s
i=0 t1
t1
p t2 2n −1
a n ,a n
i i+1 n ] (Ms ) fs dWs Ys I(ain ,ai+1 C E
i=0
t1
t2 2n −1
p
a n ,a n 1
i i+1 p− 2 n ] (Ms ) gs ds + |t1 − t2 | + E Ys I(ain ,ai+1
(by (20))
i=0
t1
t22n −1 2 p/2 a n ,a n i i+1 2 n ] (Ms ) C E Ys I(ain ,ai+1 |fs | ds i=0
t1
t2 2n −1
p
a n ,a n 1
p− i i+1 2 n ] (Ms ) gs ds + |t1 − t2 | + E Ys I(ain ,ai+1
t1
i=0
C |t1 − t2 |
p/2−1
p t2 2n −1
a n ,a n
i i+1 p n ] (Ms ) |fs | ds E Ys I(ain ,ai+1
t1
+ |t1 − t2 |
p−1
p t2 2n −1
a n ,a n
1
i i+1 p p− n ] (Ms ) |gs | ds Ys I(ain ,ai+1 + |t1 − t2 | 2 E
i=0
t1
= C |t1 − t2 |
p/2−1
t2 2n −1 a n ,a n p i p i+1
Ys
I(a n ,a n ] (Ms ) |fs | ds E i i+1 t1
+ |t1 − t2 |
p−1
i=0
i=0
t2 2n −1 a n ,a n p i i+1 p p
Ys
I(a n ,a n ] (Ms ) |fs | |ξs | ds E i i+1 t1
i=0
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
101
+ |t1 − t2 |
p− 21
(by (2))
2n −1 t2 1/2 t2 1/2 n ,a n
a 2p E Ys i i+1 ds E |fs |2p ds C |t1 − t2 |p/2−1
i=0
t1
t1
2n −1 t2 1/2 t2 1/4 n
ain ,ai+1 2p p−1 4p
ds + |t1 − t2 | E Ys E |fs | ds i=0
t1
t1
t2 1/4 4p p− 12 + |t1 − t2 | × E |ξs | ds
t1
C |t1 − t2 |p/2−1 2n 2−np/2 |t1 − t2 |1/4
1 (by (13)) + |t1 − t2 |p−1 2n 2−np/2 |t1 − t2 |1/8 |t1 − t2 |1/8 + |t1 − t2 |p− 2 p 3 p 3 3 1 C |t1 − t2 | 2 − 4 + |t1 − t2 |p− 4 + |t1 − t2 |p− 2 C|t1 − t2 | 2 − 4 .
The last one goes as follows. p 2n −1 a n n n n ai+1 ain ain ,ai+1 ain ain ,ai+1 i+1 − φt2 − φt2 Yt2 φt1 − φt1 Yt1 i=0 p
2n −1
a n a n a n ,a n an a n a n ,a n
φt1i+1 − φt1i Yt1i i+1 − Yt2i i+1 + φt1i+1 − φt2i+1 = E
i=0
p n ain ain ain ,ai+1
− φt1 − φt2 Yt2
p 2n −1
an n n ain ,ai+1 ain ain ,ai+1
i+1 φt1 − φt1 Yt1 C E − Yt2
i=0
p
2n −1
a n n n n a n ,a n a a a
φt1i+1 − φt2i+1 − φt1i − φt2i Yt2i i+1 + E
i=0 n −1
an n p−1 2 a n ,a n p a n p a n ,a n E φt i+1 − φt i Yt i i+1 − Yt i i+1 C 2 1
1
1
2
i=0 −1
a n an n p−1 2 an a n p a n ,a n p + 2 E φt1i+1 − φt2i+1 − φt1i − φt2i Yt2i i+1 n
i=0
−1 n n p−1 2 ai+1 a n ,a n 2p 1/2 a n 2p 1/2 ain ,ai+1 C 2n E φt1 − φt1i E Yt1 − Yt2i i+1 n
i=0
102
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104 −1 n ai+1 an p−1 2 an a n 2p 1/4 E φt1 − φt2i+1 − φt1i − φt2i + 2n n
i=0 n a n an an a n 2p 1/4 ain ,ai+1
2p 1/2 × E φt i+1 − φt i − φt i+1 − φt i E Yt 1
1
2
2
2
p−1 n −np p−1 2 2 |t1 − t2 | 2 C 2n p−1 n p−1 np np (by (11)–(13) and (17)) + 2n 2 |t1 − t2 | 4 2− 2 2− 2 p−1 p−1 p−1 = C |t1 − t2 | 2 + |t1 − t2 | 4 C|t1 − t2 | 4 . So we get finally −n X 2 , t1 − X 2−n , t2 p p p 2n −1 a n n ai+1 ain 2 ain 2 i+1 φt 1 − φt 1 − φt 2 − φt 2 C i=0 p p 2n −1 a n ,a n a n ,a n 2 2 Yt1i i+1 − Yt2i i+1 + i=0 p p 2n −1 a n n n n ai+1 ain ain ,ai+1 ain ain ,ai+1 i+1 φt1 − φt1 Yt1 + − φt2 − φt2 Yt2 i=0
C |t1 − t2 |
p−1 2
+ |t1 − t2 |
p 3 2 −4
+ |t1 − t2 |
p−1 4
p
C|t1 − t2 |
p−1 4
as desired. ✷ Note that X is piecewise linear in s, by (21)–(23) we conclude that: Corollary 3.5. For every p 2, there exists a constant C = C(p) > 0 such that p−1 p−1 X(s1 , t1 ) − X(s2 , t2 )p C |s1 − s2 | 2 + |t1 − t2 | 4 . p
(24)
So (s, t) → X(s, t) admit a continuous modification, and the convergence: lim
n→∞
n −1 2
an Lt i+1
a n 2 − Lt i
i=0
b =4
Lxt dx a
holds uniformly in t ∈ [0, T ], a.s. and Lp for every p 1. Lemma 3.6. There exists a constant C = C(p, β) > 0 such that supX(s, t)p,β C s,t
for every p > 1, 0 < β < 1/6.
(25)
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
Proof. For 0 < ε < 1/2, b x X(0, t) = 4 L dx t p,ε
b C
p,ε
a
x L dx C t p,ε
i=0
C
p,ε
n n ai+1 Lt − Lat i 2 2p,ε
i=0 n −2nγ
C2 2
(by (5)),
a
and for 0 < γ < 1/2 − ε, 2n −1 an −n ain 2 i+1 X 2 , t = L − L t t p,ε n −1 2
= C2n(1−2γ )
(by (7)) (by (3)).
Then we have −n X 2 , t − X(0, t) C2n(1−2γ ) . p,ε For 0 < β < 1/6, choose q p ∨ 2 such that β < ξ :=
(6q − 2)β . q −1
δ :=
ξq 2qβ = > 0. 3q − 1 q − 1
(26) q−1 6q−2
and let ξ ∈ (0, 1) be defined by
Set
Then β +δ=
1 (3q − 1)β ξ = < . q −1 2 2
Let now 1 η 1 − β − δ − < − β − δ, 2 2 2 where 0 < η < 1 − ξ . Then we have 1 β β 1 δ − + (2γ − 1) = 2γ > 0. 2 2q β + δ β +δ β +δ γ :=
Thus by (8) we obtain −n X 2 , t − X(0, t) X 2−n , t − X(0, t) p,β q,β β δ β+δ C X 2−n , t − X(0, t)qβ+δ X 2−n , t − X(0, t)q,β+δ
C2 = C2
β 1 δ −n( 12 − 2q ) β+δ −n(2γ −1) β+δ
2
β 1 δ −n ( 21 − 2q ) β+δ +(2γ −1) β+δ
103
(by (21) and (26))
C.
104
K. He, J. Ren / Bull. Sci. math. 128 (2004) 91–104
So X(2−n , t) p,β C.
✷
At last, we can now give Proof of Theorem 3.1. By (9), (24) and (25), for any fixed n ∈ N, we have X(s1 , t1 ) − X(s2 , t2 ) n p,α nσ nα α+σ X(s , t ) − X(s , t ) α+σ C X(s1 , t1 ) − X(s2 , t2 )np 1 1 2 2 np,α+σ np−1 σ np−1 σ C |s1 − s2 | 2p α+σ + |t1 − t2 | 4p α+σ , where p 2, 0 < α < 1/6, 0 < σ < 1/6 − α. Choosing n sufficiently big such that np−1 σ 4p α+σ > 2, we conclude that (s, t) → X(s, t) admit a (p, α)-modification by the quasi sure version of Kolmogorov’s criterion (see [9, Theorem 3.1]) and the first part of the theorem is therefore proved. Combining the first part and [4, Theorem 2.19] gives the second part and the proof is thus finished. ✷ Acknowledgements We are very grateful to Guilan Cao, Xicheng Zhang, and Jicheng Liu for their helps and valuable discussions.
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