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Wong–Zakai approximations of backward doubly stochastic differential equations
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ScienceDirect Stochastic Processes and their Applications xx (xxxx) xxx–xxx www.elsevier.com/locate/spa
Wong–Zakai approximations of backward doubly stochastic differential equations Ying Hu a , Anis Matoussi b , Tusheng Zhang c,∗
Q1
a Universit´e de Rennes 1, campus Beaulieu, 35042 Rennes Cedex, France b Laboratoire Manceau de Math´ematiques, Universit´e du Maine, Avenue Olivier Messiaen, 72 085 LE MANS Cedex,
France c School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England, UK
Received 12 January 2015; received in revised form 18 June 2015; accepted 3 July 2015
1 2
Abstract
3
In this paper we obtain a Wong–Zakai approximation to solutions of backward doubly stochastic differential equations (BDSDEs). The situation is quite different from the case of stochastic differential equations because the integrands in the stochastic integral are also part of the solutions. c 2015 Published by Elsevier B.V. ⃝
7
MSC: primary 60H15; secondary 60H30
8
Keywords: Backward stochastic differential equations; Backward doubly stochastic differential equations; Wong–Zakai approximations; Backward integrals
4 5 6
9 10
1. Framework and introduction
11
Let {Wt , 0 ≤ t ≤ T } and {Bt , 0 ≤ t ≤ T } be two independent standard Brownian motions with values respectively in R l and in R m , defined on a probability space (Ω , F, P). Let N denote the class of P-null sets. For each t ∈ [0, T ], we define
13
B Ft = FtW ∨ Ft,T , ∗ Corresponding author.
E-mail addresses: [email protected] (Y. Hu), [email protected] (A. Matoussi), [email protected] (T. Zhang). http://dx.doi.org/10.1016/j.spa.2015.07.003 c 2015 Published by Elsevier B.V. 0304-4149/⃝
12
14
15
2
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx η
η
η
2
where for any process {ηt }, Fs,t = σ {ηr − ηs ; s ≤ r ≤ t} ∨ N , Ft = F0,t . Note that Ft , t ≥ 0 is not a filtration.
3
(H.1) Let f : R d × R d×l → R d be a bounded measurable function satisfying
1
4
5 6 7
8
9
10
| f (y1 , z 1 ) − f (y2 , z 2 )| ≤ c(|y1 − y2 | + ∥z 1 − z 2 ∥) for (y1 ,√z 1 ), (y2 , z 2 ) ∈ R d × R d×l , where |y| stands for the Euclidean norm of the vector y and ∥z∥ = T r (zz ∗ ) for a d × l matrix z. Let g(·) = (gi, j (·)) ∈ Cb1 (R d , R d×m ). For n ≥ 1, define the linear interpolation B n of B as k+1 k k+1 n n Bt = B k+2 +2 t − n B k+2 − B k+1 , for t ∈ n , n . (1.1) 2n 2n 2n 2 2 2 Let gg ′ : R d → R d be defined as m d ∂gi, j (y)
(gg ′ (y))i =
j=1 k=1 11 12
13 14
15 16
17 18
19 20
21
∂ yk
gk, j (y),
23 24 25 26
27
(1.2)
For any positive integer p ∈ N , let M 2 ([0, T ]; R p ) denote the set of p dimensional jointly measurable stochastic processes {φt ; t ∈ [0, T ]} which satisfy T (i) E[ 0 |φt |2 dt] < ∞, (ii) φt is Ft measurable for a.e. t ∈ [0, T ]. Similarly denote by S 2 ([0, T ]; R p ) the set of continuous p dimensional stochastic processes which satisfy (i) E[sup0≤t≤T |φt |2 ] < ∞, (ii) φt is Ft measurable for any t ∈ [0, T ]. Let ξ ∈ L 2 (Ω ; R d ) be FT -measurable. Consider the following backward doubly stochastic differential equations (BDSDE): Yt = ξ +
T
f (Ys , Z s )ds +
Ytn
=ξ+
T
t 22
i = 1, . . . , d.
g(Ys )dBs +
t
t
T
f (Ysn ,
Z sn )ds +
t
T
g(Ysn )dBns
1 2
T
gg ′ (Ys )ds −
t
− t
T
Z s dW s .
(1.3)
t T
Z sn dW s .
(1.4)
Here dBs stands for the backward Ito integral and the integrals against W are the forward Ito − → integrals. Correspondingly, here by dBns we mean B˙ sn ds , where and also in the sequel, B˙ sn denotes − → the piecewise derivative of B n and ds stands for the backward integral against the Lebesgue measure, namely, b − → h ′ (s) ds = h(a) − h(b), for h ∈ C 1 (R). a
28 29
Under the assumption (H.1), Eqs. (1.3) and (1.4) were shown in [8] to admit unique solutions (Y, Z ), (Y n , Z n ) ∈ S 2 ([0, T ]; R d ) × M 2 ([0, T ]; R d×l ).
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
3
Backward doubly stochastic differential equations were first studied by Pardoux and Peng in [8]. It is now a powerful tool to study stochastic partial differential equations with singular coefficients (see [2,5,6]). Our purpose of this paper is to obtain the convergence of the Wong–Zakai approximation to the backward doubly stochastic differential equations, namely we will prove that (Y n , Z n ) converges to (Y, Z ) in L 2 . The convergence of Wong–Zakai approximations to stochastic differential equations is now well known, see e.g. [4]. Because of the nature of the BDSDEs, the integrands Z n , Z in the stochastic integrals against the Brownian motion W are also part of the solutions. We could not impose conditions on Z n , Z in advance. This makes the problem drastically different from the Wong–Zakai approximation for stochastic differential equations. Another difficulty comes from the fact that the H¨older type estimate E[|Ytn − Ysn | p ] ≤ C|t − s|α
1 2 3 4 5 6 7 8 9 10
11
is no longer available for these BSDEs. However it plays a crucial role in the proof of the Wong–Zakai approximations for stochastic differential equations. We overcome this by carefully exploiting the independence of the two Brownian motions B and W . The rest of the paper is entirely devoted to the proof of the convergence result. The proof is involved and delicate. We need to complete the squares in last stage of the proof. Therefore we have to be very careful with the signs of each term in the estimates. Finally we hope to apply our results to obtain the support theorem of backward doubly stochastic differential equations and also to get some Wong–Zakai approximations of stochastic partial differential equations. We like to leave this study in a forthcoming paper given the length of the current article. 2. Main results
12 13 14 15 16 17 18 19 20 21
22
For the simplicity of the exposition, we assume d = m = l = 1. The same proofs carry over to the multi-dimensional case without extra difficulties. Throughout the paper, C, Ci , i = 1, . . . denote generic constants whose values may change from line to line. We start with an a priori estimate for the family {(Y n , Z n ), n ≥ 1}. Proposition 2.1. Assume (H.1). Then there exists a constant C such that T n 2 n 2 sup sup E[(Yt ) ] + E (Z s ) ds ≤ C. n 0≤t≤T
23 24 25 26
27
(2.5)
28
t
Proof. By Ito’s formula, we have T n 2 n 2 2 (Yt ) + (Z s ) ds = (ξ ) + 2 t
t
−2 t
T
29
T
Ysn f (Ysn , Z sn )ds + 2
Ysn Z sn dW s .
t
T
Ysn g(Ysn )dBns
30
(2.6)
k+2 k−1 + − For s ∈ [ 2kn , k+1 2n ], set s = 2n and s = 2n . In view of (H.1), it is easy to see that there exists C1 > 0 such that T T 1 T n 2 n n n n 2 2 Ys f (Ys , Z s )ds ≤ C1 (Ys ) ds + (Z s ) ds + C1 . (2.7) 4 t t t
31
32 33
34
4 1
2
3
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
Now, the third term on the right side of (2.6) can be written as T T T 2 Ysn g(Ysn )dBns = 2 Ysn+ g(Ysn+ )dBns + 2 (Ysn − Ysn+ )g(Ysn )dBns t t t T n n n +2 Ys + (g(Ys ) − g(Ys + ))dBns t
:= I1 + I2 + I3 .
4
5
6
As a stochastic integral, we have E[I1 ] = 0. By Eq. (1.4) it follows that T s + T s + n n n n n n I2 = 2 f (Yu , Z u )du g(Ys )dBs + 2 g(Yu )dBu g(Ysn )dBns t
s
t
T
t
s
t
≤C
11
13
14
1 2n
1
2
,
and T
E[I2.2 ] ≤ C E s
t
16
T
≤C
T
t 18
s T
≤C
20
1
1
1
du(2n ) 2 (2n ) 2 ≤ C,
(2.11)
s
t
=2 = 0.
s
T
E t
22
1
du(E[| B˙ un |2 ]) 2 (E[| B˙ sn |2 ]) 2
where C is a constant independent of n. For the term I2.3 , we have T s + n n ˙n E[I2.3 ] = 2 E Z u dW u g(Ys ) Bs ds
21
s+
ds t
19
s+
ds
n n ˙ ˙ | Bu |du | Bs |ds
duE[| B˙ un | | B˙ sn |]
s
≤C
s+
s+
ds t
17
(2.10)
here, as in Section 1, B˙ sn is the piecewise derivative of Bsn , i.e., k k+1 , B˙ sn = 2n B k+2 − B for s ∈ k+1 , 2n 2n 2n 2n
15
(2.9)
By the boundedness of f and g, we have T s + du E[| B˙ sn |]ds E[I2.1 ] ≤ C
12
Z un dW u g(Ysn )dBns
s
:= I2.1 + I2.2 + I2.3 .
8
10
s
s+
−2
7
9
(2.8)
g(Ysn ) B˙ sn E
s
s+
Z un dW u
Fs ds (2.12)
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
5
Putting together (2.10)–(2.12) we get
1
sup E[I2 ] ≤ C,
(2.13)
2
n
for some constant C. To bound I3 in (2.8), we write T 1 I3 = 2 Ysn+ dλg ′ (Ysn+ + λ(Ysn − Ysn+ ))(Ysn − Ysn+ )dBns t 0 + 1
T
Ysn+
= 2 t
0
T
+2 t
dλg
Ysn+
Ysn+
T
−2 t
′
(Ysn+
1
dλg
′
0 1
dλg 0
′
+ λ(Ysn
− Ysn+ ))
(Ysn+
+ λ(Ysn
− Ysn+ ))
(Ysn+
+ λ(Ysn
− Ysn+ ))
s
f (Yun ,
s
s+ s
4
Z un )du
dBns
5
g(Yun )dBnu
dBns
6
s+ s
3
Z un dW u
dBns
:= I3.1 + I3.2 + I3.3 .
7
(2.14)
For I3.1 , we have
8
9
T
E[I3.1 ] ≤ C t
T
≤C t
T
≤C t
1 E |Ysn+ | n | B˙ sn | ds 2 T 2 1 n 2 n 2 ˙ E[(Ys + ) ]ds + C E | Bs | ds 2n t
E[(Ysn+ )2 ]ds + C.
10
11
(2.15)
Similarly we have
12
13
T
E[I3.2 ] ≤ C
s+
duE[|Ysn+ | | B˙ un | | B˙ sn |]ds
ds t
s T
≤C
s+
ds
t
s T
≤C
s+
ds
t
s T
≤C t
1
14
1
du(E[(Ysn+ )2 ]) 2 (E[| B˙ un |2 | B˙ sn |2 ]) 2 ds
15
1
du(E[(Ysn+ )2 ]) 2 2n
E[(Ysn+ )2 ]ds + C.
By virtue of the independence of Ysn+ and B˙ sn , we have + s T E[I3.3 ] ≤ C E |Ysn+ | | B˙ sn | Z un dW u ds t s 2 12 T s + 1 ≤C (E[(Ysn+ )2 | B˙ sn |2 ]) 2 E Z un dW u ds s t
16
(2.16)
17
18
19
20
6
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
1
T
=C t
2
T
=C t
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
1 ≤ 4
1 1 (E[(Ysn+ )2 ]) 2 (E[| B˙ sn |2 ]) 2
2 E s
t
1 E 4
1 ≤ E 4
=
1 2
s+
s
n
T
t
t
T
s+
(Z un )2 du
(Z un )2 du2n (Z un )2 du
T
+ C2 t
T
+ C2 t
ds
2
t
u−
2
(Z un )2 du
ds
1 (Z un )2 du 1
ds + C2
u
s+
s
(E[(Ysn+ )2 ]) (2n ) E 1 2
T
E
ds
E[(Ysn+ )2 ]ds T
E[(Ysn+ )2 ]ds
E[(Ysn+ )2 ]ds.
(2.17)
(2.15)–(2.17) yield that T T 1 (Z un )2 du + C2 E[(Ysn+ )2 ]ds + C. E[I3 ] ≤ E 4 t t
(2.18)
It follows from (2.6), (2.7), (2.13) and (2.18) that T T 1 n 2 n 2 2 E[(Yt ) ] + E (Z s ) ds ≤ E[(ξ ) ] + C E[(Ysn+ )2 ]ds + C. 2 t t Applying Gronwall’s inequality, we complete the proof of the Proposition.
(2.19)
We also need the following estimate. Lemma 2.2. There is a constant C such that T 1 n n 2 E |Ys + − Ys | ds ≤ C n . 2 0 T 1 E |Ys + − Ys |2 ds ≤ C n . 2 0 Proof. Let us prove (2.20). The proof of (2.21) is similar. Indeed, we have 2 T T s + E |Ysn+ − Ysn |2 ds ≤ C E f (Yun , Z un )du ds 0 0 s 2 T s + +CE g(Yun )dBnu ds 0 s 2 T s + n +CE Z u dW u ds 0 s s+ T 1 (s + − s) | B˙ un |2 duds ≤ C n + CE 2 0 s
(2.20) (2.21)
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
T
s+
+CE 0
s
7
(Z un )2 duds
1
≤C
T 1 1 n 2 + sup E (Z ) du u n 2 2n n 0
≤C
1 . 2n
2
(2.22)
Applying H¨older’s inequality it follows immediately from (2.20) and (2.21) that
T
|Ysn+
E 0
T
E
|Ys + 0
1 1 2 ≤C . 2n 1 1 2 − Ys |ds ≤ C . 2n − Ysn |ds
4
(2.23)
5
(2.24)
6
To establish the main result, we need a strengthened version of Proposition 2.1. We introduce the following condition. (H.2). Suppose yg(y),
y 3 g(y)
∈
Cb1 (R).
7 8
9
Remark. (H.2) clearly holds if g has a compact support.
10
Proposition 2.3. Assume ξ ∈ L 4 (Ω ; R d ) and suppose that (H.1), (H.2) hold. Then, there exists a constant C such that 2 T n 4 n 2 sup E[ sup |Yt | ] + E (Z s ) ds ≤ C. (2.25) n
3
11 12
13
t
0≤t≤T
Proof. By Ito’s formula, T |Ytn |4 + 6 (Ysn )2 (Z sn )2 ds = (ξ )4 + 4 t
+4 t
T
14
T t
(Ysn )3 f (Ysn , Z sn )ds
(Ysn )3 g(Ysn )dBns
:= (ξ )4 + I1n + I2n + I3n .
T
−4 t
15
(Ysn )3 Z sn dW s
16
(2.26)
By the linear growth of f , we have T n I1 = 4 (Ysn )3 f (Ysn , Z sn )ds
17
18
19
t
T
≤C t
|Ysn |3 (1 + |Ysn | + |Z sn |) T
≤ C +C t
(Ysn )4 ds +
t
T
(Ysn )2 (Z sn )2 ds.
20
(2.27)
21
8
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
Set F(y) = 4y 3 g(y). By (H.2) and (2.20), we have
1
E[I2n ] = E
2
T
t
T
= E
3
t
F(Ysn )dBns
F(Ysn+ )dBns
t
≤C E
T
t
≤C
6
1 2n
+E t
1
2
− Ysn+ |2 ds
|Ysn
T
|Ysn − Ysn+ | | B˙ sn |ds
≤ CE
5
T
4
(F(Ysn ) −
F(Ysn+ ))dBns
21 E
T
t
| B˙ sn |2 ds
21
1
(2n ) 2 ≤ C.
(2.28)
Substituting (2.28), (2.27) into (2.26) and taking expectation we get
7
E[|Ytn |4 ] + 5E
8
t
T
(Ysn )2 (Z sn )2 ds
≤ C + E[(ξ ) ] + C 4
T
t
E[|Ysn |4 ]ds.
(2.29)
In particular,
9
≤ C + E[(ξ ) ] + C
E[|Ytn |4 ]
10
4
T
t
E[|Ysn |4 ]ds,
where the constant C is independent of n. By Gronwall’s inequality we obtain
11
sup sup E[|Ytn |4 ] ≤ C(1 + E[(ξ )4 ]),
12
n 0≤t≤T
here again C denotes a generic constant whose value changes from line to line. The above inequality together with (2.29) further implies
13 14
sup
15
n
16
Q2
sup E[|Ytn |4 ] + E
0≤t≤T
0
|Ysn |2 (Z sn )2 ds
E[ sup 0≤t≤T
|I2n (t)|]
= E
sup
0≤t≤T
T t
18
(2.30)
≤ C.
Now applying Burkholder’s inequality and taking into account (H.2) and (2.20), we have
17
T
≤ C + CE 0
F(Ysn+ )dBns T
+ E
T
0
(F(Ysn ) −
F(Ysn+ )) B˙ sn ds
12
(Ysn+ )6 g 2 (Ysn+ )ds
≤ C,
(2.31)
9
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
and
1
E
sup 4
0≤t≤T
t
T n 3 n (Ys ) Z s dW s ≤ C E
T 0
(Ysn )6 (Z sn )2 ds
≤ C E sup (Ytn )2
T
0
0≤t≤T
2
(Ysn )2 (Z sn )2 ds
1 E[ sup (Y n )4 ] + C E 4 0≤t≤T t
≤
21
T
0
21
3
(Ysn )2 (Z sn )2 ds . (2.32)
By virtue of (2.30)–(2.32), taking supremum in (2.26) we obtain
5
sup E[ sup (Ytn )4 ] < ∞. n
4
(2.33)
6
0≤t≤T
Next we show that 2 T n 2 sup E (Z t ) dt < ∞. n
7
(2.34)
8
0
It follows from Ito’s formula that T (Z sn )2 ds = (ξ )2 − (Y0n )2 + 2 0
9
T
0
+2 0
T
Ysn f (Ysn , Z sn )ds
Ysn g(Ysn )dBns − 2
T
0
10
Ysn Z sn dW s .
(2.35)
Hence,
11
12
E 0
2 T n 2 4 n 4 (Z s ) ds ≤ C E[(ξ ) ] + C E[(Y0 ) ] + C E
T
0
+CE 0
T
Ysn g(Ysn )dBns
2
T
+ CE 0
By the linear growth of f , it is easy to see that 2 T n n n ≤ CE CE Ys f (Ys , Z s )ds
0
0
Applying Burkholder’s inequality, we have 2 T n n CE Ys Z s dW s = CE 0
2
.
13
(2.36)
14
15
T
(Ysn )4 ds +
1 E 4
0
T
(Z sn )2 ds
2
. (2.37)
16
17
T
0
(Ysn )2 (Z sn )2 ds
≤ CE
Ysn Z sn dW s
Ysn f (Ysn , Z sn )ds
2
sup (Ytn )2
0≤t≤T
0
T
18
(Z sn )2 ds
1 ≤ C E[ sup (Ytn )4 ] + E 4 0≤t≤T
19
T 0
(Z sn )2 ds
2
.
(2.38)
20
10
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
T T To bound E[( 0 Ysn g(Ysn )dBns )2 ], we set h nt = t Ysn g(Ysn )dBns . By the chain rule, T (h nt )2 = 2 h ns Ysn g(Ysn )dBns
1
2
t
T
h ns+ Ysn+ g(Ysn+ )dBns + 2
=2
3
t
T
+2
4
t 5
Q3
6
7
8
9
10
Q4
T
t
(h ns − h ns+ )Ysn+ g(Ysn+ )dBns
h ns+ (Ysn g(Ysn ) − Ysn+ g(Ysn+ ))dBns .
(2.39)
T As a stochastic integral, E[ t h ns+ Ysn+ g(Ysn+ )dBns ] = 0. Applying H¨older’s inequality, we have T (h ns − h ns+ )Ysn+ g(Ysn+ )dBns 2E t + T s n n n n n n ˙ ≤ 2E Yu g(Yu )dBu |Ys g(Ys )| | Bs |ds t s T s + ˙n n ˙ (2.40) | Bu |du | Bs |ds ≤ C. ≤ CE t s In view of the independence of h ns+ and B˙ sn , keeping in mind the assumption (H.2) we have T 2E h ns+ (Ysn g(Ysn ) − Ysn+ g(Ysn+ ))dBns t
T
|h ns+ | | B˙ sn | |Ysn
11
≤ CE
12
≤C E
t
T
|Ysn
t
13
T
≤C t
14
15
− Ysn+ |ds
− Ysn+ |2 ds
E[|h ns+ |2 ]ds
21 t
12
T
12
E[|h ns+ |2 ]E[| B˙ sn |2 ]ds T
≤C +C t
E[|h ns+ |2 ]ds
.
(2.41)
Taking expectation on both sides of (2.39) and collecting the terms in (2.40), (2.41), we get T n 2 n 2 E[(h t ) ] ≤ C + C E[|h s + | ]ds . t
16
17
This yields that sup E[(h n0 )2 ] < ∞.
(2.42)
n
18
Combining (2.36)–(2.38) and (2.42) together, we arrive at 2 T
19
sup E n
20
0
(Z sn )2 ds
The proof is complete.
< ∞.
(2.43)
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
11
The following theorem is the main result of the paper.
1
Theorem 2.4. Assume E[|ξ |4 ] < ∞ and that (H.1) and (H.2) are in place. Then it holds that T n 2 n 2 (Z s − Z s ) ds = 0. lim sup E[(Yt − Yt ) ] + E (2.44) n→∞ 0≤t≤T
2
3
t
Proof. By Ito’s formula, we have T n 2 (Yt − Yt ) + (Z sn − Z s )2 ds
4
5
t
T
=2 t
(Ysn − Ys )( f (Ysn , Z sn ) − f (Ys , Z s ))ds T
(Ysn − Ys )g(Ysn )dBns +
+2 t T
(Ysn − Ys )g(Ys )dBs − 2
−2 t T
T
6
g 2 (Ys )ds
7
t T
t
(Ysn − Ys )(Z sn − Z s )dW s
8
(Ysn − Ys )gg ′ (Ys )ds
− t
9
:= I1n + I2n + I3n + I4n + I5n + I6n .
(2.45)
Crucially we need to bound the term I2n . We write T n I2 = 2 (Ysn − Ys )g(Ysn )dBns
10
11
12
t
T
[(Ysn − Ys ) − (Ysn+ − Ys + )]g(Ysn )dBns
= 2 t
T
+2 t
T
+2 t
13
(Ysn+ − Ys + )(g(Ysn ) − g(Ysn+ ))dBns
14
(Ysn+ − Ys + )g(Ysn+ )dBns
15
(2.46)
:= A + B + C.
As a stochastic integral, we have E[C] = 0. We need the precise bounds (including the signs) for E[A] and E[B]. These precise bounds will be used to complete squares in the proof. We will give these estimates in Lemmas 2.5 and 2.6. Because the proofs of these lemmas are quite involved and long, let us first complete the proof of the theorem by admitting Lemmas 2.5 and 2.6. Taking expectation in (2.45) we obtain T n 2 n 2 E[(Yt − Yt ) ] + E (Z s − Z s ) ds
16
17 18 19 20 21 22
23
t
= 2E t
T
(Ysn − Ys )( f (Ysn , Z sn ) − f (Ys , Z s ))ds
24
12
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx T
1
+ 2E t
2
T
−E t
3
4
(Ysn
(Ysn
− Ys )g(Ysn )dBns
T
+E
g (Ys )ds 2
t
− Ys )gg (Ys )ds . ′
(2.47)
Collecting the estimates in Lemmas 2.5 and 2.6 we obtain from (2.47) that for any δ < 12 , T (Z sn − Z s )2 ds E[(Ytn − Yt )2 ] + E t
5
6
7
1 −δ
T 1 2 n 2 + C E |Y − Y | ds s s 2n t T T 1 + E |Z sn − Z s |2 ds + E g 2 (Ys )ds 2 t t T T ′ n n ′ n −E (Ys − Ys )gg (Ys )ds + E (Ys + − Ys + )gg (Ys + )ds
≤C
t
t
T
8
− 2E t
9
10
g(Ys + )g(Ysn+ )ds + E
T
g 2 (Ysn+ )ds .
t
(2.48)
In view of (2.20) and (2.21), we can now replace s + by s on the right side of (2.48) to obtain T E[(Ytn − Yt )2 ] + E (Z sn − Z s )2 ds t
11
12
13
1 −δ
T 1 n 2 + C E |Y − Y | ds s s 2n t T T 1 n 2 2 + E |Z s − Z s | ds + E g (Ys )ds 2 t t T T n ′ n ′ n −E (Ys − Ys )gg (Ys )ds + E (Ys − Ys )gg (Ys )ds
2
≤C
t
t
T
14
− 2E t
g(Ys )g(Ysn )ds
T
+E
g t
2
(Ysn )ds
(2.49) 1
15 16
17
by simply changing the value of the constant C in front of ( 21n ) 2 −δ . Completing the square in (2.49) we get that T E[(Ytn − Yt )2 ] + E (Z sn − Z s )2 ds t
18
19
20
1 −δ
T 1 n 2 + C E |Y − Y | ds s s 2n t T T 1 n 2 n 2 + E |Z s − Z s | ds + E (g(Ys ) − g(Ys )) ds 2 t t T +E (Ysn − Ys )(gg ′ (Ysn ) − gg ′ (Ys ))ds .
2
≤C
t
(2.50)
13
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
By the Lipschitz continuity of gg ′ and g, it follows from (2.50) that T 1 E[(Ytn − Yt )2 ] + E (Z sn − Z s )2 ds 2 t 1 −δ T 1 2 n 2 + C E |Y − Y | ds ≤C . s s 2n t
1
2
(2.51)
Application of Gronwall’s inequality completes the proof of Theorem 2.4.
3
4
The remaining part of the paper is devoted to the proofs of Lemmas 2.5 and 2.6.
5
Lemma 2.5. For any given δ < 12 , we have T E[A] = 2E [(Ysn − Ys ) − (Ysn+ − Ys + )]g(Ysn )dBns
6
7
t
≤C
1 2n
1 −δ
2
T
− 2E t
g(Ys + )g(Ysn+ )ds
+E
T
g t
2
(Ysn+ )ds
.
(2.52)
Proof. By the equations satisfied by Y n and Y we have T A = 2 [(Ysn − Ys ) − (Ysn+ − Ys + )]g(Ysn )dBns t T s + T s + n n n n = 2 g(Yu )dBu g(Ys )dBs − 2 g(Yu )dBu g(Ysn )dBns t s t s + T
+2 t
T
−2 t
8
9
10
11
s
[ f (Yun , Z un ) − f (Yu , Z u )]du g(Ysn )dBns s + T s + s n n n ′ (Z u − Z u )dW u g(Ys )dBs − gg (Yu )du g(Ysn )dBns s
t
12
13
s
:= A1 + A2 + A3 + A4 + A5 .
(2.53)
Clearly,
14
15
E[|A5 |] ≤ C
1 2n
T
E[| B˙ sn |]ds ≤ C
t
1 2n
1
2
,
(2.54)
and also
16
17
E[|A3 |] ≤ C
1 2n
1 2
.
(2.55)
By conditioning on Fs , we find that T s + n n ˙n E[A4 ] = −2 E (Z u − Z u )dW u g(Ys ) Bs ds t s T s + n n ˙n = −2 E E (Z u − Z u )dW u Fs g(Ys ) Bs ds t
= 0.
18
19
20
21
s
(2.56)
22
14 1
2
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
To bound A1 , we write it as T s + n n A1 = 2 g(Yu )dBu g(Ysn )dBns t
s
3
T
= 2 t
s
4
T
+2 t
5
T
+2 t
6
7
8
11
12
(2.57) [ 2kn , k+1 2n ]
we see that Splitting the interval [t, T ] into subintervals − k+1 2n → n 2n B k+2 A13 = 2 − B g 2 Y k+2 B ds k+3 k+2 − B k+1 n n n n +2
k 2n
2n
k+1 2n
k 2n
16
n g 2 Y k+2
B k+2 − B k+1 n n 2
2 − → ds
2
(2.58)
2
2
2
2
2
Integrating with respect to s, we get that E[A13 ] = E[A13,2 ] 2 2 n B k+2 − B k+1 = E g Y k+2 n n = E
2n
g
2
n
Y k+2
2
2n
T
= E
2
B k+2 − B k+1 n n 2
2
2
1 − n 2
+E
T
g t
2
(Ysn+ )ds
g 2 (Ysn+ )ds ,
(2.60)
where the fact that the sequence {(B k+2 − B k+1 )2 − 21n , k ≥ 0} is a martingale has been used. 2n 2n For the term A11 in (2.57), we have 1 T s+ A11 = 2 dλ g ′ (Ysn+ + λ(Yun − Ysn+ ))(Yun − Ysn+ )dBnu g(Ysn )dBns 0
t
1
22
k+1 2n
2
(2.59)
t
21
2
= 0.
20
(2n )2 s −
2n
k
19
2
we have Conditioning on F k+2 2n n B E[A13,1 ] = −2 E g 2 Y k+2 B k+2 − B E k+2 − B k+1 F k+2 k+3 n n n n n
k
18
2n
17
2
:= A13,1 + A13,2 .
13
15
g(Ysn )dBns
g(Ysn+ )(Bsn − Bsn+ )g(Ysn+ )dBns
k
14
g(Ysn+ ))dBnu
g(Ysn+ )(Bsn − Bsn+ )(g(Ysn ) − g(Ysn+ ))dBns
k 10
(g(Yun ) −
:= A11 + A12 + A13 .
k 9
s+
= 2
s
T
s+
dλ 0
t
s
g ′ (Ysn+ + λ(Yun − Ysn+ ))
u
s+
f (Yvn , Z vn )dv dBnu g(Ysn )dBns
15
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
1
T
+2
s+
g ′ (Ysn+ + λ(Yun − Ysn+ ))
dλ 0
t 1
s T
−2
s+
dλ
g
0
t
s
′
(Ysn+
+ λ(Yun
− Ysn+ ))
s+ u
g(Yvn )dBnv dBnu g(Ysn )dBns
s+
Z vn dW v
u
dBnu g(Ysn )dBns
:= A11,1 + A11,2 + A11,3 .
2
(2.61)
The first two terms on the right can be bounded as follows. E[A11,1 ] ≤ ≤ E[A11,2 ] ≤ ≤
(2.62)
5
(2.63)
6
The last term A11,3 can be estimated as follows. + s+ s T E[A11,3 ] ≤ C E ds | B˙ un | | B˙ sn | Z vn dW v du t s u T
s+
≤C
ds s
t
n
T
t T
du E
ds
≤ C2
≤C
ds E
s+
s
T
≤C
s+
dsE t
s
≤ C sup E
T
n
s+
u
s
t
7
8
2 12 s+ 1 Z vn dW v du(E[| B˙ un |2 | B˙ sn |2 ]) 2 E u s+
t
9
1
2
(Z vn )2 dv
10
1 2
(Z vn )2 dv
11
1 2
(Z vn )2 dv
(Z vn )2 dv
21
≤C E t
1 2n
1
2
.
T
(Z vn )2 dv
v
v−
12 ds
12
(2.64)
Putting together (2.62)–(2.64) we get E[A11 ] ≤ C
1 2n
1
2
.
3
4
T s+ 1 E[| B˙ un | | B˙ sn |]du ds C n 2 t s 1 1 1 1 1 C n (2n ) 2 (2n ) 2 n ≤ C n . 2 2 2 T s+ s+ dv E[| B˙ un | | B˙ vn | | B˙ sn |] du C ds u s t 2 1 3 1 1 2 n 2 C(2 ) . ≤ C 2n 2n
1
13
14
(2.65)
15
16 1
2
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
Similarly the term A12 can be decomposed as s+ 1 T n dBnu g ′ (Ysn+ + λ(Ysn − Ysn+ ))(Ysn − Ysn+ )dBns A12 = 2 dλ g(Ys + ) s 0 t + + 1
T
= 2
3
dλ 0
t 1
+2
4
t
1
−2
9
t
11
1 , 2n
E[A12, j ] ≤ C
13
14
E[A12 ] ≤ C
1 2n
1 2
E[A1 ] ≤ C
1 2n
T
+ λ(Ysn
− Ysn+ ))
s
s
g(Yvn )dBnv dBns
s+
Z vn dW v
dBns (2.66)
.
(2.68)
T
+E
g t
s+
s T
−2 T
−2 t
2
(Ysn+ )ds
.
(2.69)
(g(Yu ) − g(Ys + ))dBu g(Ysn )dBns
g(Ys + )(Bs − Bs + )(g(Ysn ) − g(Ysn+ ))dBns g(Ys + )(Bs − Bs + )g(Ysn+ )dBns
:= A21 + A22 + A23 .
(2.70)
Splitting the interval [t, T ] into subintervals [ 2kn , k+1 2n ], k+1 − 2n → n n A23 = −2 g Y k+2 g Y B − B B ds k+2 2 k+2 − B k+1 s k+2 n n n n k
21
s
dBnu g ′ (Ysn+
s+
(2.67)
= −2
t
20
s+
dBnu g ′ (Ysn+ + λ(Ysn − Ysn+ ))
dBns
s
19
s
s
Z vn )dv
j = 1, 2, 3.
2
t
18
s+
f (Yvn ,
Now we turn to A2 which can be written as T s + g(Yu )dBu g(Ysn )dBns A2 = −2
17
g(Ysn+ )
1
t
16
g(Ysn+ )
s
− Ysn+ ))
Combining (2.60), (2.65) and (2.68) we get
15
s
+ λ(Ysn
Hence,
12
dBnu g ′ (Ysn+
Using the similar arguments as for (2.62) and (2.63) we can show that
10
s
:= A12,1 + A12,2 + A12,3 .
6
8
T
dλ 0
7
T
dλ 0
5
g(Ysn+ )
= −2
k
k 2n
k+1 2n k 2n
2
2n
2
2
2
− → n n g Y k+2 g Y B − B B ds k+2 − B k+1 k+1 2 s k+2 n n n n
2
2n
2
2
2
17
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
2 n −2 g Y k+2 g Y − B B k+1 k+2 k+2 n n n 2n
2
k
2
1
2
:= A23,1 + A23,2 .
(2.71)
it follows that Using the independence of the increments of B and conditioning on F k+2 n
2
3
2
E[A23,1 ] = −2
k+1 2n
k 2n
k
× E
n 2 ds E g Y k+2 g Y k+2 n → n−
Bs − B k+1 n 2
= −2
k+1 2n
× E
B k+2 − B k+1 F k+2 2n 2n 2n
5
− → n g Y 2n ds E g Y k+2 k+2 n
k 2n
k
4
2n
2
Bs − B k+1 n 2
B k+2 − B k+1 n n 2
6
2n
2
7
2
= 0.
(2.72)
Now,
8
9
A23,2
2 1 n B = −2 g Y k+2 − B − g Y k+2 k+1 k+2 2n 2n 2n 2n 2n k T
−2 t
10
g(Ys + )g(Ysn+ )ds.
(2.73)
By conditioning on F k+2 we see that the expectation of the first term of the above equation 2n vanishes. Hence, (2.71)–(2.73) together yield E[A23 ] = E[A23,2 ] = −2E t
T
g(Ys + )g(Ysn+ )ds .
(2.74)
To bound the term A21 we write 1 T s+ dλ g ′ (Ys + + λ(Yu − Ys + ))(Yu − Ys + )dBu g(Ysn )dBns A21 = −2 0
t
1
s+
0
t 1
−2
T
+2
t 1
T
t
s+
u s+
g (Ys + + λ(Yu − Ys + )) ′
s
dλ 0
s
dλ 0
g (Ys + + λ(Yu − Ys + )) ′
dλ
= −2
12 13
14
15
16
s
T
11
s+
u s+
g (Ys + + λ(Yu − Ys + )) ′
s
:= A21,1 + A21,2 + A21,3 .
u
s+
f (Yv , Z v )dv dBu g(Ysn )dBns
17
g(Yv )dBv dBu g(Ysn )dBns
18
Z v dW v dBu g(Ysn )dBns
19
(2.75)
20
18
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
We will estimate each of the terms on the right. First we have 1 T s + g ′ (Ys + + λ(Yu − Ys + )) E[A21,1 ] ≤ C dλ E s 0 t + s ˙n × f (Yv , Z v )dv dBu | Bs | ds u 1 T s + g ′ (Ys + + λ(Yu − Ys + )) E ≤C dλ s 0 t 2 21 + s 1 f (Yv , Z v )dv dBu (E[| B˙ sn |2 ]) 2 ds × u 1 T s+ n 12 E ≤ C(2 ) dλ g ′ (Ys + + λ(Yu − Ys + ))2
1
2
3
4
5
6
0
t
f (Yv , Z v )dv
u 1
≤ C(2n ) 2
8
T
1
2
s+
×
7
s
2
du ds 1
[(s + − s)(s + − s)2 ] 2 ds
t
≤C
9
10
Q5
12
13
15
T
2 21 1 2 × g(Yv )dBv dBu E | B˙ sn |2 ds u 1 T s+ 1 E ≤ C(2n ) 2 dλ g ′ (Ys + + λ(Yu − Ys + ))2
s+
× u
s+
0
16
(2.76)
+ s g ′ (Ys + + λ(Yu − Ys + )) E dλ E[A21,2 ] ≤ C s 0 t + s ˙n × g(Yv )dBv dBu | Bs | ds u 1 T s + E ≤C dλ g ′ (Ys + + λ(Yu − Ys + )) s 0 t 1
14
.
Similarly,
11
1 2n
t
s
2 g(Yv )dBv
1 2
du ds
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
1 2
≤ C(2 ) n
T
1 2
≤ C(2 )
s+
s+
T
t
≤C
s+
u
2
1
2 g(Yv )dBv
1 2 du E
s
1
1 2n
E
s
t
n
s+ s
19
2
du ds
1
1 2
g (Yv )dv 2
ds
.
2
(2.77)
By H¨older’s inequality and Ito’s isometry, we have 1 T s + g ′ (Ys + + λ(Yu − Ys + )) E[A21,3 ] ≤ C dλ E s 0 t + s ˙n Z v dW v dBu | Bs | ds × u 1 T s + E ≤C dλ g ′ (Ys + + λ(Yu − Ys + )) s 0 t
Z v dW v
× u 1
≤ C(2n ) 2
1
dλ 0
1 2
t
≤ C(2 ) n
1 2
s+
s+
t
T
≤C t
T t
≤C
s+
1 2n
1
2
.
Z v2 dv
8
9
10
2
s+
Z v dW v
1 2 du E
s+
s
s
≤C E
E
u
E
7
2
s
6
du ds
s
T
5
1
2
T
4
s
Z v dW v
u
≤ C(2 )
t
s+
×
n
T
Q6
2 21 1 dBu (E[| B˙ sn |2 ]) 2 ds s+ E g ′ (Ys + + λ(Yu − Ys + ))2
s+
3
1 2
du ds
11
1 2
Z v2 dv
ds
12
1 2
Z v2 dv ds
21
v
v−
13
ds
14
(2.78)
15
20 1
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
It follows from (2.76)–(2.78) that E[A21 ] ≤ C
2
3
4
5
6
7
14
16
1 E[A22,2 ] ≤ C n 2
1 2n
≤C
≤C
T
1 2n
1
1
(E[|Bs − Bs + |2 ]) 2 (E[sup | B˙ un |4 ]) 2 ds u
t
1 2n
1
2
(E[ sup
t
1 −δ 2
(2n )2
T
|r −v|≤ 21n
1
|Br − Bv |4 ]) 2 ds
.
(2.82)
By H¨older’s inequality, T
E[A22,3 ] ≤ C ≤C ≤C
2 12 s + (E[|Bs − Bs + |4 ]) (E[| B˙ sn |4 ]) E Z un dW u ds s 1 4
T
s+
E t
19
(2.81)
and
t
18
(2.80)
T 1 1 1 (E[|Bs − Bs + |2 ]) 2 (E[| B˙ sn |2 ]) 2 ds n 2 t 1 ≤ C n, 2
17
(2.79)
E[A22,1 ] ≤ C
15
.
Now,
11
13
2
:= A22,1 + A22,2 + A22,3 .
10
12
1
Let us turn to the term A22 . We have T A22 ≤ C |Bs − Bs + | |Ysn − Ysn+ | | B˙ sn |ds t + T s n n f (Yu , Z u )du | B˙ sn |ds ≤ C |Bs − Bs + | s t T s + g(Yun )dBnu | B˙ sn |ds +C |Bs − Bs + | s t + T s Z un dW u | B˙ sn |ds +C |Bs − Bs + | s t
8
9
1 2n
1 . 2n
s
1 4
1
(Z un )2 du
2
ds (2.83)
21
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
It follows from (2.80)–(2.83) that E[A22 ] ≤ C
1 −δ
1 2n
2
1
.
(2.84)
2
Collecting (2.74), (2.79) and (2.80) we obtain E[A2 ] ≤ C
1 2n
1 −δ
2
− 2E t
T
3
g(Ys + )g(Ysn+ )ds .
(2.85)
4
The proof is now completed by putting (2.54)–(2.56), (2.69) and (2.85) together.
5
Lemma 2.6. For any given δ < 12 , we have T
E[B] = 2E t
≤C
1 2n
6
(Ysn+ − Ys + )(g(Ysn ) − g(Ysn+ ))dBns
1 −δ
2
+E t
T
(Ysn+
− Ys + )gg
′
7
(Ysn+ )ds
.
(2.86)
8
Proof. Write
9
T
(Ysn+ − Ys + )(g(Ysn ) − g(Ysn+ ))dBns
B = 2 t 1
T
= 2
(Ysn+ − Ys + )g ′ (Ysn+ + λ(Ysn − Ysn+ ))(Ysn − Ysn+ )dBns +
dλ 0
t 1
T
= 2
(Ysn+ − Ys + )g ′ (Ysn+ + λ(Ysn − Ysn+ ))
dλ 0
t
1
+2
T
dλ 0
−2
t 1
T
dλ 0
10
t
(Ysn+
− Ys + )g
(Ysn+
− Ys + )g
′
′
(Ysn+
+ λ(Ysn
− Ysn+ ))
(Ysn+
+ λ(Ysn
− Ysn+ ))
:= B1 + B2 + B3 . By Proposition 2.2, we have T 1 E[B1 ] ≤ C n E |Ysn+ − Ys + | | B˙ sn |ds 2 t T 1 1 1 ≤C n (E[|Ysn+ − Ys + |2 ]) 2 (E[| B˙ sn |2 ]) 2 ds 2 t 1 1 2 ≤C . 2n
s
11
f (Yvn , Z vn )dv dBns
s
s+
s
s
12
g(Yvn )dBnv
dBns
13
s+
Z vn dW v
dBns
14
(2.87)
15
Q7
16
17
18
(2.88)
19
22 1
2
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
B2 is further written as follows. 1 T B2 = 2 dλ (Ysn+ − Ys + )[g ′ (Ysn+ + λ(Ysn − Ysn+ )) − g ′ (Ysn+ )] 0 t + s
×
3
s
1
+2
4
g(Yun )dBnu dBns
dλ 0
t 1
+2
5
9
10
11
14
16
18
19
20
s
s+
(g(Yun ) −
g(Ysn+ ))dBnu
dBns
(Ysn+ − Ys + )g ′ (Ysn+ )g(Ysn+ )(Bsn − Bsn+ )dBns (2.89)
:= B21,1 + B21,2 + B21,3 .
(2.90)
The following two inequalities will be used frequently in sequel. sup | B˙ un | ≤ 2n
sup
(2.91)
|Br − Bs |.
|r −s|≤ 21n
For any δ > 0 and p ≥ 1, there exists a constant C p,δ such that E[ sup
p
|Br − Bs | ] ≤ C p,δ
|r −s|≤ 21n 17
(Ysn+ )
By the Lipschitz continuity of it follows that + T s B21 ≤ C |Ysn+ − Ys + | |Ysn − Ysn+ | g(Yun )dBnu | B˙ sn |ds t s T s + s + n n ˙ n n n n g(Yu )dBu | Bs |ds f (Yv , Z v )dv ≤ C |Ys + − Ys + | s s t 2 T s + g(Yun )dBnu | B˙ sn |ds +C |Ysn+ − Ys + | s t + + T s s +C |Ysn+ − Ys + | Z vn dW v g(Yun )dBnu | B˙ sn |ds s s t
u
15
− Ys + )g
′
g′ ,
12
13
t
(Ysn+
:= B21 + B22 + B23 .
6
8
T
dλ
0
7
T
1 2n
p −δ 2
.
By H¨older’s inequality and (2.91), (2.92), we have + s T 1 ˙n n n E[B21,1 ] ≤ C n E |Ys + − Ys + | B |du| | B˙ s |ds s u 2 t + T s 1 |Ysn+ − Ys + | du sup |Br − Bs |2 ds ≤ C n (2n )2 E s |r −s|≤ 1 2 t n 2 T ≤ CE |Ysn+ − Ys + | sup |Br − Bs |2 ds t
|r −s|≤ 21n
(2.92)
23
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx T
≤C t
≤C
1
1
(E[|Ysn+ − Ys + |2 ]) 2 (E[ sup
|Br − Bs |4 ]) 2 ds
1
|r −s|≤ 21n
1 2n
1−δ
.
(2.93)
Similarly, in view of (2.92), we have T E[B21,2 ] ≤ C E |Ysn+ − Ys + | t
2
3
+ 2 s du sup | B˙ sn |3 ds s s
4
1 2 n 3 T n ≤C (2 ) E |Ys + − Ys + | sup |Br − Bs |3 ds 2n t |r −s|≤ 21n T 1 1 ≤ C2n (E[|Ysn+ − Ys + |2 ]) 2 (E[ sup |Br − Bs |6 ]) 2 ds
t
≤C
1 2n
5
6
|r −s|≤ 21n
1 −δ 2
,
(2.94)
and
7
8
+ + s s T n n n n ˙ ˙ | Bu |du | Bs |ds Z v dW v E[B21,3 ] ≤ C E |Ys + − Ys + | s s t + s 1 n 2 T n ≤ C n (2 ) E |Ys + − Ys + | Z vn dW v sup |Br − Bs |2 ds s |r −s|≤ 1 2 t
9
10
2n
≤ C2n
T
t
2 12 s+ 1 Z vn dW v − Ys + |4 ]) 4 E s
(E[|Ysn+
11
1
× (E[ sup
|Br − Bs |8 ]) 4 ds
12
|r −s|≤ 21n
≤C
1 2n
1−δ
n
2
t
T
(E[|Ysn+
− Ys + | ]) E
s+
1 4
4
s
1 2
(Z vn )2 dv
ds
13
21 T 1 1−δ n n 4 12 ≤C 2 (E[|Ys + − Ys + | ]) ds 2n t + 1 2 T s n 2 × E (Z v ) dv ds
t
≤C ≤C
14
15
s
1 2n
1−δ
1 2n
1 −δ
2 E n
t 2
,
T
(Z vn )2 dv
12
v
v−
ds
16
(2.95)
17
24 1
Q8
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
where the estimate (2.25) has been used. (2.93) yield E[B21 ] ≤ C
2
3
Q9
1 2n
1 −δ 2
.
(2.96)
By the Lipschitz continuity of g, we have + T
4
B22 ≤ C t
t
T
+C
6
t T
+C
7
t
11
E[B22, j ] ≤ C
13
14
E[B22 ] ≤ C
17
B23 = 2
2
+2
k 2n
(2.97)
,
j = 1, 2, 3.
(2.98)
.
(2.99)
n
Y k+2 − Y k+2 n
gg
2n
k+1 2n k 2n
′
Y k+2
− → 2n B k+2 B ds − B k+3 k+2 − B k+1 n n n n
2n
2
n Y k+2 − Y k+2 n 2n
n
2
2
n gg ′ Y k+2
2
2
2
2n
2 − k+1 → × (2n )2 s − n B k+2 − B k+1 ds n n 2 2 2 n ′ n = −2 Y k+2 − Y k+2 gg Y B k+2 − B k+3 B k+2 − B k+1 k+2 n n n n n 2n
+
+ t
T
2n
2
n Y k+2 − Y k+2 n 2n
k
19
1−δ
1 −δ
k+1 2n
k 18
1 2n
1 2n
k 16
s+
For the term B23 , we split the interval [t, T ] into subintervals [ 2kn , k+1 2n ] to get
k 15
Hence,
12
| B˙ sn |ds
By the similar arguments as above, we have
10
s
− Ysn+ | | B˙ un |du
:= B22,1 + B22,2 + B22,3 .
8
9
− Ys + |
|Ysn+
≤ C
5
|Yun
+ s ˙n n n f (Yv , Z v )dv | Bu |du | B˙ sn |ds − Ys + | u s + + s s n n ˙ n n g(Yv )dBv | Bu |du | B˙ sn |ds |Ys + − Ys + | u s + + s s ˙n n n Z v dW v | Bu |du | B˙ sn |ds |Ys + − Ys + | u s
T
s
|Ysn+
2
n gg ′ Y k+2
(Ysn+ − Ys + )gg ′ (Ysn+ )ds.
2n
2
2
B k+2 − B k+1 n n 2
2
2
2
−
2
1 2n
(2.100)
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
25
Conditioning on F k+2 it is easy to see that the expectation of the first two terms on the right 2n vanishes. Hence, T n ′ n E[B23 ] = E (Ys + − Ys + )gg (Ys + )ds . (2.101)
1 2
3
t
Collect the terms in (2.96), (2.99) and (2.101) to obtain E[B2 ] ≤ C
1 2n
1 −δ
T
2
+E t
4
(Ysn+ − Ys + )gg ′ (Ysn+ )ds .
(2.102)
Now we turn to the term B3 in (2.87). We split it as follows: T 1 B3 = −2 dλ [(Ysn+ − Ys + )g ′ (Ysn+ + λ(Ysn − Ysn+ )) 0
− (Ysn −2 t
6
7
t
− Ys )g
T
(Ysn
′
(Ysn )]
− Ys )g
′
s+ s
(Ysn )
Z vn dW v s
dBns
8
s+
Z vn dW v
dBns
9
:= B31 + B32 .
(2.103)
First we note that
10
11
E[B32 ] = −2 t
T
E (Ysn − Ys )g ′ (Ysn ) B˙ sn E
s
s+
n ds Z v dW v Fs
= 0.
12
(2.104)
By the Lipschitz continuity of g ′ , we have + T s n n n B31 ≤ C |Ys + − Ys | Z v dW v | B˙ sn |ds s t + T s n +C |Ys + − Ys | Z v dW v | B˙ sn |ds t s T s + n n n n +C |Ys − Ys | |Ys + − Ys | Z v dW v | B˙ sn |ds s t := B31,1 + B31,2 + B31,3 . Furthermore, B31,1
5
+ T s + s n n n ≤ C f (Yv , Z v )dv Z v dW v | B˙ sn |ds s t s + T s + s +C g(Yvn )dBnv Z vn dW v | B˙ sn |ds s t s
13
14
15
16
17
(2.105)
18
19
20
21
26
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx T
+C
1
t
:= B31,11 + B31,12 + B31,13 .
2
3
4
5
6
7
8
9
10
11
12
13
2 + s n Z v dW v | B˙ sn |ds s
Now, interchanging the order of integration, we have 2 21 T s + 1 1 E (Z vn )2 dv (E[| B˙ sn |2 ]) 2 ds E[B31,11 ] ≤ C n 2 t s 1 . (2.107) 2n In view of (2.91), (2.92), we have + T s 1 E[B31,12 ] ≤ C n Z vn dW v ds E sup | B˙ sn |2 2 t s s 2 12 s + 1 1 n 2 T Z vn dW v ds ≤ C n (2 ) (E[ sup |Br − Bv |4 ]) 2 E 2 1 s t |r −v|≤ 2n 1 T s+ 2 1 n 2 1 1−δ n 2 (Z ) dv ds ≤ C n (2 ) E v 2 2n s t 12 1−δ 1 T 1 n 2 1 1 2 n 2 ≤ C n (2 ) E[(Z v ) ]dv 2 2n 2n t 1 −δ 1 2 ≤C . (2.108) 2n ≤C
Q10 Using
the fact that | B˙ sn | is Fs -measurable we have 2 T s + Z vn dW v | B˙ sn |ds E[B31,13 ] ≤ C E s
t
14
= CE
T
t
15
(2.106)
s+
s T
s+
= CE t
s
(| B˙ sn |) Z vn dW v
≤ C(2n )E ( sup
≤ C(2n )(E[ sup |r −v|≤ 21n
18
≤C
1 2n
1 −δ 2
,
T
|r −v|≤ 21n
17
ds
(| B˙ sn |)(Z vn )2 dvds
16
2
1 2
|Br − Bv |)
t
s+
s
(Z vn )2 dvds
|Br − Bv |2 ]) E 1 2
t
T
s
s+
2 12 (Z vn )2 dvds
(2.109)
27
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
where (2.91), (2.92) again were used. (2.107)–(2.109) imply that E[B31,1 ] ≤ C
1 2n
1 −δ 2
1
.
(2.110)
For the term B31,2 we have + T s + s Z vn dW v | B˙ sn |ds f (Yv , Z v )dv B31,2 ≤ C s s t T s + s + +C g(Yv )dBv Z vn dW v | B˙ sn |ds s t s + T s + s +C Z v dW v Z vn dW v | B˙ sn |ds t s s
3
4
5
6
:= B31,21 + B31,22 + B31,23 .
(2.111)
By a similar argument as for (2.107), we have 1 E[B31,21 ] ≤ C . 2n
(2.112)
T
E[B31,22 ] ≤ C t
≤C
1 2n
1 2
1 2
2 2 + s s + 1 E g(Yu )dBu ds Z un (| B˙ sn |) 2 dW u E s s 1 2
T
E s
t
s+
E[B31,22 ] ≤ C
1 2n
1 −δ 2
2
(2.113)
ds.
.
The term B31,23 is bounded as + T
t
| B˙ sn |ds + C
t
T
s
2
s+
Z v dW v
| B˙ sn |ds,
(2.115)
which, together with the similar arguments as for the derivation of (2.23), yields
1 2n
1 −δ 2
.
1 −δ 2
.
16
17
(2.116)
(2.112)–(2.116) together give that 1 2n
14
15
Z vn dW v
s
12
13
(2.114)
2
s
B31,23 ≤ C
11
1 (Z un )2 | B˙ sn |du
From here following the same arguments as the derivation of (2.109) we obtain
9
10
E[B31,2 ] ≤ C
7
8
As for B31,22 , we have
E[B31,23 ] ≤ C
2
18
19
(2.117)
20
28 1
2
3
4
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
Finally we need to find a bound for E[B31,3 ]. Notice that + + T s s n n n n Z v dW v | B˙ sn |ds f (Yv , Z v )dv B31,3 ≤ C |Ys − Ys | s s t + + T s s Z vn dW v | B˙ sn |ds g(Yvn )dBnv +C |Ysn − Ys | s s t 2 T s + Z vn dW v | B˙ sn |ds +C |Ysn − Ys | s t := B31,31 + B31,32 + B31,33 .
5
6
7
8
9
Q11 We
have 1 E[B31,31 ] ≤ C n 2 1 ≤C n 2
≤C
11
≤C
12
T
t
2 12 s + 1 (| B˙ sn |) 2 Z vn dW v ds (E[|Ysn − Ys |2 ]) E s 1 2
T
E[|Ysn
t
≤C
1 2n
1
1 2n
1 −δ 2 E
1 2n
1 −δ
2
1 2n
T
t
s
2 |Br − Bv | ]) E 1 4
(E[ sup |r −v|≤ 21n
T
(Z vn )2 dv
s+
t
T t
2
1 + 2 s n n 2 | B˙ s |(Z v ) dv ds E s t 1 2 T s+ n 2 |Br − Bv |) (Z v ) dv ds
21 − Ys | ]ds 2
1 n 1 ≤ C n (2 ) 2 E ( sup 2 |r −v|≤
10
(2.118)
s
2 14 (Z vn )2 dvds
2 41
,
(2.119)
where (2.5), (2.25) have been used. Keeping in mind that |Ysn − Ys |, | B˙ sn | are Fs measurable we 14 Q12 have + 2 T s E[B31,33 ] = C E |Ysn − Ys | 15 Z vn dW v | B˙ sn | ds s t T s+ 16 =C E |Ysn − Ys | (Z vn )2 dv | B˙ sn | ds 13
t
s
17
≤ C2n E ( sup |Ysn − Ys |)( sup 0≤s≤T
|r −v|≤ 21n
|Br − Bv |)
1
18
≤ C2n (E[ sup |Ysn − Ys |4 ]) 4 (E[ sup 0≤s≤T
T
|r −v|≤ 21n
t
s+ s
1
|Br − Bv |4 ]) 4
(Z vn )2 dv ds
29
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
× E
T
t
≤ C2n
n
≤ C2
≤C
1 2n
s+
s
1 −δ 2 E
1 2n
1 −δ
t
1 −δ 2
1 2n
T
(Z vn )2
E
T t
1
v
v−
(Z vn )2 dv
2 12 ds dv 2 21
3
(2.120)
t
5
+ 2 s Z vn dW v | B˙ sn | ds s
The second term above is bounded by 2 + T s g(Yvn )dBnv | B˙ sn | ds E |Ysn − Ys | s t T n 3 3 n ≤ C(2 ) E ( sup |Br − Bv | ) |Ys − Ys | t
|r −v|≤ 21n
≤ C(2 )(E[ sup |r −v|≤ 21n
≤C
2
s
|Br − Bv | ]) E 6
n
1 −δ
6
2 + s g(Yvn )dBnv | B˙ sn | ds. E |Ysn − Ys | s
t
1 2n
1 2
t
.
E[B31,32 ] ≤ C
1 −δ 2
.
s+
2 |g(Yvn )|dv
ds
10
2 21 − Ys |ds
11
1 2n
12
13
(2.123)
(2.119)–(2.123) yield that
E[B31,3 ] ≤ C
8
9
(2.122)
1 2n
7
Q13
T
|Ysn
(2.121)
It follows now from (2.120)–(2.122) that
4
T
+C
2
.
As for the term B31.32 we have T E[B31,32 ] ≤ C E |Ysn − Ys |
(Z vn )2 dv ds
1 2n
2
2 12
14
15
1 −δ 2
.
(2.124)
16
30
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
It follows from (2.110), (2.117) and (2.124) that
1
E[B31 ] ≤ C
2
1 2n
1 −δ 2
.
(2.125)
This together with (2.104) yields
3
E[B3 ] ≤ C
4
1 2n
1 −δ 2
.
The lemma now follows from (2.88), (2.102) and (2.126).
5
(2.126)
Uncited references
6
Q14 7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
[1], [3], [7], [9], [10] and [11]. Acknowledgments We thank the referees for very useful comments and suggestions. References [1] S. Aida, K. Sasaki, Wong–Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces, Stochastic Process. Appl. 123 (10) (2013) 3800–3827. [2] L. Denis, A. Matoussi, J. Zhang, The obstacle problem for quasilinear stochastic PDEs: an analytical approach, Ann. Probab. 42 (3) (2014) 865–905. [3] L.C. Evans, D.W. Stroock, An approximation scheme for reflected stochastic differential equations, Stochastic Process. Appl. 121 (2011) 1464–1491. [4] N. Ikeda, S. Watanable, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981. [5] A. Matoussi, L. Stoica, The obstacle problem for quasilinear stochastic PDEs, Ann. Probab. 38 (3) (2010) 1143–1179. [6] A. Matoussi, M. Xu, Sobolev solution for semilinear PDE with obstacle under monotonicity conditions, Electron. J. Probab. 13 (35) (2008) 1035–1067. [7] E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990) 55–61. [8] E. Pardoux, S. Peng, Backward doubly SDEs and systems of quasilinear SPDEs, Probab. Theory Related Fields 98 (1994) 209–227. [9] E. Wong, M. Zakai, On the convegence of ordinary integrals to stochastic integrals, Ann. Math. Statist. 36 (1965) 1560–1564. [10] E. Wong, M. Zakai, On the relation between ordinary and stochastic differential equations, Internat. J. Engrg. Sci. 3 (1965) 213–229. [11] Tusheng Zhang, Wong–Zakai approximations to SDEs with reflection, Potential Anal. 41 (3) (2014) 783–815.
ScienceDirect Stochastic Processes and their Applications xx (xxxx) xxx–xxx www.elsevier.com/locate/spa
Wong–Zakai approximations of backward doubly stochastic differential equations Ying Hu a , Anis Matoussi b , Tusheng Zhang c,∗
Q1
a Universit´e de Rennes 1, campus Beaulieu, 35042 Rennes Cedex, France b Laboratoire Manceau de Math´ematiques, Universit´e du Maine, Avenue Olivier Messiaen, 72 085 LE MANS Cedex,
France c School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England, UK
Received 12 January 2015; received in revised form 18 June 2015; accepted 3 July 2015
1 2
Abstract
3
In this paper we obtain a Wong–Zakai approximation to solutions of backward doubly stochastic differential equations (BDSDEs). The situation is quite different from the case of stochastic differential equations because the integrands in the stochastic integral are also part of the solutions. c 2015 Published by Elsevier B.V. ⃝
7
MSC: primary 60H15; secondary 60H30
8
Keywords: Backward stochastic differential equations; Backward doubly stochastic differential equations; Wong–Zakai approximations; Backward integrals
4 5 6
9 10
1. Framework and introduction
11
Let {Wt , 0 ≤ t ≤ T } and {Bt , 0 ≤ t ≤ T } be two independent standard Brownian motions with values respectively in R l and in R m , defined on a probability space (Ω , F, P). Let N denote the class of P-null sets. For each t ∈ [0, T ], we define
13
B Ft = FtW ∨ Ft,T , ∗ Corresponding author.
E-mail addresses: [email protected] (Y. Hu), [email protected] (A. Matoussi), [email protected] (T. Zhang). http://dx.doi.org/10.1016/j.spa.2015.07.003 c 2015 Published by Elsevier B.V. 0304-4149/⃝
12
14
15
2
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx η
η
η
2
where for any process {ηt }, Fs,t = σ {ηr − ηs ; s ≤ r ≤ t} ∨ N , Ft = F0,t . Note that Ft , t ≥ 0 is not a filtration.
3
(H.1) Let f : R d × R d×l → R d be a bounded measurable function satisfying
1
4
5 6 7
8
9
10
| f (y1 , z 1 ) − f (y2 , z 2 )| ≤ c(|y1 − y2 | + ∥z 1 − z 2 ∥) for (y1 ,√z 1 ), (y2 , z 2 ) ∈ R d × R d×l , where |y| stands for the Euclidean norm of the vector y and ∥z∥ = T r (zz ∗ ) for a d × l matrix z. Let g(·) = (gi, j (·)) ∈ Cb1 (R d , R d×m ). For n ≥ 1, define the linear interpolation B n of B as k+1 k k+1 n n Bt = B k+2 +2 t − n B k+2 − B k+1 , for t ∈ n , n . (1.1) 2n 2n 2n 2 2 2 Let gg ′ : R d → R d be defined as m d ∂gi, j (y)
(gg ′ (y))i =
j=1 k=1 11 12
13 14
15 16
17 18
19 20
21
∂ yk
gk, j (y),
23 24 25 26
27
(1.2)
For any positive integer p ∈ N , let M 2 ([0, T ]; R p ) denote the set of p dimensional jointly measurable stochastic processes {φt ; t ∈ [0, T ]} which satisfy T (i) E[ 0 |φt |2 dt] < ∞, (ii) φt is Ft measurable for a.e. t ∈ [0, T ]. Similarly denote by S 2 ([0, T ]; R p ) the set of continuous p dimensional stochastic processes which satisfy (i) E[sup0≤t≤T |φt |2 ] < ∞, (ii) φt is Ft measurable for any t ∈ [0, T ]. Let ξ ∈ L 2 (Ω ; R d ) be FT -measurable. Consider the following backward doubly stochastic differential equations (BDSDE): Yt = ξ +
T
f (Ys , Z s )ds +
Ytn
=ξ+
T
t 22
i = 1, . . . , d.
g(Ys )dBs +
t
t
T
f (Ysn ,
Z sn )ds +
t
T
g(Ysn )dBns
1 2
T
gg ′ (Ys )ds −
t
− t
T
Z s dW s .
(1.3)
t T
Z sn dW s .
(1.4)
Here dBs stands for the backward Ito integral and the integrals against W are the forward Ito − → integrals. Correspondingly, here by dBns we mean B˙ sn ds , where and also in the sequel, B˙ sn denotes − → the piecewise derivative of B n and ds stands for the backward integral against the Lebesgue measure, namely, b − → h ′ (s) ds = h(a) − h(b), for h ∈ C 1 (R). a
28 29
Under the assumption (H.1), Eqs. (1.3) and (1.4) were shown in [8] to admit unique solutions (Y, Z ), (Y n , Z n ) ∈ S 2 ([0, T ]; R d ) × M 2 ([0, T ]; R d×l ).
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
3
Backward doubly stochastic differential equations were first studied by Pardoux and Peng in [8]. It is now a powerful tool to study stochastic partial differential equations with singular coefficients (see [2,5,6]). Our purpose of this paper is to obtain the convergence of the Wong–Zakai approximation to the backward doubly stochastic differential equations, namely we will prove that (Y n , Z n ) converges to (Y, Z ) in L 2 . The convergence of Wong–Zakai approximations to stochastic differential equations is now well known, see e.g. [4]. Because of the nature of the BDSDEs, the integrands Z n , Z in the stochastic integrals against the Brownian motion W are also part of the solutions. We could not impose conditions on Z n , Z in advance. This makes the problem drastically different from the Wong–Zakai approximation for stochastic differential equations. Another difficulty comes from the fact that the H¨older type estimate E[|Ytn − Ysn | p ] ≤ C|t − s|α
1 2 3 4 5 6 7 8 9 10
11
is no longer available for these BSDEs. However it plays a crucial role in the proof of the Wong–Zakai approximations for stochastic differential equations. We overcome this by carefully exploiting the independence of the two Brownian motions B and W . The rest of the paper is entirely devoted to the proof of the convergence result. The proof is involved and delicate. We need to complete the squares in last stage of the proof. Therefore we have to be very careful with the signs of each term in the estimates. Finally we hope to apply our results to obtain the support theorem of backward doubly stochastic differential equations and also to get some Wong–Zakai approximations of stochastic partial differential equations. We like to leave this study in a forthcoming paper given the length of the current article. 2. Main results
12 13 14 15 16 17 18 19 20 21
22
For the simplicity of the exposition, we assume d = m = l = 1. The same proofs carry over to the multi-dimensional case without extra difficulties. Throughout the paper, C, Ci , i = 1, . . . denote generic constants whose values may change from line to line. We start with an a priori estimate for the family {(Y n , Z n ), n ≥ 1}. Proposition 2.1. Assume (H.1). Then there exists a constant C such that T n 2 n 2 sup sup E[(Yt ) ] + E (Z s ) ds ≤ C. n 0≤t≤T
23 24 25 26
27
(2.5)
28
t
Proof. By Ito’s formula, we have T n 2 n 2 2 (Yt ) + (Z s ) ds = (ξ ) + 2 t
t
−2 t
T
29
T
Ysn f (Ysn , Z sn )ds + 2
Ysn Z sn dW s .
t
T
Ysn g(Ysn )dBns
30
(2.6)
k+2 k−1 + − For s ∈ [ 2kn , k+1 2n ], set s = 2n and s = 2n . In view of (H.1), it is easy to see that there exists C1 > 0 such that T T 1 T n 2 n n n n 2 2 Ys f (Ys , Z s )ds ≤ C1 (Ys ) ds + (Z s ) ds + C1 . (2.7) 4 t t t
31
32 33
34
4 1
2
3
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
Now, the third term on the right side of (2.6) can be written as T T T 2 Ysn g(Ysn )dBns = 2 Ysn+ g(Ysn+ )dBns + 2 (Ysn − Ysn+ )g(Ysn )dBns t t t T n n n +2 Ys + (g(Ys ) − g(Ys + ))dBns t
:= I1 + I2 + I3 .
4
5
6
As a stochastic integral, we have E[I1 ] = 0. By Eq. (1.4) it follows that T s + T s + n n n n n n I2 = 2 f (Yu , Z u )du g(Ys )dBs + 2 g(Yu )dBu g(Ysn )dBns t
s
t
T
t
s
t
≤C
11
13
14
1 2n
1
2
,
and T
E[I2.2 ] ≤ C E s
t
16
T
≤C
T
t 18
s T
≤C
20
1
1
1
du(2n ) 2 (2n ) 2 ≤ C,
(2.11)
s
t
=2 = 0.
s
T
E t
22
1
du(E[| B˙ un |2 ]) 2 (E[| B˙ sn |2 ]) 2
where C is a constant independent of n. For the term I2.3 , we have T s + n n ˙n E[I2.3 ] = 2 E Z u dW u g(Ys ) Bs ds
21
s+
ds t
19
s+
ds
n n ˙ ˙ | Bu |du | Bs |ds
duE[| B˙ un | | B˙ sn |]
s
≤C
s+
s+
ds t
17
(2.10)
here, as in Section 1, B˙ sn is the piecewise derivative of Bsn , i.e., k k+1 , B˙ sn = 2n B k+2 − B for s ∈ k+1 , 2n 2n 2n 2n
15
(2.9)
By the boundedness of f and g, we have T s + du E[| B˙ sn |]ds E[I2.1 ] ≤ C
12
Z un dW u g(Ysn )dBns
s
:= I2.1 + I2.2 + I2.3 .
8
10
s
s+
−2
7
9
(2.8)
g(Ysn ) B˙ sn E
s
s+
Z un dW u
Fs ds (2.12)
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
5
Putting together (2.10)–(2.12) we get
1
sup E[I2 ] ≤ C,
(2.13)
2
n
for some constant C. To bound I3 in (2.8), we write T 1 I3 = 2 Ysn+ dλg ′ (Ysn+ + λ(Ysn − Ysn+ ))(Ysn − Ysn+ )dBns t 0 + 1
T
Ysn+
= 2 t
0
T
+2 t
dλg
Ysn+
Ysn+
T
−2 t
′
(Ysn+
1
dλg
′
0 1
dλg 0
′
+ λ(Ysn
− Ysn+ ))
(Ysn+
+ λ(Ysn
− Ysn+ ))
(Ysn+
+ λ(Ysn
− Ysn+ ))
s
f (Yun ,
s
s+ s
4
Z un )du
dBns
5
g(Yun )dBnu
dBns
6
s+ s
3
Z un dW u
dBns
:= I3.1 + I3.2 + I3.3 .
7
(2.14)
For I3.1 , we have
8
9
T
E[I3.1 ] ≤ C t
T
≤C t
T
≤C t
1 E |Ysn+ | n | B˙ sn | ds 2 T 2 1 n 2 n 2 ˙ E[(Ys + ) ]ds + C E | Bs | ds 2n t
E[(Ysn+ )2 ]ds + C.
10
11
(2.15)
Similarly we have
12
13
T
E[I3.2 ] ≤ C
s+
duE[|Ysn+ | | B˙ un | | B˙ sn |]ds
ds t
s T
≤C
s+
ds
t
s T
≤C
s+
ds
t
s T
≤C t
1
14
1
du(E[(Ysn+ )2 ]) 2 (E[| B˙ un |2 | B˙ sn |2 ]) 2 ds
15
1
du(E[(Ysn+ )2 ]) 2 2n
E[(Ysn+ )2 ]ds + C.
By virtue of the independence of Ysn+ and B˙ sn , we have + s T E[I3.3 ] ≤ C E |Ysn+ | | B˙ sn | Z un dW u ds t s 2 12 T s + 1 ≤C (E[(Ysn+ )2 | B˙ sn |2 ]) 2 E Z un dW u ds s t
16
(2.16)
17
18
19
20
6
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
1
T
=C t
2
T
=C t
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
1 ≤ 4
1 1 (E[(Ysn+ )2 ]) 2 (E[| B˙ sn |2 ]) 2
2 E s
t
1 E 4
1 ≤ E 4
=
1 2
s+
s
n
T
t
t
T
s+
(Z un )2 du
(Z un )2 du2n (Z un )2 du
T
+ C2 t
T
+ C2 t
ds
2
t
u−
2
(Z un )2 du
ds
1 (Z un )2 du 1
ds + C2
u
s+
s
(E[(Ysn+ )2 ]) (2n ) E 1 2
T
E
ds
E[(Ysn+ )2 ]ds T
E[(Ysn+ )2 ]ds
E[(Ysn+ )2 ]ds.
(2.17)
(2.15)–(2.17) yield that T T 1 (Z un )2 du + C2 E[(Ysn+ )2 ]ds + C. E[I3 ] ≤ E 4 t t
(2.18)
It follows from (2.6), (2.7), (2.13) and (2.18) that T T 1 n 2 n 2 2 E[(Yt ) ] + E (Z s ) ds ≤ E[(ξ ) ] + C E[(Ysn+ )2 ]ds + C. 2 t t Applying Gronwall’s inequality, we complete the proof of the Proposition.
(2.19)
We also need the following estimate. Lemma 2.2. There is a constant C such that T 1 n n 2 E |Ys + − Ys | ds ≤ C n . 2 0 T 1 E |Ys + − Ys |2 ds ≤ C n . 2 0 Proof. Let us prove (2.20). The proof of (2.21) is similar. Indeed, we have 2 T T s + E |Ysn+ − Ysn |2 ds ≤ C E f (Yun , Z un )du ds 0 0 s 2 T s + +CE g(Yun )dBnu ds 0 s 2 T s + n +CE Z u dW u ds 0 s s+ T 1 (s + − s) | B˙ un |2 duds ≤ C n + CE 2 0 s
(2.20) (2.21)
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
T
s+
+CE 0
s
7
(Z un )2 duds
1
≤C
T 1 1 n 2 + sup E (Z ) du u n 2 2n n 0
≤C
1 . 2n
2
(2.22)
Applying H¨older’s inequality it follows immediately from (2.20) and (2.21) that
T
|Ysn+
E 0
T
E
|Ys + 0
1 1 2 ≤C . 2n 1 1 2 − Ys |ds ≤ C . 2n − Ysn |ds
4
(2.23)
5
(2.24)
6
To establish the main result, we need a strengthened version of Proposition 2.1. We introduce the following condition. (H.2). Suppose yg(y),
y 3 g(y)
∈
Cb1 (R).
7 8
9
Remark. (H.2) clearly holds if g has a compact support.
10
Proposition 2.3. Assume ξ ∈ L 4 (Ω ; R d ) and suppose that (H.1), (H.2) hold. Then, there exists a constant C such that 2 T n 4 n 2 sup E[ sup |Yt | ] + E (Z s ) ds ≤ C. (2.25) n
3
11 12
13
t
0≤t≤T
Proof. By Ito’s formula, T |Ytn |4 + 6 (Ysn )2 (Z sn )2 ds = (ξ )4 + 4 t
+4 t
T
14
T t
(Ysn )3 f (Ysn , Z sn )ds
(Ysn )3 g(Ysn )dBns
:= (ξ )4 + I1n + I2n + I3n .
T
−4 t
15
(Ysn )3 Z sn dW s
16
(2.26)
By the linear growth of f , we have T n I1 = 4 (Ysn )3 f (Ysn , Z sn )ds
17
18
19
t
T
≤C t
|Ysn |3 (1 + |Ysn | + |Z sn |) T
≤ C +C t
(Ysn )4 ds +
t
T
(Ysn )2 (Z sn )2 ds.
20
(2.27)
21
8
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
Set F(y) = 4y 3 g(y). By (H.2) and (2.20), we have
1
E[I2n ] = E
2
T
t
T
= E
3
t
F(Ysn )dBns
F(Ysn+ )dBns
t
≤C E
T
t
≤C
6
1 2n
+E t
1
2
− Ysn+ |2 ds
|Ysn
T
|Ysn − Ysn+ | | B˙ sn |ds
≤ CE
5
T
4
(F(Ysn ) −
F(Ysn+ ))dBns
21 E
T
t
| B˙ sn |2 ds
21
1
(2n ) 2 ≤ C.
(2.28)
Substituting (2.28), (2.27) into (2.26) and taking expectation we get
7
E[|Ytn |4 ] + 5E
8
t
T
(Ysn )2 (Z sn )2 ds
≤ C + E[(ξ ) ] + C 4
T
t
E[|Ysn |4 ]ds.
(2.29)
In particular,
9
≤ C + E[(ξ ) ] + C
E[|Ytn |4 ]
10
4
T
t
E[|Ysn |4 ]ds,
where the constant C is independent of n. By Gronwall’s inequality we obtain
11
sup sup E[|Ytn |4 ] ≤ C(1 + E[(ξ )4 ]),
12
n 0≤t≤T
here again C denotes a generic constant whose value changes from line to line. The above inequality together with (2.29) further implies
13 14
sup
15
n
16
Q2
sup E[|Ytn |4 ] + E
0≤t≤T
0
|Ysn |2 (Z sn )2 ds
E[ sup 0≤t≤T
|I2n (t)|]
= E
sup
0≤t≤T
T t
18
(2.30)
≤ C.
Now applying Burkholder’s inequality and taking into account (H.2) and (2.20), we have
17
T
≤ C + CE 0
F(Ysn+ )dBns T
+ E
T
0
(F(Ysn ) −
F(Ysn+ )) B˙ sn ds
12
(Ysn+ )6 g 2 (Ysn+ )ds
≤ C,
(2.31)
9
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
and
1
E
sup 4
0≤t≤T
t
T n 3 n (Ys ) Z s dW s ≤ C E
T 0
(Ysn )6 (Z sn )2 ds
≤ C E sup (Ytn )2
T
0
0≤t≤T
2
(Ysn )2 (Z sn )2 ds
1 E[ sup (Y n )4 ] + C E 4 0≤t≤T t
≤
21
T
0
21
3
(Ysn )2 (Z sn )2 ds . (2.32)
By virtue of (2.30)–(2.32), taking supremum in (2.26) we obtain
5
sup E[ sup (Ytn )4 ] < ∞. n
4
(2.33)
6
0≤t≤T
Next we show that 2 T n 2 sup E (Z t ) dt < ∞. n
7
(2.34)
8
0
It follows from Ito’s formula that T (Z sn )2 ds = (ξ )2 − (Y0n )2 + 2 0
9
T
0
+2 0
T
Ysn f (Ysn , Z sn )ds
Ysn g(Ysn )dBns − 2
T
0
10
Ysn Z sn dW s .
(2.35)
Hence,
11
12
E 0
2 T n 2 4 n 4 (Z s ) ds ≤ C E[(ξ ) ] + C E[(Y0 ) ] + C E
T
0
+CE 0
T
Ysn g(Ysn )dBns
2
T
+ CE 0
By the linear growth of f , it is easy to see that 2 T n n n ≤ CE CE Ys f (Ys , Z s )ds
0
0
Applying Burkholder’s inequality, we have 2 T n n CE Ys Z s dW s = CE 0
2
.
13
(2.36)
14
15
T
(Ysn )4 ds +
1 E 4
0
T
(Z sn )2 ds
2
. (2.37)
16
17
T
0
(Ysn )2 (Z sn )2 ds
≤ CE
Ysn Z sn dW s
Ysn f (Ysn , Z sn )ds
2
sup (Ytn )2
0≤t≤T
0
T
18
(Z sn )2 ds
1 ≤ C E[ sup (Ytn )4 ] + E 4 0≤t≤T
19
T 0
(Z sn )2 ds
2
.
(2.38)
20
10
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
T T To bound E[( 0 Ysn g(Ysn )dBns )2 ], we set h nt = t Ysn g(Ysn )dBns . By the chain rule, T (h nt )2 = 2 h ns Ysn g(Ysn )dBns
1
2
t
T
h ns+ Ysn+ g(Ysn+ )dBns + 2
=2
3
t
T
+2
4
t 5
Q3
6
7
8
9
10
Q4
T
t
(h ns − h ns+ )Ysn+ g(Ysn+ )dBns
h ns+ (Ysn g(Ysn ) − Ysn+ g(Ysn+ ))dBns .
(2.39)
T As a stochastic integral, E[ t h ns+ Ysn+ g(Ysn+ )dBns ] = 0. Applying H¨older’s inequality, we have T (h ns − h ns+ )Ysn+ g(Ysn+ )dBns 2E t + T s n n n n n n ˙ ≤ 2E Yu g(Yu )dBu |Ys g(Ys )| | Bs |ds t s T s + ˙n n ˙ (2.40) | Bu |du | Bs |ds ≤ C. ≤ CE t s In view of the independence of h ns+ and B˙ sn , keeping in mind the assumption (H.2) we have T 2E h ns+ (Ysn g(Ysn ) − Ysn+ g(Ysn+ ))dBns t
T
|h ns+ | | B˙ sn | |Ysn
11
≤ CE
12
≤C E
t
T
|Ysn
t
13
T
≤C t
14
15
− Ysn+ |ds
− Ysn+ |2 ds
E[|h ns+ |2 ]ds
21 t
12
T
12
E[|h ns+ |2 ]E[| B˙ sn |2 ]ds T
≤C +C t
E[|h ns+ |2 ]ds
.
(2.41)
Taking expectation on both sides of (2.39) and collecting the terms in (2.40), (2.41), we get T n 2 n 2 E[(h t ) ] ≤ C + C E[|h s + | ]ds . t
16
17
This yields that sup E[(h n0 )2 ] < ∞.
(2.42)
n
18
Combining (2.36)–(2.38) and (2.42) together, we arrive at 2 T
19
sup E n
20
0
(Z sn )2 ds
The proof is complete.
< ∞.
(2.43)
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
11
The following theorem is the main result of the paper.
1
Theorem 2.4. Assume E[|ξ |4 ] < ∞ and that (H.1) and (H.2) are in place. Then it holds that T n 2 n 2 (Z s − Z s ) ds = 0. lim sup E[(Yt − Yt ) ] + E (2.44) n→∞ 0≤t≤T
2
3
t
Proof. By Ito’s formula, we have T n 2 (Yt − Yt ) + (Z sn − Z s )2 ds
4
5
t
T
=2 t
(Ysn − Ys )( f (Ysn , Z sn ) − f (Ys , Z s ))ds T
(Ysn − Ys )g(Ysn )dBns +
+2 t T
(Ysn − Ys )g(Ys )dBs − 2
−2 t T
T
6
g 2 (Ys )ds
7
t T
t
(Ysn − Ys )(Z sn − Z s )dW s
8
(Ysn − Ys )gg ′ (Ys )ds
− t
9
:= I1n + I2n + I3n + I4n + I5n + I6n .
(2.45)
Crucially we need to bound the term I2n . We write T n I2 = 2 (Ysn − Ys )g(Ysn )dBns
10
11
12
t
T
[(Ysn − Ys ) − (Ysn+ − Ys + )]g(Ysn )dBns
= 2 t
T
+2 t
T
+2 t
13
(Ysn+ − Ys + )(g(Ysn ) − g(Ysn+ ))dBns
14
(Ysn+ − Ys + )g(Ysn+ )dBns
15
(2.46)
:= A + B + C.
As a stochastic integral, we have E[C] = 0. We need the precise bounds (including the signs) for E[A] and E[B]. These precise bounds will be used to complete squares in the proof. We will give these estimates in Lemmas 2.5 and 2.6. Because the proofs of these lemmas are quite involved and long, let us first complete the proof of the theorem by admitting Lemmas 2.5 and 2.6. Taking expectation in (2.45) we obtain T n 2 n 2 E[(Yt − Yt ) ] + E (Z s − Z s ) ds
16
17 18 19 20 21 22
23
t
= 2E t
T
(Ysn − Ys )( f (Ysn , Z sn ) − f (Ys , Z s ))ds
24
12
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx T
1
+ 2E t
2
T
−E t
3
4
(Ysn
(Ysn
− Ys )g(Ysn )dBns
T
+E
g (Ys )ds 2
t
− Ys )gg (Ys )ds . ′
(2.47)
Collecting the estimates in Lemmas 2.5 and 2.6 we obtain from (2.47) that for any δ < 12 , T (Z sn − Z s )2 ds E[(Ytn − Yt )2 ] + E t
5
6
7
1 −δ
T 1 2 n 2 + C E |Y − Y | ds s s 2n t T T 1 + E |Z sn − Z s |2 ds + E g 2 (Ys )ds 2 t t T T ′ n n ′ n −E (Ys − Ys )gg (Ys )ds + E (Ys + − Ys + )gg (Ys + )ds
≤C
t
t
T
8
− 2E t
9
10
g(Ys + )g(Ysn+ )ds + E
T
g 2 (Ysn+ )ds .
t
(2.48)
In view of (2.20) and (2.21), we can now replace s + by s on the right side of (2.48) to obtain T E[(Ytn − Yt )2 ] + E (Z sn − Z s )2 ds t
11
12
13
1 −δ
T 1 n 2 + C E |Y − Y | ds s s 2n t T T 1 n 2 2 + E |Z s − Z s | ds + E g (Ys )ds 2 t t T T n ′ n ′ n −E (Ys − Ys )gg (Ys )ds + E (Ys − Ys )gg (Ys )ds
2
≤C
t
t
T
14
− 2E t
g(Ys )g(Ysn )ds
T
+E
g t
2
(Ysn )ds
(2.49) 1
15 16
17
by simply changing the value of the constant C in front of ( 21n ) 2 −δ . Completing the square in (2.49) we get that T E[(Ytn − Yt )2 ] + E (Z sn − Z s )2 ds t
18
19
20
1 −δ
T 1 n 2 + C E |Y − Y | ds s s 2n t T T 1 n 2 n 2 + E |Z s − Z s | ds + E (g(Ys ) − g(Ys )) ds 2 t t T +E (Ysn − Ys )(gg ′ (Ysn ) − gg ′ (Ys ))ds .
2
≤C
t
(2.50)
13
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
By the Lipschitz continuity of gg ′ and g, it follows from (2.50) that T 1 E[(Ytn − Yt )2 ] + E (Z sn − Z s )2 ds 2 t 1 −δ T 1 2 n 2 + C E |Y − Y | ds ≤C . s s 2n t
1
2
(2.51)
Application of Gronwall’s inequality completes the proof of Theorem 2.4.
3
4
The remaining part of the paper is devoted to the proofs of Lemmas 2.5 and 2.6.
5
Lemma 2.5. For any given δ < 12 , we have T E[A] = 2E [(Ysn − Ys ) − (Ysn+ − Ys + )]g(Ysn )dBns
6
7
t
≤C
1 2n
1 −δ
2
T
− 2E t
g(Ys + )g(Ysn+ )ds
+E
T
g t
2
(Ysn+ )ds
.
(2.52)
Proof. By the equations satisfied by Y n and Y we have T A = 2 [(Ysn − Ys ) − (Ysn+ − Ys + )]g(Ysn )dBns t T s + T s + n n n n = 2 g(Yu )dBu g(Ys )dBs − 2 g(Yu )dBu g(Ysn )dBns t s t s + T
+2 t
T
−2 t
8
9
10
11
s
[ f (Yun , Z un ) − f (Yu , Z u )]du g(Ysn )dBns s + T s + s n n n ′ (Z u − Z u )dW u g(Ys )dBs − gg (Yu )du g(Ysn )dBns s
t
12
13
s
:= A1 + A2 + A3 + A4 + A5 .
(2.53)
Clearly,
14
15
E[|A5 |] ≤ C
1 2n
T
E[| B˙ sn |]ds ≤ C
t
1 2n
1
2
,
(2.54)
and also
16
17
E[|A3 |] ≤ C
1 2n
1 2
.
(2.55)
By conditioning on Fs , we find that T s + n n ˙n E[A4 ] = −2 E (Z u − Z u )dW u g(Ys ) Bs ds t s T s + n n ˙n = −2 E E (Z u − Z u )dW u Fs g(Ys ) Bs ds t
= 0.
18
19
20
21
s
(2.56)
22
14 1
2
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
To bound A1 , we write it as T s + n n A1 = 2 g(Yu )dBu g(Ysn )dBns t
s
3
T
= 2 t
s
4
T
+2 t
5
T
+2 t
6
7
8
11
12
(2.57) [ 2kn , k+1 2n ]
we see that Splitting the interval [t, T ] into subintervals − k+1 2n → n 2n B k+2 A13 = 2 − B g 2 Y k+2 B ds k+3 k+2 − B k+1 n n n n +2
k 2n
2n
k+1 2n
k 2n
16
n g 2 Y k+2
B k+2 − B k+1 n n 2
2 − → ds
2
(2.58)
2
2
2
2
2
Integrating with respect to s, we get that E[A13 ] = E[A13,2 ] 2 2 n B k+2 − B k+1 = E g Y k+2 n n = E
2n
g
2
n
Y k+2
2
2n
T
= E
2
B k+2 − B k+1 n n 2
2
2
1 − n 2
+E
T
g t
2
(Ysn+ )ds
g 2 (Ysn+ )ds ,
(2.60)
where the fact that the sequence {(B k+2 − B k+1 )2 − 21n , k ≥ 0} is a martingale has been used. 2n 2n For the term A11 in (2.57), we have 1 T s+ A11 = 2 dλ g ′ (Ysn+ + λ(Yun − Ysn+ ))(Yun − Ysn+ )dBnu g(Ysn )dBns 0
t
1
22
k+1 2n
2
(2.59)
t
21
2
= 0.
20
(2n )2 s −
2n
k
19
2
we have Conditioning on F k+2 2n n B E[A13,1 ] = −2 E g 2 Y k+2 B k+2 − B E k+2 − B k+1 F k+2 k+3 n n n n n
k
18
2n
17
2
:= A13,1 + A13,2 .
13
15
g(Ysn )dBns
g(Ysn+ )(Bsn − Bsn+ )g(Ysn+ )dBns
k
14
g(Ysn+ ))dBnu
g(Ysn+ )(Bsn − Bsn+ )(g(Ysn ) − g(Ysn+ ))dBns
k 10
(g(Yun ) −
:= A11 + A12 + A13 .
k 9
s+
= 2
s
T
s+
dλ 0
t
s
g ′ (Ysn+ + λ(Yun − Ysn+ ))
u
s+
f (Yvn , Z vn )dv dBnu g(Ysn )dBns
15
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
1
T
+2
s+
g ′ (Ysn+ + λ(Yun − Ysn+ ))
dλ 0
t 1
s T
−2
s+
dλ
g
0
t
s
′
(Ysn+
+ λ(Yun
− Ysn+ ))
s+ u
g(Yvn )dBnv dBnu g(Ysn )dBns
s+
Z vn dW v
u
dBnu g(Ysn )dBns
:= A11,1 + A11,2 + A11,3 .
2
(2.61)
The first two terms on the right can be bounded as follows. E[A11,1 ] ≤ ≤ E[A11,2 ] ≤ ≤
(2.62)
5
(2.63)
6
The last term A11,3 can be estimated as follows. + s+ s T E[A11,3 ] ≤ C E ds | B˙ un | | B˙ sn | Z vn dW v du t s u T
s+
≤C
ds s
t
n
T
t T
du E
ds
≤ C2
≤C
ds E
s+
s
T
≤C
s+
dsE t
s
≤ C sup E
T
n
s+
u
s
t
7
8
2 12 s+ 1 Z vn dW v du(E[| B˙ un |2 | B˙ sn |2 ]) 2 E u s+
t
9
1
2
(Z vn )2 dv
10
1 2
(Z vn )2 dv
11
1 2
(Z vn )2 dv
(Z vn )2 dv
21
≤C E t
1 2n
1
2
.
T
(Z vn )2 dv
v
v−
12 ds
12
(2.64)
Putting together (2.62)–(2.64) we get E[A11 ] ≤ C
1 2n
1
2
.
3
4
T s+ 1 E[| B˙ un | | B˙ sn |]du ds C n 2 t s 1 1 1 1 1 C n (2n ) 2 (2n ) 2 n ≤ C n . 2 2 2 T s+ s+ dv E[| B˙ un | | B˙ vn | | B˙ sn |] du C ds u s t 2 1 3 1 1 2 n 2 C(2 ) . ≤ C 2n 2n
1
13
14
(2.65)
15
16 1
2
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
Similarly the term A12 can be decomposed as s+ 1 T n dBnu g ′ (Ysn+ + λ(Ysn − Ysn+ ))(Ysn − Ysn+ )dBns A12 = 2 dλ g(Ys + ) s 0 t + + 1
T
= 2
3
dλ 0
t 1
+2
4
t
1
−2
9
t
11
1 , 2n
E[A12, j ] ≤ C
13
14
E[A12 ] ≤ C
1 2n
1 2
E[A1 ] ≤ C
1 2n
T
+ λ(Ysn
− Ysn+ ))
s
s
g(Yvn )dBnv dBns
s+
Z vn dW v
dBns (2.66)
.
(2.68)
T
+E
g t
s+
s T
−2 T
−2 t
2
(Ysn+ )ds
.
(2.69)
(g(Yu ) − g(Ys + ))dBu g(Ysn )dBns
g(Ys + )(Bs − Bs + )(g(Ysn ) − g(Ysn+ ))dBns g(Ys + )(Bs − Bs + )g(Ysn+ )dBns
:= A21 + A22 + A23 .
(2.70)
Splitting the interval [t, T ] into subintervals [ 2kn , k+1 2n ], k+1 − 2n → n n A23 = −2 g Y k+2 g Y B − B B ds k+2 2 k+2 − B k+1 s k+2 n n n n k
21
s
dBnu g ′ (Ysn+
s+
(2.67)
= −2
t
20
s+
dBnu g ′ (Ysn+ + λ(Ysn − Ysn+ ))
dBns
s
19
s
s
Z vn )dv
j = 1, 2, 3.
2
t
18
s+
f (Yvn ,
Now we turn to A2 which can be written as T s + g(Yu )dBu g(Ysn )dBns A2 = −2
17
g(Ysn+ )
1
t
16
g(Ysn+ )
s
− Ysn+ ))
Combining (2.60), (2.65) and (2.68) we get
15
s
+ λ(Ysn
Hence,
12
dBnu g ′ (Ysn+
Using the similar arguments as for (2.62) and (2.63) we can show that
10
s
:= A12,1 + A12,2 + A12,3 .
6
8
T
dλ 0
7
T
dλ 0
5
g(Ysn+ )
= −2
k
k 2n
k+1 2n k 2n
2
2n
2
2
2
− → n n g Y k+2 g Y B − B B ds k+2 − B k+1 k+1 2 s k+2 n n n n
2
2n
2
2
2
17
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
2 n −2 g Y k+2 g Y − B B k+1 k+2 k+2 n n n 2n
2
k
2
1
2
:= A23,1 + A23,2 .
(2.71)
it follows that Using the independence of the increments of B and conditioning on F k+2 n
2
3
2
E[A23,1 ] = −2
k+1 2n
k 2n
k
× E
n 2 ds E g Y k+2 g Y k+2 n → n−
Bs − B k+1 n 2
= −2
k+1 2n
× E
B k+2 − B k+1 F k+2 2n 2n 2n
5
− → n g Y 2n ds E g Y k+2 k+2 n
k 2n
k
4
2n
2
Bs − B k+1 n 2
B k+2 − B k+1 n n 2
6
2n
2
7
2
= 0.
(2.72)
Now,
8
9
A23,2
2 1 n B = −2 g Y k+2 − B − g Y k+2 k+1 k+2 2n 2n 2n 2n 2n k T
−2 t
10
g(Ys + )g(Ysn+ )ds.
(2.73)
By conditioning on F k+2 we see that the expectation of the first term of the above equation 2n vanishes. Hence, (2.71)–(2.73) together yield E[A23 ] = E[A23,2 ] = −2E t
T
g(Ys + )g(Ysn+ )ds .
(2.74)
To bound the term A21 we write 1 T s+ dλ g ′ (Ys + + λ(Yu − Ys + ))(Yu − Ys + )dBu g(Ysn )dBns A21 = −2 0
t
1
s+
0
t 1
−2
T
+2
t 1
T
t
s+
u s+
g (Ys + + λ(Yu − Ys + )) ′
s
dλ 0
s
dλ 0
g (Ys + + λ(Yu − Ys + )) ′
dλ
= −2
12 13
14
15
16
s
T
11
s+
u s+
g (Ys + + λ(Yu − Ys + )) ′
s
:= A21,1 + A21,2 + A21,3 .
u
s+
f (Yv , Z v )dv dBu g(Ysn )dBns
17
g(Yv )dBv dBu g(Ysn )dBns
18
Z v dW v dBu g(Ysn )dBns
19
(2.75)
20
18
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
We will estimate each of the terms on the right. First we have 1 T s + g ′ (Ys + + λ(Yu − Ys + )) E[A21,1 ] ≤ C dλ E s 0 t + s ˙n × f (Yv , Z v )dv dBu | Bs | ds u 1 T s + g ′ (Ys + + λ(Yu − Ys + )) E ≤C dλ s 0 t 2 21 + s 1 f (Yv , Z v )dv dBu (E[| B˙ sn |2 ]) 2 ds × u 1 T s+ n 12 E ≤ C(2 ) dλ g ′ (Ys + + λ(Yu − Ys + ))2
1
2
3
4
5
6
0
t
f (Yv , Z v )dv
u 1
≤ C(2n ) 2
8
T
1
2
s+
×
7
s
2
du ds 1
[(s + − s)(s + − s)2 ] 2 ds
t
≤C
9
10
Q5
12
13
15
T
2 21 1 2 × g(Yv )dBv dBu E | B˙ sn |2 ds u 1 T s+ 1 E ≤ C(2n ) 2 dλ g ′ (Ys + + λ(Yu − Ys + ))2
s+
× u
s+
0
16
(2.76)
+ s g ′ (Ys + + λ(Yu − Ys + )) E dλ E[A21,2 ] ≤ C s 0 t + s ˙n × g(Yv )dBv dBu | Bs | ds u 1 T s + E ≤C dλ g ′ (Ys + + λ(Yu − Ys + )) s 0 t 1
14
.
Similarly,
11
1 2n
t
s
2 g(Yv )dBv
1 2
du ds
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
1 2
≤ C(2 ) n
T
1 2
≤ C(2 )
s+
s+
T
t
≤C
s+
u
2
1
2 g(Yv )dBv
1 2 du E
s
1
1 2n
E
s
t
n
s+ s
19
2
du ds
1
1 2
g (Yv )dv 2
ds
.
2
(2.77)
By H¨older’s inequality and Ito’s isometry, we have 1 T s + g ′ (Ys + + λ(Yu − Ys + )) E[A21,3 ] ≤ C dλ E s 0 t + s ˙n Z v dW v dBu | Bs | ds × u 1 T s + E ≤C dλ g ′ (Ys + + λ(Yu − Ys + )) s 0 t
Z v dW v
× u 1
≤ C(2n ) 2
1
dλ 0
1 2
t
≤ C(2 ) n
1 2
s+
s+
t
T
≤C t
T t
≤C
s+
1 2n
1
2
.
Z v2 dv
8
9
10
2
s+
Z v dW v
1 2 du E
s+
s
s
≤C E
E
u
E
7
2
s
6
du ds
s
T
5
1
2
T
4
s
Z v dW v
u
≤ C(2 )
t
s+
×
n
T
Q6
2 21 1 dBu (E[| B˙ sn |2 ]) 2 ds s+ E g ′ (Ys + + λ(Yu − Ys + ))2
s+
3
1 2
du ds
11
1 2
Z v2 dv
ds
12
1 2
Z v2 dv ds
21
v
v−
13
ds
14
(2.78)
15
20 1
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
It follows from (2.76)–(2.78) that E[A21 ] ≤ C
2
3
4
5
6
7
14
16
1 E[A22,2 ] ≤ C n 2
1 2n
≤C
≤C
T
1 2n
1
1
(E[|Bs − Bs + |2 ]) 2 (E[sup | B˙ un |4 ]) 2 ds u
t
1 2n
1
2
(E[ sup
t
1 −δ 2
(2n )2
T
|r −v|≤ 21n
1
|Br − Bv |4 ]) 2 ds
.
(2.82)
By H¨older’s inequality, T
E[A22,3 ] ≤ C ≤C ≤C
2 12 s + (E[|Bs − Bs + |4 ]) (E[| B˙ sn |4 ]) E Z un dW u ds s 1 4
T
s+
E t
19
(2.81)
and
t
18
(2.80)
T 1 1 1 (E[|Bs − Bs + |2 ]) 2 (E[| B˙ sn |2 ]) 2 ds n 2 t 1 ≤ C n, 2
17
(2.79)
E[A22,1 ] ≤ C
15
.
Now,
11
13
2
:= A22,1 + A22,2 + A22,3 .
10
12
1
Let us turn to the term A22 . We have T A22 ≤ C |Bs − Bs + | |Ysn − Ysn+ | | B˙ sn |ds t + T s n n f (Yu , Z u )du | B˙ sn |ds ≤ C |Bs − Bs + | s t T s + g(Yun )dBnu | B˙ sn |ds +C |Bs − Bs + | s t + T s Z un dW u | B˙ sn |ds +C |Bs − Bs + | s t
8
9
1 2n
1 . 2n
s
1 4
1
(Z un )2 du
2
ds (2.83)
21
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
It follows from (2.80)–(2.83) that E[A22 ] ≤ C
1 −δ
1 2n
2
1
.
(2.84)
2
Collecting (2.74), (2.79) and (2.80) we obtain E[A2 ] ≤ C
1 2n
1 −δ
2
− 2E t
T
3
g(Ys + )g(Ysn+ )ds .
(2.85)
4
The proof is now completed by putting (2.54)–(2.56), (2.69) and (2.85) together.
5
Lemma 2.6. For any given δ < 12 , we have T
E[B] = 2E t
≤C
1 2n
6
(Ysn+ − Ys + )(g(Ysn ) − g(Ysn+ ))dBns
1 −δ
2
+E t
T
(Ysn+
− Ys + )gg
′
7
(Ysn+ )ds
.
(2.86)
8
Proof. Write
9
T
(Ysn+ − Ys + )(g(Ysn ) − g(Ysn+ ))dBns
B = 2 t 1
T
= 2
(Ysn+ − Ys + )g ′ (Ysn+ + λ(Ysn − Ysn+ ))(Ysn − Ysn+ )dBns +
dλ 0
t 1
T
= 2
(Ysn+ − Ys + )g ′ (Ysn+ + λ(Ysn − Ysn+ ))
dλ 0
t
1
+2
T
dλ 0
−2
t 1
T
dλ 0
10
t
(Ysn+
− Ys + )g
(Ysn+
− Ys + )g
′
′
(Ysn+
+ λ(Ysn
− Ysn+ ))
(Ysn+
+ λ(Ysn
− Ysn+ ))
:= B1 + B2 + B3 . By Proposition 2.2, we have T 1 E[B1 ] ≤ C n E |Ysn+ − Ys + | | B˙ sn |ds 2 t T 1 1 1 ≤C n (E[|Ysn+ − Ys + |2 ]) 2 (E[| B˙ sn |2 ]) 2 ds 2 t 1 1 2 ≤C . 2n
s
11
f (Yvn , Z vn )dv dBns
s
s+
s
s
12
g(Yvn )dBnv
dBns
13
s+
Z vn dW v
dBns
14
(2.87)
15
Q7
16
17
18
(2.88)
19
22 1
2
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
B2 is further written as follows. 1 T B2 = 2 dλ (Ysn+ − Ys + )[g ′ (Ysn+ + λ(Ysn − Ysn+ )) − g ′ (Ysn+ )] 0 t + s
×
3
s
1
+2
4
g(Yun )dBnu dBns
dλ 0
t 1
+2
5
9
10
11
14
16
18
19
20
s
s+
(g(Yun ) −
g(Ysn+ ))dBnu
dBns
(Ysn+ − Ys + )g ′ (Ysn+ )g(Ysn+ )(Bsn − Bsn+ )dBns (2.89)
:= B21,1 + B21,2 + B21,3 .
(2.90)
The following two inequalities will be used frequently in sequel. sup | B˙ un | ≤ 2n
sup
(2.91)
|Br − Bs |.
|r −s|≤ 21n
For any δ > 0 and p ≥ 1, there exists a constant C p,δ such that E[ sup
p
|Br − Bs | ] ≤ C p,δ
|r −s|≤ 21n 17
(Ysn+ )
By the Lipschitz continuity of it follows that + T s B21 ≤ C |Ysn+ − Ys + | |Ysn − Ysn+ | g(Yun )dBnu | B˙ sn |ds t s T s + s + n n ˙ n n n n g(Yu )dBu | Bs |ds f (Yv , Z v )dv ≤ C |Ys + − Ys + | s s t 2 T s + g(Yun )dBnu | B˙ sn |ds +C |Ysn+ − Ys + | s t + + T s s +C |Ysn+ − Ys + | Z vn dW v g(Yun )dBnu | B˙ sn |ds s s t
u
15
− Ys + )g
′
g′ ,
12
13
t
(Ysn+
:= B21 + B22 + B23 .
6
8
T
dλ
0
7
T
1 2n
p −δ 2
.
By H¨older’s inequality and (2.91), (2.92), we have + s T 1 ˙n n n E[B21,1 ] ≤ C n E |Ys + − Ys + | B |du| | B˙ s |ds s u 2 t + T s 1 |Ysn+ − Ys + | du sup |Br − Bs |2 ds ≤ C n (2n )2 E s |r −s|≤ 1 2 t n 2 T ≤ CE |Ysn+ − Ys + | sup |Br − Bs |2 ds t
|r −s|≤ 21n
(2.92)
23
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx T
≤C t
≤C
1
1
(E[|Ysn+ − Ys + |2 ]) 2 (E[ sup
|Br − Bs |4 ]) 2 ds
1
|r −s|≤ 21n
1 2n
1−δ
.
(2.93)
Similarly, in view of (2.92), we have T E[B21,2 ] ≤ C E |Ysn+ − Ys + | t
2
3
+ 2 s du sup | B˙ sn |3 ds s s
4
1 2 n 3 T n ≤C (2 ) E |Ys + − Ys + | sup |Br − Bs |3 ds 2n t |r −s|≤ 21n T 1 1 ≤ C2n (E[|Ysn+ − Ys + |2 ]) 2 (E[ sup |Br − Bs |6 ]) 2 ds
t
≤C
1 2n
5
6
|r −s|≤ 21n
1 −δ 2
,
(2.94)
and
7
8
+ + s s T n n n n ˙ ˙ | Bu |du | Bs |ds Z v dW v E[B21,3 ] ≤ C E |Ys + − Ys + | s s t + s 1 n 2 T n ≤ C n (2 ) E |Ys + − Ys + | Z vn dW v sup |Br − Bs |2 ds s |r −s|≤ 1 2 t
9
10
2n
≤ C2n
T
t
2 12 s+ 1 Z vn dW v − Ys + |4 ]) 4 E s
(E[|Ysn+
11
1
× (E[ sup
|Br − Bs |8 ]) 4 ds
12
|r −s|≤ 21n
≤C
1 2n
1−δ
n
2
t
T
(E[|Ysn+
− Ys + | ]) E
s+
1 4
4
s
1 2
(Z vn )2 dv
ds
13
21 T 1 1−δ n n 4 12 ≤C 2 (E[|Ys + − Ys + | ]) ds 2n t + 1 2 T s n 2 × E (Z v ) dv ds
t
≤C ≤C
14
15
s
1 2n
1−δ
1 2n
1 −δ
2 E n
t 2
,
T
(Z vn )2 dv
12
v
v−
ds
16
(2.95)
17
24 1
Q8
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
where the estimate (2.25) has been used. (2.93) yield E[B21 ] ≤ C
2
3
Q9
1 2n
1 −δ 2
.
(2.96)
By the Lipschitz continuity of g, we have + T
4
B22 ≤ C t
t
T
+C
6
t T
+C
7
t
11
E[B22, j ] ≤ C
13
14
E[B22 ] ≤ C
17
B23 = 2
2
+2
k 2n
(2.97)
,
j = 1, 2, 3.
(2.98)
.
(2.99)
n
Y k+2 − Y k+2 n
gg
2n
k+1 2n k 2n
′
Y k+2
− → 2n B k+2 B ds − B k+3 k+2 − B k+1 n n n n
2n
2
n Y k+2 − Y k+2 n 2n
n
2
2
n gg ′ Y k+2
2
2
2
2n
2 − k+1 → × (2n )2 s − n B k+2 − B k+1 ds n n 2 2 2 n ′ n = −2 Y k+2 − Y k+2 gg Y B k+2 − B k+3 B k+2 − B k+1 k+2 n n n n n 2n
+
+ t
T
2n
2
n Y k+2 − Y k+2 n 2n
k
19
1−δ
1 −δ
k+1 2n
k 18
1 2n
1 2n
k 16
s+
For the term B23 , we split the interval [t, T ] into subintervals [ 2kn , k+1 2n ] to get
k 15
Hence,
12
| B˙ sn |ds
By the similar arguments as above, we have
10
s
− Ysn+ | | B˙ un |du
:= B22,1 + B22,2 + B22,3 .
8
9
− Ys + |
|Ysn+
≤ C
5
|Yun
+ s ˙n n n f (Yv , Z v )dv | Bu |du | B˙ sn |ds − Ys + | u s + + s s n n ˙ n n g(Yv )dBv | Bu |du | B˙ sn |ds |Ys + − Ys + | u s + + s s ˙n n n Z v dW v | Bu |du | B˙ sn |ds |Ys + − Ys + | u s
T
s
|Ysn+
2
n gg ′ Y k+2
(Ysn+ − Ys + )gg ′ (Ysn+ )ds.
2n
2
2
B k+2 − B k+1 n n 2
2
2
2
−
2
1 2n
(2.100)
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
25
Conditioning on F k+2 it is easy to see that the expectation of the first two terms on the right 2n vanishes. Hence, T n ′ n E[B23 ] = E (Ys + − Ys + )gg (Ys + )ds . (2.101)
1 2
3
t
Collect the terms in (2.96), (2.99) and (2.101) to obtain E[B2 ] ≤ C
1 2n
1 −δ
T
2
+E t
4
(Ysn+ − Ys + )gg ′ (Ysn+ )ds .
(2.102)
Now we turn to the term B3 in (2.87). We split it as follows: T 1 B3 = −2 dλ [(Ysn+ − Ys + )g ′ (Ysn+ + λ(Ysn − Ysn+ )) 0
− (Ysn −2 t
6
7
t
− Ys )g
T
(Ysn
′
(Ysn )]
− Ys )g
′
s+ s
(Ysn )
Z vn dW v s
dBns
8
s+
Z vn dW v
dBns
9
:= B31 + B32 .
(2.103)
First we note that
10
11
E[B32 ] = −2 t
T
E (Ysn − Ys )g ′ (Ysn ) B˙ sn E
s
s+
n ds Z v dW v Fs
= 0.
12
(2.104)
By the Lipschitz continuity of g ′ , we have + T s n n n B31 ≤ C |Ys + − Ys | Z v dW v | B˙ sn |ds s t + T s n +C |Ys + − Ys | Z v dW v | B˙ sn |ds t s T s + n n n n +C |Ys − Ys | |Ys + − Ys | Z v dW v | B˙ sn |ds s t := B31,1 + B31,2 + B31,3 . Furthermore, B31,1
5
+ T s + s n n n ≤ C f (Yv , Z v )dv Z v dW v | B˙ sn |ds s t s + T s + s +C g(Yvn )dBnv Z vn dW v | B˙ sn |ds s t s
13
14
15
16
17
(2.105)
18
19
20
21
26
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx T
+C
1
t
:= B31,11 + B31,12 + B31,13 .
2
3
4
5
6
7
8
9
10
11
12
13
2 + s n Z v dW v | B˙ sn |ds s
Now, interchanging the order of integration, we have 2 21 T s + 1 1 E (Z vn )2 dv (E[| B˙ sn |2 ]) 2 ds E[B31,11 ] ≤ C n 2 t s 1 . (2.107) 2n In view of (2.91), (2.92), we have + T s 1 E[B31,12 ] ≤ C n Z vn dW v ds E sup | B˙ sn |2 2 t s s 2 12 s + 1 1 n 2 T Z vn dW v ds ≤ C n (2 ) (E[ sup |Br − Bv |4 ]) 2 E 2 1 s t |r −v|≤ 2n 1 T s+ 2 1 n 2 1 1−δ n 2 (Z ) dv ds ≤ C n (2 ) E v 2 2n s t 12 1−δ 1 T 1 n 2 1 1 2 n 2 ≤ C n (2 ) E[(Z v ) ]dv 2 2n 2n t 1 −δ 1 2 ≤C . (2.108) 2n ≤C
Q10 Using
the fact that | B˙ sn | is Fs -measurable we have 2 T s + Z vn dW v | B˙ sn |ds E[B31,13 ] ≤ C E s
t
14
= CE
T
t
15
(2.106)
s+
s T
s+
= CE t
s
(| B˙ sn |) Z vn dW v
≤ C(2n )E ( sup
≤ C(2n )(E[ sup |r −v|≤ 21n
18
≤C
1 2n
1 −δ 2
,
T
|r −v|≤ 21n
17
ds
(| B˙ sn |)(Z vn )2 dvds
16
2
1 2
|Br − Bv |)
t
s+
s
(Z vn )2 dvds
|Br − Bv |2 ]) E 1 2
t
T
s
s+
2 12 (Z vn )2 dvds
(2.109)
27
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
where (2.91), (2.92) again were used. (2.107)–(2.109) imply that E[B31,1 ] ≤ C
1 2n
1 −δ 2
1
.
(2.110)
For the term B31,2 we have + T s + s Z vn dW v | B˙ sn |ds f (Yv , Z v )dv B31,2 ≤ C s s t T s + s + +C g(Yv )dBv Z vn dW v | B˙ sn |ds s t s + T s + s +C Z v dW v Z vn dW v | B˙ sn |ds t s s
3
4
5
6
:= B31,21 + B31,22 + B31,23 .
(2.111)
By a similar argument as for (2.107), we have 1 E[B31,21 ] ≤ C . 2n
(2.112)
T
E[B31,22 ] ≤ C t
≤C
1 2n
1 2
1 2
2 2 + s s + 1 E g(Yu )dBu ds Z un (| B˙ sn |) 2 dW u E s s 1 2
T
E s
t
s+
E[B31,22 ] ≤ C
1 2n
1 −δ 2
2
(2.113)
ds.
.
The term B31,23 is bounded as + T
t
| B˙ sn |ds + C
t
T
s
2
s+
Z v dW v
| B˙ sn |ds,
(2.115)
which, together with the similar arguments as for the derivation of (2.23), yields
1 2n
1 −δ 2
.
1 −δ 2
.
16
17
(2.116)
(2.112)–(2.116) together give that 1 2n
14
15
Z vn dW v
s
12
13
(2.114)
2
s
B31,23 ≤ C
11
1 (Z un )2 | B˙ sn |du
From here following the same arguments as the derivation of (2.109) we obtain
9
10
E[B31,2 ] ≤ C
7
8
As for B31,22 , we have
E[B31,23 ] ≤ C
2
18
19
(2.117)
20
28 1
2
3
4
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
Finally we need to find a bound for E[B31,3 ]. Notice that + + T s s n n n n Z v dW v | B˙ sn |ds f (Yv , Z v )dv B31,3 ≤ C |Ys − Ys | s s t + + T s s Z vn dW v | B˙ sn |ds g(Yvn )dBnv +C |Ysn − Ys | s s t 2 T s + Z vn dW v | B˙ sn |ds +C |Ysn − Ys | s t := B31,31 + B31,32 + B31,33 .
5
6
7
8
9
Q11 We
have 1 E[B31,31 ] ≤ C n 2 1 ≤C n 2
≤C
11
≤C
12
T
t
2 12 s + 1 (| B˙ sn |) 2 Z vn dW v ds (E[|Ysn − Ys |2 ]) E s 1 2
T
E[|Ysn
t
≤C
1 2n
1
1 2n
1 −δ 2 E
1 2n
1 −δ
2
1 2n
T
t
s
2 |Br − Bv | ]) E 1 4
(E[ sup |r −v|≤ 21n
T
(Z vn )2 dv
s+
t
T t
2
1 + 2 s n n 2 | B˙ s |(Z v ) dv ds E s t 1 2 T s+ n 2 |Br − Bv |) (Z v ) dv ds
21 − Ys | ]ds 2
1 n 1 ≤ C n (2 ) 2 E ( sup 2 |r −v|≤
10
(2.118)
s
2 14 (Z vn )2 dvds
2 41
,
(2.119)
where (2.5), (2.25) have been used. Keeping in mind that |Ysn − Ys |, | B˙ sn | are Fs measurable we 14 Q12 have + 2 T s E[B31,33 ] = C E |Ysn − Ys | 15 Z vn dW v | B˙ sn | ds s t T s+ 16 =C E |Ysn − Ys | (Z vn )2 dv | B˙ sn | ds 13
t
s
17
≤ C2n E ( sup |Ysn − Ys |)( sup 0≤s≤T
|r −v|≤ 21n
|Br − Bv |)
1
18
≤ C2n (E[ sup |Ysn − Ys |4 ]) 4 (E[ sup 0≤s≤T
T
|r −v|≤ 21n
t
s+ s
1
|Br − Bv |4 ]) 4
(Z vn )2 dv ds
29
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
× E
T
t
≤ C2n
n
≤ C2
≤C
1 2n
s+
s
1 −δ 2 E
1 2n
1 −δ
t
1 −δ 2
1 2n
T
(Z vn )2
E
T t
1
v
v−
(Z vn )2 dv
2 12 ds dv 2 21
3
(2.120)
t
5
+ 2 s Z vn dW v | B˙ sn | ds s
The second term above is bounded by 2 + T s g(Yvn )dBnv | B˙ sn | ds E |Ysn − Ys | s t T n 3 3 n ≤ C(2 ) E ( sup |Br − Bv | ) |Ys − Ys | t
|r −v|≤ 21n
≤ C(2 )(E[ sup |r −v|≤ 21n
≤C
2
s
|Br − Bv | ]) E 6
n
1 −δ
6
2 + s g(Yvn )dBnv | B˙ sn | ds. E |Ysn − Ys | s
t
1 2n
1 2
t
.
E[B31,32 ] ≤ C
1 −δ 2
.
s+
2 |g(Yvn )|dv
ds
10
2 21 − Ys |ds
11
1 2n
12
13
(2.123)
(2.119)–(2.123) yield that
E[B31,3 ] ≤ C
8
9
(2.122)
1 2n
7
Q13
T
|Ysn
(2.121)
It follows now from (2.120)–(2.122) that
4
T
+C
2
.
As for the term B31.32 we have T E[B31,32 ] ≤ C E |Ysn − Ys |
(Z vn )2 dv ds
1 2n
2
2 12
14
15
1 −δ 2
.
(2.124)
16
30
Y. Hu et al. / Stochastic Processes and their Applications xx (xxxx) xxx–xxx
It follows from (2.110), (2.117) and (2.124) that
1
E[B31 ] ≤ C
2
1 2n
1 −δ 2
.
(2.125)
This together with (2.104) yields
3
E[B3 ] ≤ C
4
1 2n
1 −δ 2
.
The lemma now follows from (2.88), (2.102) and (2.126).
5
(2.126)
Uncited references
6
Q14 7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
[1], [3], [7], [9], [10] and [11]. Acknowledgments We thank the referees for very useful comments and suggestions. References [1] S. Aida, K. Sasaki, Wong–Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces, Stochastic Process. Appl. 123 (10) (2013) 3800–3827. [2] L. Denis, A. Matoussi, J. Zhang, The obstacle problem for quasilinear stochastic PDEs: an analytical approach, Ann. Probab. 42 (3) (2014) 865–905. [3] L.C. Evans, D.W. Stroock, An approximation scheme for reflected stochastic differential equations, Stochastic Process. Appl. 121 (2011) 1464–1491. [4] N. Ikeda, S. Watanable, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981. [5] A. Matoussi, L. Stoica, The obstacle problem for quasilinear stochastic PDEs, Ann. Probab. 38 (3) (2010) 1143–1179. [6] A. Matoussi, M. Xu, Sobolev solution for semilinear PDE with obstacle under monotonicity conditions, Electron. J. Probab. 13 (35) (2008) 1035–1067. [7] E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990) 55–61. [8] E. Pardoux, S. Peng, Backward doubly SDEs and systems of quasilinear SPDEs, Probab. Theory Related Fields 98 (1994) 209–227. [9] E. Wong, M. Zakai, On the convegence of ordinary integrals to stochastic integrals, Ann. Math. Statist. 36 (1965) 1560–1564. [10] E. Wong, M. Zakai, On the relation between ordinary and stochastic differential equations, Internat. J. Engrg. Sci. 3 (1965) 213–229. [11] Tusheng Zhang, Wong–Zakai approximations to SDEs with reflection, Potential Anal. 41 (3) (2014) 783–815.