On approximate continuity and the support of reflected ... - arXiv

arXiv:1606.01618v1 [math.PR] 6 Jun 2016

The Annals of Probability 2016, Vol. 44, No. 3, 2064–2116 DOI: 10.1214/15-AOP1018 c Institute of Mathematical Statistics, 2016

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS1 By Jiagang Ren and Jing Wu Sun Yat-sen University In this paper we prove an approximate continuity result for stochastic differential equations with normal reflections in domains satisfying Saisho’s conditions, which together with the Wong–Zakai approximation result completes the support theorem for such diffusions in the uniform convergence topology. Also by adapting Millet and Sanz-Sol´e’s idea, we characterize in H¨ older norm the support of diffusions reflected in domains satisfying the Lions–Sznitman conditions by proving limit theorems of adapted interpolations. Finally we apply the support theorem to establish a boundary-interior maximum principle for subharmonic functions.

1. Introduction. The support theorem for diffusion processes defined by stochastic differential equations has been a much studied topic for probabilists and analysts since the seminal work of Stroock and Varadhan [13]. The typical approach to a support theorem in the norm of uniform convergence consists of two steps. One step is to establish a limit theorem for SDEs, meaning that the solution of an SDE can be approximated by a sequence of solutions of ODEs, obtained by regularizing the Brownian paths [15]; the other is to prove a Denjoy-type approximate continuity theorem, stating that the solution of an SDE is approximately continuous at points in a dense set of the Cameron–Martin space. Millet and Sanz-Sol´e [7, 8] proposed a simple approach to characterizing in H¨older spaces the support of diffusions described by general SDEs, obtained by approximating Brownian motions with linear adapted interpolations, and proved the two inclusions through approximation results. Received October 2014; revised March 2015. Supported by NSFC (Nos. 11171358,11301553, 11471340) and the Fundamental Research Funds for the Central Universities (No. 13lgpy64). AMS 2000 subject classifications. Primary 60H10, 60H99; secondary 60F99. Key words and phrases. Reflected stochastic differential equation, approximate continuity, support, limit theorem, maximum principle. 1

This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2016, Vol. 44, No. 3, 2064–2116. This reprint differs from the original in pagination and typographic detail. 1

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J. REN AND J. WU

In this work we are concerned with the support problem of diffusions constrained in a domain D with normal reflection boundary. Such diffusions have been constructed by Anderson and Orey [2] if D has smooth boundary and by Tanaka [14] if D is convex. Correspondingly the support theorem has been established by Doss and Priouret [3] if D has smooth boundary, and a limit theorem has been proved by Pettersson [9] when D is a convex domain and the diffusion coefficient is constant. Recently in [10], a support theorem was proved for stochastic variational inequalities; this means, in particular, that the support theorem holds true for diffusions normally reflected in convex domains. However, normally reflected diffusions have been constructed for domains much wider than convex domains and smooth domains (see Lions and Sznitman [6] and Saisho [12]), so a natural (and application-motivated) question is whether or not the support theorem continues to hold true for such diffusions. The first step in this respect was taken by Evans and Stroock [4] who proved, under the set of conditions given by Lions and Sznitman, that a weak limit theorem holds. Very recently this result was improved by Aida and Sasaki [1], and independently by Zhang [16], who used an adapted version of the Wong–Zakai approximations rather than the usual one, by removing the admissibility condition from the set of conditions and proving that the convergence takes place, in fact, in Lp (and they obtained the convergence speed). Roughly speaking, they proved a strong limit theorem for the reflected diffusions studied by Saisho in [12]. To date, this was the widest, well-studied situation. On the other hand, however, approximate continuity has not yet been touched in such situations. Our first result fills this gap, and it, together with the Wong–Zakai convergence result in [1] and [16], will yield the support theorem in the locally uniform convergence topology for normally reflected SDEs in domains, satisfying the conditions of Lions and Sznitman [6], except the admissibility. The second contribution of this paper is to present a characterization of the support for reflected diffusions in H¨older spaces in domains satisfying the conditions in [4], by extending the idea of Millet and Sanz-Sol´e [8] to SDEs with normal reflections. We recall the Skorohod problem here. Let D be a domain in Rd and ¯ A pair of continuous functions (x, k) w· ∈ C([0, +∞); Rd ) such that w0 ∈ D. is a solution of the Skorohod problem if: ¯ for all t ≥ 0 and x0 = w0 ; • xt ∈ D • for all t ≥ 0, xt = wt + kt ; • k(0) = 0, and k is of bounded variation on each finite interval and satisfies Z t Z t kt = ns d|k|s , |k|t = 1∂D (xs ) d|k|s , 0

0

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

3

where ns ∈ Nxs and Nx is the set of inward normal unit vectors at x ∈ ∂D defined by [ Nx = Nx,r , r>0

Nx,r = {n ∈ Rd ; |n| = 1, B(x − rn, r) ∩ D = ∅}.

Here and in what follows B(a, r) = {y ∈ Rd ; |y − a| < r}, a ∈ Rd , r > 0 and |k|t denotes the total variation of k on [0, t]. Let Ω = C0 ([0, ∞), Rd1 ) be the space consisting of continuous functions from [0, ∞) to Rd1 vanishing at 0. Let F be the completion of the Borel σalgebra on Ω associated with the locally uniform convergence topology and P the distribution of an d1 -dimensional Brownian motion. Then (Ω, F, P) is a complete probability space, and the coordinate process t≥0

wt (ω) := ω(t),

is a d1 -dimensional standard Brownian motion. The natural filtration generated by (wt )t≥0 is denoted by (Ft )t≥0 . We consider the following reflected SDE:  Z t Z t  ¯  b(Xs ) ds + Kt , X0 = x ∈ D, σ(Xs ) ◦ dws +  Xt = X0 + 0 0 Z t Z t    |K|t = ξs d|K|s , 1∂D (Xs ) d|K|s , Kt = 0

0

(1.1) where ξs ∈ NXs . In Itˆo’s notation, it takes the following form:  Z t Z t  ˜b(Xs ) ds + Kt , ¯  X0 = x ∈ D, σ(Xs ) dws +  Xt = X0 + 0 Z t Z t 0    |K|t = ξs d|K|s 1∂D (Xs ) d|K|s , Kt = 0

0

with

d

d

1 XX ˜bi (x) := bi (x) + 1 [σki (x)]j σkj (x). 2

j=1 k=1

Throughout the paper we will assume that σ : Rd 7→ Rd ⊗ Rd1 and b : Rd 7→ Rd are Cb2 and Cb1 functions, respectively. Then by Saisho [12] this equation has a unique solution (X, K). Let Wd (resp., Wd1 ) denote the space of all Rd (resp., Rd1 )-valued continuous functions defined on [0, ∞), and for each α ∈ (0, 12 ), Wdα denote the subspace of Wd consisting of locally α-H¨older continuous functions. Then

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J. REN AND J. WU

for every α ∈ [0, 12 ), Wdα is a Fr´echet space with the topology defined by the system of seminorms {k · kT,α , T > 0}, where for x ∈ Wd , kxkT := sup |xt |,

kxkT,α := kxkT +

0≤t≤T

|x(t) − x(s)| . |t − s|α 0≤s,t≤T,s6=t sup

Denote H := {h : h ∈ Wd1 ; h(0) = 0, h(·) is absolutely continuous and h˙ ∈ L2 ([0, ∞); Rd1 ), ∀T > 0},

S := {h ∈ Wd1 ; h(0) = 0, t → h(t) is smooth},

Sp := {h ∈ Wd1 ; h(0) = 0, t → h(t) is piecewise smooth}. H will be endowed with the topology given by the family of seminorms RT 2 ˙ {khkHT := ( 0 |ht | dt)1/2 , T > 0}. Given h ∈ H, denote by (Z(h), ψ(h)) the solution to the following deterministic Skorohod problem: Z t Z t ˙ (1.2) b(Zs ) ds + ψt . σ(Zs )hs ds + Zt = x + 0

0

Let S (H) := {Z(h), h ∈ H}; α

S := {Z(h), h ∈ S};

Sp := {Z(h), h ∈ Sp }.

Denote by S (H) the closure of S (H) in Wdα , and S , Sp and S (H) the closures of S , Sp and S (H) in Wd , respectively. We are going to prove in Section 2 the approximate continuity theorem, which together with the result in [1] and [16] yields that the support of P ◦ X −1 in Wd coincides with S . We also prove in Section 3 an enhanced version of the support theorem by showing that for every α ∈ (0, 12 ), the support of P ◦ X −1 in Wdα coincides α with S (H) . The paper is organized as follows: in Section 2 an approximate continuity theorem for normally reflected diffusions is proved, and this result combined with the main result in [1] and [16] implies, of course, the support theorem for such diffusions. Next, we provide in Section 3 an alternate approach to solving the support problem in H¨older spaces. Finally in Section 4, we give a first application of our support theorem to maximum principle for Lsubharmonic functions in domains having nonsmooth boundaries and with possibly degenerate L. Throughout the paper we use C to denote a generic constant which may be different in different places, and we use summation convention for repeated indices. Finally A . B means that there exists a C ≥ 0 such that A ≤ CB.

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

5

2. Approximate continuity. In this section we will work in the setup of [6]. But, as in [12], we will not need the admissibility condition on the domain. Precisely, we assume that we are given a domain D ⊂ Rd satisfying: ¯ and ξ ∈ Nx , (H1 ) There exists c0 > 0 such that for any x ∈ ∂D, y ∈ D (y − x, ξ) + c0 |x − y|2 ≥ 0, where Nx denotes the set of unit inward normals at x; (H2 ) There exist a function ϕ ∈ Cb3 (Rd ; R) and a constant α > 0 such that Dϕ(x) · ξ ≥ αc0

∀x ∈ ∂D, ξ ∈ Nx .

It is obvious that under the conditions(H1 )–(H2 ), S = Sp = S (H). To see this, we only need to show S ⊃ S (H). In fact, for any h ∈ H, we can take a sequence hn ∈ S such that hn → h in H. Denote by (Z, Ψ) and (Z n , Ψn ) the corresponding solutions of the Skorohod problem (1.2). Set n ρ(t) := e−(2/α)(ϕ(Zt )+ϕ(Zt )) . Then for any t ≥ 0, by (H2 ) and the assumptions b ∈ Cb1 and σ ∈ Cb2 , we have n

|Ztn − Zt |2 e−(2/α)(ϕ(Zt )+ϕ(Zt )) Z t Z t 2 n n ˙ ˙ |h˙ ns − h˙ s |2 ds, ρ(s)|Zs − Zs | (1 + |hs | + |hs |) ds + C ≤C 0

0

which implies by Gronwall’s lemma that sup0≤t≤T |Ztn − Zt |2 → 0 as n → ∞ and thus Z ∈ S , yielding that S ⊃ S (H). Before we proceed, a few words about these conditions are in order. The constant c0 appearing in condition (H1 ) is also allowed to equal to zero in [6]. Then the function ϕ in condition (H2 ) can be taken to be identically zero, and it turns out that some arguments below will break down, and different treatments will be needed. But in this case D is a convex domain, and thus the equation is a special case of stochastic variational inequalities already treated in [10]. Hence we simply assume c0 > 0 here. For convenience we record here some basic facts which will be used below; see [5]. Set for i, j = 1, . . . , d1 , Z t Z 1 t i κij (t) := wsi ◦ dwsj . [ws dwsj − wsj dwsi ], ζ ij (t) := 2 0 0 Let T > 0 be arbitrarily fixed. Lemma 2.1.

(i) There exist two positive constants c1 and c2 such that   c2 as δ ↓ 0. P(kwkT < δ) ∼ c1 exp − 2 δ

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J. REN AND J. WU

(ii) For all i, j = 1, . . . , d1 , lim sup P(kκij kT > M δ|kwkT < δ) = 0.

M ↑∞ 0 M δ|kwkT < δ) = 0.

M ↑∞ 0 0 and α ∈ (0, 1), P(kζ ij kT > εδα |kwkT < δ) → 0

(2.1)

as δ ↓ 0.

In fact, for arbitrary M > 0, take δ0 > 0 such that εδ0α−1 ≥ M . Then for any 0 < δ < δ0 , P(kζ ij kT > εδα |kwkT < δ) ≤ P(kζ ij kT > M δ|kwkT < δ).

Thus

lim sup P(kζ ij kT > εδα |kwkT < δ) δ↓0

≤ sup P(kζ ij kT > M δ|kwkT < δ). 0 0 such that

Proposition 2.1.

2

E[eβ(|K|T ) ] < ∞,

2

E[eβkXkT ] < ∞.

Proof. By Itˆo’s formula and (H2 ) we have Z t Z t Dϕ(Xs )˜b(Xs ) ds (Dϕ)(Xs )σ(Xs ) dws − αc0 |K|t ≤ ϕ(Xt ) − ϕ(X0 ) − −

Since

ϕ ∈ Cb2 ,

0

0

(2.3)

1 2

Z

t

tr[D 2 ϕ(Xs )(σσ ∗ )(Xs )] ds.

0

there exists a β ′ > 0 such that

2    Z ·

′ E exp β (Dϕ)(Xs )σ(Xs ) dws < ∞.

0

T

From this the first inequality follows immediately, and the second follows from the first together with equation (1.1). 

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

Lemma 2.2.

7

limδ↓0 P(|K|T ≥ εδ−1/2 |kwkT < δ) = 0.

Proof. We have by Lemma 2.1 and Proposition 2.1 that (2.4)

lim P(|K|T ≥ εδ−3/2 |kwkT < δ) . lim δ↓0

δ↓0

exp{−ε2 δ−3 β} = 0. exp{−Cδ−2 }

Next we prove that for f ∈ Cb2 (Rd ; R) and 1 ≤ k ≤ d1 we have

 Z · 

k −1/2

lim P f (Xs ) ◦ dws ≥ εδ (2.5) kwkT < δ = 0. δ↓0

0

T

∂f ∂xi (x).

By Itˆo’s formula we have Set fi (x) := Z t Z t [fi σji ](Xs )wsk ◦ dwsj f (Xs ) ◦ dwsk = f (Xt )wtk − 0

0

−

Z

t

i

k

[fi b ](Xs )w ds −

0

Z

t

0

fi (Xs )wsk dKsi

=: I1 (t) − I2 (t) − I3 (t) + I4 (t). We need to prove lim P(kIi kT ≥ εδ−1/2 |kwkT < δ) = 0, δ↓0

i = 1, 2, 3, 4.

This is obvious for I1 and I3 . To show this for I2 we notice that Z t Z 1 t [fi σji ](Xs )wsk dwsj + I2 (t) = [fi σji ](Xs )δkj ds 2 0 0 Z t 1 [fi σji ]q σlq (Xs )wsk δlj ds + 2 0 := I21 (t) + I22 (t) + I23 (t).

Noticing that f and σ are bounded, the sets {kI2i kT > εδ−1/2 } ∩ {kwkT < δ}, i = 2, 3 will be empty for small δ and thus lim{P(kI22 kT ≥ εδ−1/2 |kwkT < δ) + P(kI23 kT ≥ εδ−1/2 |kwkT < δ)} = 0. δ↓0

Since for t ∈ [0, T ], hI21 , I21 i(t) =

d1 Z X j=1

0

t

[fi σji ]2 (Xs )(wsk )2 ds . kwk2t .

By the exponential inequality (cf. [11], Exercise IV.3.16) we have lim P(kI21 kT ≥ εδ−1/2 |kwkT < δ) . lim δ↓0

δ↓0

exp{−ε2 δ−3 } = 0. exp{−Cδ−2 }

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J. REN AND J. WU

Hence lim P(kI2 kT ≥ εδ−1/2 |kwkT ≤ δ) = 0. δ↓0

Finally, since we have by using (2.4) that

kI4 kT . kwkT |K|T ,

lim P(kI4 kT ≥ εδ−1/2 |kwkT < δ) = 0. δ↓0

Thus (2.5) has been proved. Now the result follows from (2.3) and (2.5).  Corollary 2.1.

For every ε > 0, lim P(kζ ij kT |K|T > ε|kwkT < δ) = 0,

(2.6)

δ↓0

 Z · 

ij

lim P ζ (s) dKs > ε kwkT < δ → 0.

(2.7)

δ↓0

0

T

Proof. It suffices to prove (2.6). Using (2.1) with α = lemma we have

1 2

and the above

P(kζ ij kT |K|T > ε|kwkT < δ)

≤ P(kζ ij kT > δ1/2 |kwkT < δ) + P(|K|T > εδ−1/2 |kwkT < δ) → 0,

δ ↓ 0.



Now we can prove the following: Lemma 2.3. Suppose f ∈ Cb (Rd ; R) is uniformly continuous. Then for all ε > 0 and i, j = 1, 2, . . . , d1 ,

 Z · 

ij

lim P f (Xs ) dζ (s) > ε kwkT < δ → 0. (2.8) δ↓0 0

T

Proof. First we assume that f ∈ Cb2 (Rd ; R). Itˆo’s formula gives us Z t Z t ij ij ζ ij (s)fl (Xs )σkl (Xs ) dwsk f (Xs ) dζ (s) = f (Xt )ζ (t) − 0

0

−

Z

−

Z

=:

t

0 t 0

5 X q=1

(Lf )(Xs )ζ ij (s) ds − fl (Xs )ζ ij (s) dKsl

I2q ,

Z

t

0

fl (Xs )σjl (Xs )wsi ds

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

9

P P where L := 21 i,j aij ∂i ∂j + i ˜bi ∂i . It is easy to see that for q = 1, 3, 4,

lim P(kI2q kT > ε|kwkT < δ) = 0.

(2.9)

δ↓0

Since

Z ·

ij l

fl (Xs )ζ dK . kζkT |K|T , s s

0

we have

T

P(kI25 kT > ε|kwkT < δ) ≤ P (kζkT |K|T > ε|kwkT < δ).

Consequently by (2.6),

lim P(kI25 kT > ε|kwkT < δ) = 0. δ↓0

Now we deal with I22 . Set gk (x) := −fl (x)σkl (x), gk,l := by Itˆo’s formula, Z t gk (Xs )ζ ij (s) dwsk I22 =

∂ g (x). ∂xl k

We have

0

= gk (Xt )ζ

:=

−

Z

−

Z

−

Z

8 X

ij

(t)wtk

−

t

(Lgk )(Xs )ζ 0 t 0 t 0

Z

t

0

ij

gk,l (Xs )σql (Xs )ζ ij (s)wsk dwsq

(s)wsk

gj (Xs )wsi ds −

Z

t 0

ds −

Z

t

0

gk (Xs )wsk dζ ij (s)

ζ ij (s)gk,l (Xs )σql (Xs )δkq ds

gk,l (Xs )σjl (Xs )wk (s)wsi ds −

Z

t 0

gk,l (Xs )ζ ij (s)wsk dKsl

I22i .

i=1

Obviously

lim P(kI22i kT > ε|kwk < δ) = 0, δ↓0

i = 1, 3, 4, 5, 7,

and it is clear from Corollary 2.1 that it holds also for i = 8. For I222 we notice I222 (t) = Mt , where Mt =

Z

0

t

gk,l (Xs )σql (Xs )ζ ij (s)wsk dwsq .

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J. REN AND J. WU

It suffices to prove lim P(kM kT > ε|kwkT < δ) = 0.

(2.10)

δ↓0

Since hM i(t) .

Z

0

t

kζ ij k2s kwk2s ds,

we have by exponential inequality P(kM kT > ε, kζ ij kT < Aδ, kwkT < δ)

≤ P(kM kT > ε, hM i(T ) ≤ cA2 δ4 ) ≤ c exp{−cA−2 δ−4 } → 0,

δ ↓ 0.

Since P(kM kT > ε|kwkT < δ) = P(kM kT > ε, kζ ij kT > Aδ|kwkT < δ)

+ P(kM kT > ε, kζ ij kT ≤ Aδ|kwkT < δ)

≤ sup P(kM kT > ε, kζ ij kT > Aδ|kwkT < δ) 0≤δ≤1

+ P(kM kT > ε, kζ ij kT ≤ Aδ|kwk < δ), we have lim P(kM kT > ε|kwkT < δ) ≤ sup P(kM kT > ε, kζ ij kT > Aδ|kwkT < δ). δ↓0

0≤δ≤1

Hence by letting A → ∞ we have lim P(kM kT > ε|kwkT < δ) = 0. δ↓0

Now we extend the result to f ∈ Cb , which is uniformly continuous. Let ε ) sufficiently small such ε > 0 be given. For any ε′ > 0 choose an η ∈ (0, 2T that   4 ε2 ε2 < 0, exp c2 − < ε′ , c2 − 32η 2 T c1 32η 2 T where c1 and c2 are constants appearing in Lemma 2.1. Then choose a g ∈ Cb2 such that kf − gkT < η. Note that Z t Z t ij g(Xs ) dζ ij (s) f (Xs ) dζ (s) − 0

0

=

Z

t

0

(f − g)(Xs )wsi dwsj +

=: Y1 (t) + Y2 (t).

δij 2

Z

0

t

(f − g)(Xs ) ds

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

11

It is easy to see kY2 kT < 4ε . Moreover, since hY1 i(T ) ≤ η 2 kwk2T T , we have by exponential inequality and with arguments similar to the proof of (2.10) that if δ ∈ (0, 1],   ε P kY1 kT ≥ kwkT < δ 4   ε ≤ P kY1 kT ≥ , hY1 i(T ) ≤ η 2 δ2 T P (kwkT < δ)−1 4    1 ε2 4 ≤ ε′ . ≤ exp 2 c2 − c1 δ 32η 2 T Thus for such δ,

 Z · 

ij

P f (Xs ) dζ (s) ≥ ε kwkT < δ 0 T

  Z ·

ij ′

≤ ε + P g(Xs ) dζ (s) ≥ ε/2 kwk < δ . 0

T

Now we conclude by letting δ → 0 and by the arbitrariness of ε′ .  We have: (i) For all f ∈ Cb2 (Rd ; R), ε > 0 and 1 ≤ k ≤ d1 ,

  Z ·

k

lim P f (Xs ) ◦ dws ≥ ε kwkT < δ = 0.

Lemma 2.4.

δ↓0

0

T

(ii) There exists a constant c3 > 0 such that

lim P(|K|T > c3 |kwkT < δ) = 0. δ↓0

Proof. It suffices to prove (i), since then (ii) follows from (i) and (2.3). We have Z t Z t [fi σji ](Xs ) dζ kj f (Xs ) ◦ dwsk = f (Xt )wtk − 0

0

−

Z

0

t

[fi b

i

](Xs )wsk ds −

Z

t

0

fi (Xs )wsk dKsi

:= I1 (t) − I2 (t) − I3 (t) − I4 (t). Since kI4 kT . kwkT |K|T , by Lemma 2.2 we have lim P(kI4 kT ≥ ε|kwkT < δ) ≤ lim P(|K|T ≥ cεδ−1 |kwkT < δ) = 0, δ↓0

δ↓0

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J. REN AND J. WU

while by Lemma 2.3 we have lim P(kI2 kT ≥ ε|kwkT < δ) = 0. δ↓0

Finally, it is trivial that lim P(kIi kT ≥ ε|kwkT < δ) = 0,

i = 1, 3.

δ↓0

This completes the proof.  Now are ready to state our main result. Let (Y, l) denote the solution of the following deterministic Skorohod problem: Z t Z t    σ(Ys ) dhs + b(Ys ) ds + lt , Y0 = x,  Yt = Y0 + 0 (2.11) Z t Z t 0    |l|t = η(s) d|l|s , 1∂D (Ys ) d|l|s , lt = 0

0

where η(s) ∈ NYs . Theorem 2.1.

For any h ∈ S and ε > 0,

P(kX − Y kT + kK − lkT < ε|kw − hkT < δ) → 1

as δ ↓ 0.

Proof. We first assume h ≡ 0. Since (X, K) and (Y, l) are solutions to equations (1.1) and (2.11), respectively, we have Z t Z t Z t (dKs − dls ). (b(Xs ) − b(Ys )) ds + σ(Xs ) ◦ dws + Xt − Yt = 0

0

0

Set

Ψ(x) := 1 − e−|x|

2 /2

;

then Ψi (x) := Ψi,j (x) :=

2 ∂ Ψ(x) = e−|x| /2 xi ; ∂xi

∂2 2 Ψ(x) = −e−|x| /2 [xi xj + δij ]; ∂xi ∂xj

G(t) := Xt − Yt ,

ϕi (x) :=

∂ ϕ(x). ∂xi

By Itˆo’s formula we have       2 2 (ϕ(Xt ) + ϕ(Yt )) d exp − (ϕ(Xt ) + ϕ(Yt )) Ψ(G(t)) exp α α

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

13

= Ψi (G(t))σki (Xt ) ◦ dwtk + Ψi (G(t))(bi (Xt ) − bi (Yt )) dt + Ψi (G(t))(dKti − dlti ) −

2 Ψ(G(t))[ϕi (Xt )σki (Xt ) ◦ dwtk + ϕi (Xt )bi (Xt ) dt + ϕi (Xt ) dKti α + ϕi (Yt )bi (Yt ) dt + ϕi (Yt ) dlti ]

2 Ψ(G(t))Ψi (G(t))ϕi (Xt )σki σki (Xt )dt. α Using the elementary inequality 1 − e−t ≥ te−t for t ≥ 0 and conditions (H1 )– (H2 ), we have −

2 Ψi (G(t)) dKti − Ψ(G(t))ϕi (Xt ) dKti α   2 2 2 = e−|Xt −Yt | /2 (Xt − Yt )∗ ξt − (1 − e−|Xt −Yt | /2 )ϕi (Xt )ξti d|K|t α ≤ e−|Xt −Yt |

2 /2

[(Xt − Yt )∗ ξt − c0 |Xt − Yt |2 ] d|K|t ≤ 0,

2 Ψ(G(t))ϕi (Yt ) dlti ≤ 0. α Combining these with the fact |Ψi (x)xi | . Ψ(x), we have   Z t Z t 2 k exp − (ϕ(Xt ) + ϕ(Yt )) Ψ(G(t)) ≤ Ψ(G(s)) ds, ρk (s) ◦ dws + C α 0 0 −Ψi (G(t)) dlti −

where

  2 ρk (s) := exp − (ϕ(Xs ) + ϕ(Ys )) α   2 i i × Ψi (G(s))σk (Xs ) − Ψ(G(s))ϕi (Xs )σk (Xs ) . α

By Itˆo’s formula   2 exp (ϕ(Xs ) + ϕ(Ys )) ◦ dρk (s) α  2 i i (Xs ) (Xs ) + σki (Xs )Ψij (G(s)) − (Ψ(G(s))ϕi (Xs )σkj = Ψi (G(s))σkj α  i i + Ψ(G(s))ϕij (Xs )σk (Xs ) + Ψj (G(s))ϕi (Xs )σk (Xs )) × [σlj (Xs ) ◦ dwsl + bj (Xs ) ds + dKsj ]   2 i i + Ψj (G(s))ϕi (Xs )σk (Xs ) − σk (Xs )Ψij (G(s)) [bj (Ys ) ds + dlsj ] α

14

J. REN AND J. WU

  2 2 (ϕ(Xs ) + ϕ(Ys )) − ρk (s) exp α α

× [ϕj (Xs )(σlj (Xs ) ◦ dwsl + bj (Xs ) ds + dKsj )

+ ϕj (Ys )(bj (Ys ) ds + dlsj )],

i (x) := where σkl

∂ i ∂xl σk (x).

Rearranging, we write

dρk (s) = Fkl (Xs , Ys ) ◦ dwsl + Gkj (Xs , Ys )bj (Xs ) ds + Gkj (Xs , Ys ) dKsj + Hkj (Xs , Ys )bj (Ys ) ds + Hkj (Xs , Ys ) dlsj ,

where   2 Gkj (x, y) : = exp − (ϕ(x) + ϕ(y)) α  i (x) + σki (x)Ψij (x − y) × Ψi (x − y)σkj −

2 i (Ψ(x − y)ϕi (x)σkj (x) + Ψ(x − y)ϕij (x)σki (x) α + Ψj (x − y)ϕi (x)σki (x))

  2 2 − exp − (ϕ(x) + ϕ(y)) α α   2 × Ψi (x − y) − Ψ(x − y)ϕi (x) σki (x)ϕj (x), α

Fkl (x, y) : = Gkj (x, y)σlj (x),   2 i i Ψj (x − y)ϕi (x)σk (x) − σk (x)Ψij (x − y) Hkj (x, y) : = α   2 × exp − (ϕ(x) + ϕ(y)) α   2 2 − exp − (ϕ(x) + ϕ(y)) α α   2 × Ψi (x − y) − Ψ(x − y)ϕi (x) σki (x)ϕj (y). α Thus we have by Itˆo’s formula, Z t Z t ρk (s) ◦ dwsk = ρk (t)wtk − Fkl (Xs , Ys )wsk ◦ dwsl 0

0



ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

−

Z

−

Z

t

Gkj (Xs , Ys )b

0 t

0

j

(Xs )wsk ds −

Hkj (Xs , Ys )bj (Ys )wsk ds −

Z

t

0 Z t 0

15

Gkj (Xs , Ys )wsk dKsj

Hkj (Xs , Ys )wsk dlsj

=: I1 (t) − I2 (t) − I3 (t) − I4 (t) − I5 (t) − I6 (t). Obviously, X i6=2

kIi kT . (1 + |K|T )kwkT .

Thus X  lim P kIi kT ≥ ε kwkT < δ ≤ lim P((1 + |K|T )kwkT & ε|kwkT < δ) = 0. δ↓0

δ↓0

i6=2

As for I2 we have Z t Z 1 t ∂ Fkl (Xs , Ys )σpj (Xs )wsk δpl ds Fkl (Xs , Ys ) dζskl + I2 (t) = 2 ∂x j 0 0 =: I21 + I22 .

It is easily seen that lim P(kI22 kT ≥ ε|kwkT < δ) = 0, δ↓0

and by applying Lemma 2.3 to the functions Fkl (in place of f there) and the system satisfied by (X, Y ) (in place of X there), we have that lim P(kI21 kT ≥ ε|kwkT < δ) = 0. δ↓0

Consequently lim P(kI2 kT ≥ ε|kwkT < δ) = 0. δ↓0

Combining all the above and the fact that ϕ is bounded, we have Z t Ψ(G(s)) ds + A(t), Ψ(G(t)) ≤ C 0

where A(t) satisfies that for every ε > 0, lim P(kAkT > ε|kwkT < δ) = 0. δ↓0

On the set {ω; kAkT < ε}, we have

Ψ(G(t)) ≤ εeC ≤ Cε,

16

J. REN AND J. WU

that is, kX − Y kT ≤ Since ε is arbitrarily small,

p

−2 ln(1 − Cε).

P(kX − Y kT > ε|kwkT < δ) → 0

as δ ↓ 0.

P(kK − lkT < ε|kwkT < δ) → 1

as δ ↓ 0,

Finally, to see

it suffices to notice that Kt − lt = Xt − Yt −

Z

t 0

σ(Xs ) ◦ dws −

Z

0

t

(b(Xs ) − b(Ys )) ds

and use Lemma 2.4. For general h ∈ S, just as in the proof of [5], Theorem 8.2, pages 527–528, we set  Z T Z 1 T ˙ 2 ˙ |hs | ds , dP′ = M1 dP. M1 (w) := exp hs dws − 2 0 0

Then wt′ := wt − ht is a Brownian motion under P′ , and (X, K), (Y, l) satisfy the following equations, respectively: Z t Z t σ(Xt ) ◦ dws + Kt , b′ (s, Xs ) ds + Xt = x + 0

0

Yt = x +

Z

t

b′ (s, Ys ) ds + lt ,

0

b′ (s, x) := b(x) + σ(x)h˙ s .

where Therefore according to the case of h ≡ 0 we have for every ε > 0, P′ (kX − Y kT > ε|kw′ kT < δ) → 0 ′

′

P (kK − lkT > ε|kw kT < δ) → 0

as δ ↓ 0,

as δ ↓ 0,

which, together with the fact that M1 is a continuous functional of w, yields that lim P(kX − Y kT < ε|kw′ kT < δ) δ↓0

= lim δ↓0

E(M1 1{kw−hkT 0, R > 0, a1 , . . . , am ∈ Sd−1 and x1 , . . . , xm ∈ Sm ∂D such that ∂D ⊂ i=1 B(xi , R) and x ∈ ∂D ∩ B(xi , 2R) ⇒ n · ai ≥ λ, ∀n ∈ Nx . We will need some results from [1].

Lemma 3.1 ([1], Lemma 2.3). Assume (A)–(B) hold, and (x, k) is the solution to the Skorohod problem associated with a continuous function w ¯ Then for θ ∈ (0, 1], there exist constants c1 , c2 , C such that x0 = w0 ∈ D. dependent on θ, δ, β, γ0 such that for all 0 ≤ s ≤ t ≤ T , 1 |k|st ≤ C(1 + kwkc[s,t],θ (t − s))ec2 kwk[s,t] kwk[s,t] ,

where (and throughout) |k|st denotes the total variation of k on [s, t] and |wu − wv | , θ u,v∈[s,t] |u − v|

kwk[s,t],θ := sup

kwk[s,t] := sup |wu − wv |. u,v∈[s,t]

Lemma 3.2 ([1], Lemma 2.4). Assume (A) holds, and (x, k) is the solution to the Skorohod problem associated with a function w having continuous bounded variation path. Then √ |x|st ≤ 2( 2 + 1)|w|st .

18

J. REN AND J. WU

Lemma 3.3 ([1], Lemma 2.8). Assume D satisfies conditions (A)–(B), and b, σ are bounded, Lipschitz continuous functions. Then there exists a unique solution (X, K) to equation (1.1). Moreover, for all 0 ≤ s < t < ∞, E(kXk[s,t] )2p ≤ Cp |t − s|p ,

E(|K|st )2p ≤ Cp |t − s|p .

Let n ∈ N and ti = iT 2−n (here we should have used tni instead of ti to indicate the dependence on n, but in order to not surcharge the notation, we omit the superscript n), ∆ = 2−n T , and for t ∈ [ti , ti+1 ) set t¯n := ti−1 ∨ 0,

tˆn := ti ,

∆wi := wti − wti−1 ∨0 ,

wtˆn − wt¯n (t − tˆn ). ∆ Consider the following reflected equation: Z t Z t n n σ(X n (s)) dwsn + K n (t). b(X (s)) ds + X (t) = x + wtn := wt¯n +

0

Denote the solution by

0

(X n , K n ).

3.2. Support theorem. We first state our main theorem. Theorem 3.1. Suppose conditions (A)–(D) hold and σ ∈ Cb2 , b ∈ Cb1 . Then for the solution X to equation (1.1) we have the support of (P ◦ X −1 ) in Wα = S (H)

α

∀α ∈ [0, 12 ).

To prove the theorem, we will apply the following results; cf. [8]. Proposition 3.1. (X, k · k):

Let F be a measurable map from Ω to a Banach space

(1) Let Z1X : H → X be measurable and Hn : Ω → H be a sequence of random variables such that for any ε > 0, lim P(kZ1X (Hn (ω)) − F (ω)k > ε) = 0. n

Then supp(P ◦ F −1 ) ⊂ Z1X (H). (2) Let Z2X : H → X be measurable and for fixed h, Tnh : Ω → Ω be a sequence of measurable transformations such that P ◦ (Tnh )−1 ≪ P, and for any ε > 0, lim sup P(kF (Tnh (ω)) − Z2X (h)k < ε) > 0. n

Then supp(P ◦ F −1 ) ⊃ Z2X (H).

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

19

Proposition 3.2. Suppose {Xtn } is a sequence of finite dimensional processes satisfying that for every p ≥ 1 and s, t ∈ [0, T ], there exists a constant C > 0, sup E|Xtn − Xsn |2p ≤ C|t − s|p .

(3.1)

n

1 2

Then for any ε > 0 and θ
0 such that

sup P(kX n kT,θ > ε) ≤ Cε−2p . n

Moreover, besides (3.1), if for any ε > 0,   lim P sup |Xtni | > ε = 0 n

1≤i≤2n

holds as well, then for any θ ∈ [0, 1/2),

lim P(kX n kT,θ > ε) = 0, n

where k · kT,θ is defined in the Introduction. Following the idea in [8], take Hn (ω) = wn (ω),

Z1X = Z2X = Z(·),

Tnh (ω) = w − wn + h.

Then by Girsanov’s theorem, P ◦ (Tnh )−1 ≪ P. To prove Theorem 3.1, by Proposition 3.1, it suffices to prove that for every ε > 0, lim P(kX − X n kT,θ > ε) = 0

(3.2)

n

and lim P(kX(w − wn + h) − Z(h)kT,θ > ε) = 0,

(3.3)

n

where Z(h) solves the following deterministic Skorohod problem: Z t Z t ˙ b(Z(h)s ) ds + ψt . σ(Z(h)s )hs ds + Z(h)t = x + 0

0

In what follows we will use Z instead of Z(h) if no confusion is possible. (3.2) is proved in [16], so we only need to prove (3.3). Using the Riemannian sum approximation of stochastic integrals, it is easy to see that Y n := X(w − wn + h) solves the following RSDE: Z t Z t Z t Z t n ˙ n n n n ˜b(Y n ) ds + φn , σ(Ys )hs ds + σ(Ys )w˙ s ds + σ(Ys ) dws − Yt = x + s t

where ˜b := b + and We first prepare some auxiliary results. 1 2 (∇σ)σ

φn (w) = K(w

0

0

0

0

− wn

+ h).

20

J. REN AND J. WU

Lemma 3.4.

For 0 ≤ s ≤ t ≤ T , |Zt − Zs |2p ≤ Cp |t − s|p .

Proof. By Lemma 3.2, √ ≤ [2( 2 + 1)]2p

2p

|Zt − Zs |

p

Z

t s

2p ˙ ˜ |σ(Zu )hu + b(Zu )| du

≤ Cp |t − s| .



Proposition 3.3. Let p ≥ 1. Then there exists a constant Cp > 0 independent of n such that for all 0 ≤ s ≤ t ≤ T , (3.4)

E(|Ytn − Ysn |4p ) ≤ Cp |t − s|p ,

E(|φnt − φns |4p ) ≤ Cp |t − s|p .

Moreover, for all 0 ≤ s ≤ t ≤ T and for any θ ∈ (0, 14 ), (3.5)

E(kY n kp[s,t],θ ) ≤ Cp,θ ,

E(kφn kp[s,t],θ ) ≤ Cp,θ .

To prove this proposition, we need some lemmas, and without loss of generality we take T = 1. Lemma 3.5. Let λ, t > 0. Then there exists a constant C > 0 independent of λ and t such that √ 2 E(eλkwkt ) ≤ (1 + Cλ t)d1 Ceλ d1 t/2 . Proof. Set ξ = max0≤s≤t |ws |. Note that i = 1, . . . , d1 and thus Z Z ∞ s λξ e P(λξ > s) ds + 1 ≤ 2 E(e ) =

P(|wti | ∈ dx) = ∞

0

0

λ|wt |

= 2E(e

)−1≤2

d1 Y i=1

r

√ 2 ≤ (1 + Cλ t)d1 Ceλ d1 t/2 .

2 πt

Z

∞

q

2 −x2 /(2t) dx, πt e

es P(λ|wt | > s) ds + 1

eλx e−x

2 /(2t)

dx

0



Rt Lemma 3.6. Let Mt := 0 fs dws and |fs | ≤ c for some constant c. Then there exists a constant C > 0 such that for any integer m, m m/2 EkM km (t − s)m/2 . [s,t] ≤ C (m/2)

Proof. It suffices to prove the result for s = 0. Then Mt = BhM it where B is the DDS-Brownian motion of M . Note that hM it ≤ c2 t.

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

21

The result follows from Doob’s maximal inequality and that E(|Bt |2m ) ≤ (2d1 )m mm tm .

(3.6)



Set Lnt

:= x +

Z

0

t

σ(Ysn ) dws

Lemma 3.7. any p ≥ 1,

−

Z

0

t

σ(Ysn )w˙ sn ds +

Z

t

0

σ(Ysn )h˙ s ds +

Z

t

0

˜b(Y n ) ds. s

There exists a constant Cp such that for any t ∈ [0, 1] and

E(kY n k2p ) ≤ Cp ∆ p , [t¯n ,t]

¯

E(|φn |ttn )2p ≤ Cp ∆p .

Proof. By Lemma 3.1, for any θ ∈ (0, 1], n ¯ |φn |ttn ≤ C(1 + kLn kc[t¯1n ,t],θ (t − t¯n ))ec2 kL k[t¯n ,t] kLn k[t¯n ,t] .

Note that for any p ≥ 1, 2p Z t p  Z t n n n 2p n 2 kσ(Yr )k|w˙ r | dr E(kL k[t¯n ,t] ) ≤ Cp E kσ(Yr )k dr + E t¯n

t¯n

+E

Z

t t¯n

kσ(Yrn )k|h˙ r | dr

2p

 2p ¯ + (t − tn ) ≤ Cp (t − t¯n )p .

For any c, by Lemmas 3.5 and 3.6, E(ecpkL

nk [t¯n ,t]

)

≤ E(ecp maxu,v∈[t¯n,t] | × ecp|

Rt

Rv u

t¯n

σ(Yrn ) dwr +cp

Rv u

kσ(Yrn )k|h˙ r | dr+cp

≤ (1 + Cp∆1/2 )d1 eCd1

p2 ∆+Cp

Rt

σ(Yrn )w˙ rn dr|

Rt

t¯n

˙

˜ b(Yrn ) dr|

t¯n (1+|hr |) dr

)

≤ Cp < ∞.

Now combining these two estimates gives ¯

E(|φn |ttn )2p ≤ Cp ∆p . The other result follows from Ytn = Lnt + φnt and the above estimate.  Lemma 3.8. For any s, t ∈ [0, 1], 2p Z v n n σ(Yr ) dwr ≤ Cp |t − s|p , E sup u,v∈[s,t]

u

p E(kLn k2p [s,t] ) ≤ Cp |t − s| .

22

J. REN AND J. WU

Proof. When ti−1 ≤ s ≤ ti ≤ t ≤ ti+1 for some 1 ≤ i ≤ 2n , the result is trivial. For general s, t, choose 1 ≤ l < m − 1 < m ≤ 2n such that tl−1 ≤ s ≤ tl < tm−1 ≤ t ≤ tm . Note that Z t Z t Z t n n n n n σ(Yr¯nn ) dwrn (σ(Yr ) − σ(Yr¯n )) dwr + σ(Yr ) dwr = s

s

s

and Z

t s

σ(Yr¯nn ) dwrn =

tl

Z

s

σ(Yr¯nn ) dwrn +

= σ(Ytnl−2 ∨0 ) +

m−1 X

j=l+1

n n 0 σ(Yr¯n ) dwr

Z

tj

j=l+1 tj−1

Z

σ(Yr¯nn ) dwrn +

t tm−1

σ(Yr¯nn ) dwrn

wtl−1 − wtl−2 ∨0 (tl − s) ∆

σ(Ytnj−2 ∨0 )(wtj−1 − wtj−2 ∨0 )

+ σ(Ytnm−2 ∨0 ) R·

m−1 X

wtm−1 − wtm−2 ∨0 (t − tm−1 ), ∆

is the piecewise linear interpolation of Z ·−∆ n σ(Y n (πn (r))) dwr M· := 0

with πn (r) := max{tk ; tk ≤ r}, at {tk }k=0,1,...,2n −1 . Thus Z v n n sup σ(Yr¯n ) dwr ≤ sup |Mtnk − Mtnk′ | u,v∈[s,t]

u

l−2≤k,k ′ ≤m−1

≤2

sup

tl−2 ≤r≤tm−1

Using Doob’s inequality we get 2p Z v n n E sup σ(Yr¯n ) dwr ≤ Cp E u,v∈[s,t]

u

≤ Cp E

|Mrn − Mtnl |.

sup tl−2 ≤r≤tm−1

Z

tm−1 tl−2

|Mrn − Mtnl |2p

kσ(Yr¯nn )k2 dr

≤ Cp |tm−1 − tl−2 |p ≤ Cp |t − s|p .

p

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

23

By H¨older’s inequalities and Lemma 3.7, 2p Z v n n n E sup (σ(Yr ) − σ(Yr¯n )) dwr u,v∈[s,t]

≤E

u

Z

s

t

kσ(Yrn ) − σ(Yr¯nn )k2p |w˙ rn |2p dr(t − s)2p−1 2p−1

≤ (t − s)

Z

t

s

2p

(Ekσ(Yrn ) − σ(Yr¯nn )k4p )1/2 (E|w˙ rn |4p )1/2 dr

≤ Cp (t − s) . Now note that Z t Z t Z t ˜b(Y n ) dr. σ(Yrn )h˙ r dr + σ(Yrn )(dwr − dwrn ) + Lnt − Lns = r s

s

s

Trivially by the Burkholder and H¨older inequalities we have Z v 2p Z v Z v n ˙ n n ˜ σ(Yr )hr dr + b(Yr ) dr ≤ C|t − s|p . E sup σ(Yr ) dwr + u,v∈[s,t]

u

u

u

From the estimates above we deduce

E sup |Lnu − Lnv |2p ≤ C|t − s|p .

(3.7)

u,v∈[s,t]



Now we are ready to prove Proposition 3.3. Proof of Proposition 3.3. For cases of s, t ∈ [ti−1 , ti ] and ti−1 ≤ s ≤ ti < t ≤ ti+1 for some 1 ≤ i ≤ 2n , it follows from Lemmas 3.7–3.8 that (3.8)

p EkY n k2p [s,t] ≤ Cp |t − s| ,

E[(|φn |st )2p ] ≤ Cp |t − s|p .

For general cases, choose 1 ≤ l < m − 1 < m ≤ 2n such that tl−1 ≤ s ≤ tl < tm−1 ≤ t ≤ tm . We get by Itˆo’s formula, n

d(e−(2/γ)ϕ(Yt ) |Ytn − Ysn |2 )

=: Usn (t) dwt + Usn (t) dwtn + Vsn (t) dt + Zsn (t) dt + A5t ,

where according to (C),   2 n 5 −(2/γ)ϕ(Ytn ) n n n n 2 n n At := e 2hYt − Ys , dφt i − |Yt − Ys | hDϕ(Yt ), dφt i ≤ 0 γ and n Usn (t) := e−(2/γ)ϕ(Yt )



2(Ytn

− Ysn ) −

 2 n n 2 n |Y − Ys | Dϕ(Yt ) σ(Ytn ), γ t

24

J. REN AND J. WU

n Vsn (t) := e−(2/γ)ϕ(Yt )

n Zsn (t) := e−(2/γ)ϕ(Yt )

 

2(Ytn

− Ysn ) −

 2 n n 2 n |Y − Ys | Dϕ(Yt ) (σ(Ytn )h˙ t + ˜b(Ytn )), γ t

1 tr(σσ ∗ )(Ytn ) − |Ytn − Ysn |2 tr(Dϕσσ ∗ )(Ytn ) γ

4 − (Ytn − Ysn )σ(Ytn )Dϕ(Ytn )σ(Ytn ) γ  2 n n 2 n n 2 + 2 |Yt − Ys | |Dϕ(Yt )σ(Yt )| . γ

By the conditions on σ, b, ϕ,

|Usn (t)| ≤ C(|Ytn − Ysn | + |Ytn − Ysn |2 ),

|Usn (t) − Usn (t′ )| ≤ C|Ytn − Ytn′ |(1 + |Ytn − Ysn |)

+ C|Ytn − Ytn′ |(|Ytn − Ysn | + |Ytn′ − Ysn | + |Ytn′ − Ysn |2 )

≤ C|Ytn − Ytn′ |(1 + |Ytn − Ysn |) + C|Ytn − Ytn′ |2 (1 + |Ytn′ − Ysn |2 ),

|Vsn (t)| ≤ C(|Ytn − Ysn | + |Ytn − Ysn |2 )(1 + |h˙ t |), |Zsn (t)| ≤ C(1 + |Ytn − Ysn | + |Ytn − Ysn |2 ).

Thus

E|Ytn − Ysn |4p Z t Z t  Z t n n n n |Us (r)||w˙ r | dr + Vs (r) dr ≤ Cp E Us (r) dwr + s s s 2p Z t n . |Zs (r)| dr + s

Using the BDG inequality we get Z t 2p n Us (r) dwr E s

≤ Cp E

Z

t

|Usn (r)|2 dr

s

p−1

≤ Cp (t − s)

E

Z

t s

≤ Cp (t − s)p + Cp E E

Z

t s

Usn (r) dwrn

2p

p

(|Yrn

Z

s

t

− Ysn |2p

+ |Yrn

− Ysn |4p ) dr

 |Yrn − Ysn |4p dr ,



25

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

≤ Cp E

Z

t s

(Usn (r) − Usn (¯ rn

∨ s)) dwrn

+

Z

t

s

Usn (¯ rn

∨ s) dwrn

2p 

.

R· rn ) dwrn is the piecewise linear interpolation of M·n := Note that 0 Usn (¯ R ·−∆ n Us (πn (r)) dwr with 0 πn (r) := max{tk ; tk ≤ r}.

Thus by Doob’s inequality and Lemma 3.7 we get 2p Z t n n E Us (¯ rn ∨ s) dwr s

≤ Cp E

Z

t

s

|Usn (¯ rn ∨ s)|2 dr E

Z

≤ Cp |t − s|p−1 E

Z

p−1

≤ Cp |t − s|

t

s

p−1

+ Cp |t − s|

t

s

E

(|Yr¯nn ∨s

t s

(|Yrn

≤ Cp |t − s|p + Cp |t − s|p−1 ≤ Cp |t − s|p + Cp

t s

− Ysn |2p

+ |Yr¯nn ∨s

− Ysn |4p ) dr

(|Yrn − Yr¯nn ∨s |2p + |Yrn − Ysn |2p ) dr

Z

Z

p

Z

− Yr¯nn ∨s |4p t s

+ |Yrn





− Ysn |4p ) dr



E|Yrn − Ysn |4p dr

E|Yrn − Ysn |4p dr,

2p   Z t n n n rn ∨ s)) dwr E (Us (r) − Us (¯ s Z t 2p  n n n n |wrˆn − wr¯n ∨s | |Yr − Yr¯n ∨s |(1 + |Yr − Ys |) ≤ Cp E dr ∆ s 2p  Z t 2 n n n n 2 |wrˆn − wr¯n ∨s | dr |Yr − Yr¯n ∨s | (1 + |Yr − Ys | ) + Cp E ∆ s Z t 2p  |wrˆn − wr¯n ∨s | n n |Yr − Yr¯n ∨s | ≤ Cp E dr ∆ s  2p  Z t  |wrˆ − wr¯n ∨s |2 dr + Cp E |Yrn − Ysn |2 + |Yrn − Yr¯nn ∨s |2 n ∆2 s 2p  Z t 2 n n n n 2 |wrˆn − wr¯n ∨s | |Yr − Yr¯n ∨s | (1 + |Yr − Ys | ) dr + Cp E ∆ s

26

J. REN AND J. WU 2p

≤ Cp |t − s|

+ Cp (t − s)

≤ Cp |t − s|p + Cp

Z

t s

t

Z

2p−1

s

E|Yrn − Ysn |4p dr

E|Yrn − Ysn |4p dr.

Moreover, 2p Z t 2p   Z t n n E Vs (r) dr + Zs (r) dr s

s

≤E

Z

t

s

(|Yrn p

≤ Cp |t − s| + Cp

Z

s

t



− Ysn |2 )(1 + |h˙ r |) dr

− Ysn | + |Yrn

1+

E|Yrn

Z

t s

|h˙ r | dr 2

− Ysn |4p dr

Summing up we have E|Ytn − Ysn |4p ≤ Cp |t − s|p + Cp

Z

p

2p 

+ (t − s)2p

p  Z t 2 ˙ |hr | dr . 1+

t

s

s

p  Z t |h˙ r |2 dr , E|Yrn − Ysn |4p dr 1 + s

which together with Gronwall’s lemma yields

E|Ytn − Ysn |4p ≤ Cp |t − s|p . It follows from this estimate and Lemma 3.8 that E|φnt − φns |4p ≤ Cp |t − s|p . Now (3.5) holds due to Kolmogorov’s continuity criterion.  Proposition 3.4. E sup |Ytn |2p < Cp (1 + |x|2p ), t∈[0,1]

sup E[(|φn |1 )2p ] < Cp . n

Proof. Using Proposition 3.3, choose a θ ∈ [0, 14 ), and we get E sup |Ytn |2p ≤ 22p−1 E sup |Ytn − x|2p + 22p−1 |x|2p t∈[0,1]

t∈[0,1]

≤ 22p−1 E



|Ytn − x| θ |t| |t|θ t∈[0,1] sup

2p

2p−1 |x|2p ≤ 22p−1 EkY n k2p [0,1],θ + 2

≤ Cp (1 + |x|2p ).

+ 22p−1 |x|2p

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

27

Similar to [4], Theorem 3.6, by (D) we get for all 0 ≤ s < t ≤ 1, |φn |st ≤ C(|t − s|R−4 kY n k4[s,t],θ + 1)kφn k[s,t] .

From this and Proposition 3.3, E[(|φn |1 )2p ] ≤ Cp E[(R−4 kY n k4[0,1],θ + 1)2p kφn k2p [0,1] ] ≤ CR,p < ∞.



Proposition 3.5. sup 1≤k≤2n

E(|Ytnk

 Z θ/2 − Ztk | ) ≤ C ∆ + sup 2

2≤k≤2n

tk

tk−2

|h˙ s |2 ds

1/2 

,

θ ∈ (0, 1).

Proof. Set n )+ϕ(Z

µn (t) := e−(2/γ)(ϕ(Yt

t ))

,

mn (t) := µn (t)|Ytn − Zt |2 ,

a(t) := E(mn (t)). Using the condition ϕ ∈ Cb2 , Lemma 3.4 and (3.8) it is trivial to prove the following: Lemma 3.9. 

sup



sup

E E

 |µn (t) − µn (t′ )|2 ≤ C∆,

t,t′ ∈[tk−2 ,tk ]

 |mn (t) − mn (t′ )| ≤ C∆1/2 .

t,t′ ∈[tk−2 ,tk ]

For all tk−1 ≤ t ≤ tk , 2 ≤ k ≤ 2n , dµn (t)|Ytn

2

− Zt | =

11 X i=1

dIi (t) + 2µn (t)hYtn − Zt , dφnt − dψt i

2 − µn (t)|Ytn − Zt |2 (hDϕ(Ytn ), dφnt i + hDϕ(Zt ), dψt i), γ

where I1 (s) := 2

Z

I2 (s) := 2

Z

I3 (s) := 2

Z

s tk−1 s tk−1 s tk−1

µn (t)hYtn − Zt , σ(Ytn ) − σ(Zt )ih˙ t dt, µn (t)hYtn − Zt , ˜b(Ytn ) − b(Zt )i dt, µn (t)hYtn − Zt , σ(Ytn ) − σ(Yt¯n )i dwt ,

28

J. REN AND J. WU

I4 (s) := 2

Z

+ I5 (s) := 2

s tk−1

Z

Z

µn (t)(tr(σσ ∗ )(Ytn ) − tr(σσ ∗ )(Yt¯nn )) dt

s tk−1

s

tk−1

(µn (t) − µn (t¯n )) tr(σσ ∗ )(Yt¯nn ) dt,

 µn (t) hYtn − Zt , σ(Yt¯nn )(dwt − dwtn )i +

I6 (s) := −2

Z

tk−1

2 I7 (s) := − γ

Z

2 γ

Z

I8 (s) := −

2 I9 (s) := − γ

s

Z

Z

s tk−1

 µn (t¯n ) tr(σσ ∗ )(Yt¯nn ) dt,

µn (t)hYtn − Zt , (σ(Ytn ) − σ(Yt¯nn ))w˙ tn i dt,

s tk−1 s tk−1

s tk−1

µn (t)|Ytn − Zt |2 hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i, µn (t)|Ytn − Zt |2 × (hDϕ(Ytn ), σ(Ytn )h˙ t i + hDϕ(Zt ), σ(Zt )h˙ t i) dt, µn (t)|Ytn − Zt |2 

× hDϕ(Ytn ), ˜b(Ytn )i + hDϕ(Zt ), b(Zt )i  1 2 n ∗ n + tr(D ϕ(Yt )σσ (Yt )) dt, 2

 X Z 4 s n n n n hDϕ(Yt ), σ(Yt )ei ihYt − Zt , σ(Yt )ei i dt, µn (t) I10 (s) := − γ tk−1 i Z s 2 I11 (s) := 2 µn (t)|Ytn − Zt |2 |Dϕ(Ytn )σ(Ytn )|2 dt. γ tk−1 By (C), 1 hYtn − Zt , dφnt − dψt i − |Ytn − Zt |2 (hDϕ(Ytn ), dφnt i + hDϕ(Zt ), dψt i) ≤ 0. γ Thus (3.9)

µn (tk )|Ytnk − Ztk |2 ≤ µn (tk−1 )|Ytnk−1 − Ztk−1 |2 +

11 X i=1

Ii (tk ).

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

29

By the hypotheses σ ∈ Cb2 , ϕ ∈ Cb2 , |I1 (tk ) + I8 (tk )| Z t k n n ˙ 2µn (s)hYs − Zs , σ(Ys ) − σ(Zs )ihs ds ≤ tk−1

Z t k 2 µn (s)|Ysn − Zs |2 (hDϕ(Ysn ), σ(Ysn )i + γ tk−1

≤C

Z

tk

tk−1

mn (s)|h˙ s | ds.

˙ + hDϕ(Zs ), σ(Zs )i)hs ds

R tk 2hYtn − Zt , σ(Ytn ) − σ(Yt¯nn )i dwt ) = 0. For I3 , E(I3 (tk )) = E( tk−1 Throughout the proof we need several lemmas which will be proved afterward. Now we deal with the terms I2 and I6 . Note that for I2 , Z tk µn (t)hYtn − Zt , ˜b(Ytn ) − b(Zt )i dt I2 = 2 tk−1

=2

Z

tk

tk−1

µn (t)hYtn − Zt − (Yt¯nn − Zt¯n ), ˜b(Ytn ) − b(Zt )i dt

Z

tk

+2

tk

+2

Z

tk

+2

Z

tk

+2

Z

tk−1

tk−1

tk−1

tk−1

=:

5 X

(µn (t) − µn (t¯n ))hYt¯nn − Zt¯n , ˜b(Ytn ) − b(Zt )i dt µn (t¯n )hYt¯nn − Zt¯n , ˜b(Ytn ) − ˜b(Yt¯nn ) − b(Zt ) + b(Zt¯n )i dt µn (t¯n )hYt¯nn − Zt¯n , b(Yt¯nn ) − b(Zt¯n )i dt µn (tk−2 )hYtnk−2 − Ztk−2 , ˜b(Ytnk−2 ) − b(Ytnk−2 )i dt

I2,i .

i=1

Taking expectations and applying Lemmas 3.4, 3.7 and 3.9, we get 4 X I2,i E i=1

30

J. REN AND J. WU

≤ C∆

3/2

Z + E

tk

tk−1

tk

Z + C E

n n ˜ ¯ (µn (t) − µn (tn ))|Yt¯n − Zt¯n ||b(Yt ) − b(Zt )| dt

n n n ¯ µn (tn )|Yt¯n − Zt¯n ||Yt − Zt − (Yt¯n − Zt¯n )| dt + Ca(tk−2 )∆

tk−1

≤ C∆3/2 + Ca(tk−2 )∆. Note that I6 = −2

tk

Z

tk−1 tk

−2

Z

tk

−2

Z

µn (t)hYtn − Zt − (Yt¯nn − Zt¯n ), σ(Ytn ) − σ(Yt¯nn )i dwtn

tk−1

(µn (t) − µn (t¯n ))hYt¯nn − Zt¯n , σ(Ytn ) − σ(Yt¯nn )i dwtn

tk−1

µn (t¯n )hYt¯nn − Zt¯n , σ(Ytn ) − σ(Yt¯nn )i dwtn =:

X

I6,i

i

and |E(I6,1 + I6,2 )| ≤ C∆3/2 . As for I6,3 + I2,5 , note that I6,3 + I2,5 = −2Ank , where   Z tk 1 n n n n n n ¯ µn (tn ) Yt¯n − Zt¯n , (σ(Yt ) − σ(Yt¯n ))w˙ t − (∇σ)σ(Yt¯n ) dt, Ak := 2 tk−1 (3.10) and by Lemma 3.10, !  Z tk 1/2  2n X 1/2 2 n ˙ |hs | ds . Ai ≤ C ∆ + sup E 2≤k≤2n tk−2 i=1

By Lemmas 3.7 and 3.9, |EI4 (tk )| Z ≤ E

tk

µn (t)(tr(σσ

tk−1

Z + E

tk

tk−1

∗



)(Ytn ) − tr(σσ ∗ )(Yt¯nn )) dt

(µn (t) − µn (t¯n )) tr(σσ

∗



)(Yt¯nn ) dt ≤ C∆3/2 ,

|E[I9 (tk ) + I11 (tk )]| Z t k 2 (mn (s) − mn (¯ sn ) + mn (¯ sn )) ≤ E γ tk−1

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

31



× |Dϕ(Ysn )|˜b(Ysn )|+|Dϕ(Zs )||b(Zs )|  1 2 ∗ n + tr(D ϕσσ (Ys )) ds 2

Z t k 2 n n 2 + 2 E (mn (s) − mn (¯ sn ) + mn (¯ sn ))|Dϕ(Ys )σ(Ys )| ds γ tk−1

≤ C∆3/2 + Ca(tk−2 )∆. For I5 , we have Z I5 (tk ) = 2

tk

µn (t¯n )hYtn − Zt , σ(Yt¯nn )i(dwt − dwtn )

tk−1

+

tk

Z

tk−1

+2

µn (t¯n ) tr(σσ ∗ )(Yt¯nn ) dt

tk

Z

tk−1

(µn (t) − µn (t¯n ))hYtn − Zt , σ(Yt¯nn )i(dwt − dwtn )

=: I5,1 + I5,2 . However, by Lemma 3.11, ! 2n X I5,1 (ti ) ≤ C∆θ/2 E i=0

∀θ ∈ (0, 1).

With respect to I5,2 , Z Z t 4 tk I5,2 = − µn (s)hDϕ(Ysn ), σ(Ysn )(dws − dwsn )i γ tk−1 t¯n 4 − γ

4 − γ

4 − γ

Z

tk

Z

tk

Z

tk

tk−1

tk−1

tk−1

× hYtn − Zt , σ(Yt¯nn )i(dwt − dwtn )

Z Z Z

t

t¯n

µn (s)(hDϕ(Ysn ), ˜b(Ysn )i ds + hDϕ(Zs ), b(Zs )i ds)

× hYtn − Zt , σ(Yt¯nn )i(dwt − dwtn )

t t¯n

µn (s)(hDϕ(Ysn ), σ(Ysn )ih˙ s ds + hDϕ(Zs ), σ(Zs )ih˙ s ds)

× hYtn − Zt , σ(Yt¯nn )i(dwt − dwtn ) t

t¯n

µn (s)(hDϕ(Ysn ), dφns i + hDϕ(Zs ), dψs i)

32

J. REN AND J. WU

2 + 2 γ

1 − γ

=:

6 X

Z

Z

tk tk−1

tk

tk−1

× hYtn − Zt , σ(Yt¯nn )i(dwt − dwtn ) Z t µn (s)|Dϕ(Ysn )σ(Ysn )|2 ds

Z

t¯n

t t¯n

× hYtn − Zt , σ(Yt¯nn )i(dwt − dwtn )

µn (s) tr(D 2 ϕ(Ysn )σ(Ysn )) ds

× hYtn − Zt , σ(Yt¯nn )i(dwt − dwtn )

I5,2,i .

i=1

Applying the BDG inequality, the conditions σ ∈ Cb2 , b ∈ Cb1 , ϕ ∈ Cb2 , Lemmas 3.4, 3.7 and Proposition 3.4, we get Z

tk

Z + E

tk

|EI5,2,2 | ≤ CE

tk−1

C∆2 |Ytn − Zt |2 kσ(Yt¯nn )k2 dt

tk−1

C∆|Ytn



1/2

− Zt |kσ(Yt¯nn )k dwtn

≤ C∆3/2 , Z t 2 1/2 Z tk 2 n n 2 ˙ |Yt − Zt | kσ(Yt¯n )k hs ds dt |EI5,2,3 | ≤ CE t¯n

tk−1

tk

+ CE

Z

≤ C∆1/2

Z

tk

tk−1

Z t n n ˙ |Yt − Zt |kσ(Yt¯n )k hs ds |dwtn |

tk−2

t¯n

|h˙ s | ds,

|E(I5,2,5 + I5,2,6 )| ≤ C∆3/2 , Z t Z t k 4 µn (s)hDϕ(Ysn ) − Dϕ(Yt¯nn ), dφns i |EI5,2,4 | ≤ E γ tk−1 t¯n  n n n × hYt − Zt , σ(Yt¯n )i(dwt − dwt ) Z t Z t k + E µn (s)hDϕ(Zs ) − Dϕ(Zt¯n ), dψs i tk−1

t¯n

33

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

× hYtn Z + E

tk

tk−1

Z

t t¯n

− Zt , σ(Yt¯nn )i(dwt



− dwtn )

µn (s)(hDϕ(Yt¯nn ), dφns i + hDϕ(Zt¯n ), dψs i) × hYtn

− Zt , σ(Yt¯nn )i(dwt



− dwtn )

≤ C∆3/2 + CEGk , where (3.11)

Gk :=

t

t

). + |ψ|tk−2 max |Ytn − Zt ||∆wk−1 | × (|φn |tk−2 k k

t∈[tk−1 ,tk ]

Again by Lemma 3.4 and Proposition 3.4, n

2 X k=1

EGk ≤ E



max n

max |Ytn − Zt ||∆wk−1 | × (|φn |1 + |ψ|1 )

1≤k≤2 t∈[tk−1 ,tk ]



i1/2p   h  ≤ E sup |Ytn − Zt |2p E sup |∆wk |2p 1≤k≤2n

0≤t≤1

× [E(|φn |1 + |ψ|1 )q ]1/q

≤ C∆(p−1)/2p ,

p, q > 1, 1/p + 1/q = 1

and I5,2,1

4 =− γ

tk

Z

4 − γ

4 − γ

4 − γ

tk−1

Z

tk

Z

tk

Z

tk

Z

tk−1

tk−1

tk−1

t

t¯n

Z Z Z

(µn (s) − µn (t¯n ))hDϕ(Ysn ), σ(Ysn )(dws − dwsn )i

× hYtn − Zt , σ(Yt¯nn )i(dwt − dwtn ) t

t¯n

µn (t¯n )hDϕ(Ysn ), σ(Ysn )(dws − dwsn )i

× hYtn − Zt − (Yt¯nn − Zt¯n ), σ(Yt¯nn )i(dwt − dwtn )

t t¯n

µn (t¯n )hDϕ(Ysn ), (σ(Ysn ) − σ(Yt¯nn ))(dws − dwsn )i

× hYt¯nn − Zt¯n , σ(Yt¯nn )i(dwt − dwtn )

t t¯n

µn (t¯n )hDϕ(Ysn ) − Dϕ(Yt¯nn ), σ(Yt¯nn )(dws − dwsn )i

× hYt¯nn − Zt¯n , σ(Yt¯nn )i(dwt − dwtn )

34

J. REN AND J. WU

4 − γ

=:

5 X

Z

tk

tk−1

µn (t¯n )hDϕ(Yt¯nn ), σ(Yt¯nn )(wt − wt¯n − (wtn − wt¯nn )i

× hYt¯nn − Zt¯n , σ(Yt¯nn )i(dwt − dwtn )

j I5,2,1 .

j=1

Using the BDG inequality, the fact that σ ∈ Cb2 , ϕ ∈ Cb2 , Lemmas 3.4, 3.7, 3.9 and Proposition 3.4, we get 1 |EI5,2,1 | ≤ C∆(E|∆wk−1 |2 )1/2 ≤ C∆3/2 , 1/2  Z tk 2 2 −1 n n 2 1/2 |Yt − Zt − (Yt¯n − Zt¯n )| |∆wk−1 | ∆ dt |EI5,2,1 | ≤ C∆ E tk−1

≤ C∆3/2 ,

j |EI5,2,1 | ≤ C∆3/2 ,

j = 3, 4

and 5 E(I5,2,1 |Ftk−2 ) (3.12) X 2∆ µn (tk−2 ) hDϕ(Ytnk−2 ), σ(Ytnk−2 )ei ihYtnk−2 − Ztk−2 , σ(Ytnk−2 )ei i. = γ i

This estimate will be used in Lemma 3.12. As for the term I7 , Z 2 tk µn (t)|Ytn − Zt |2 hDϕ(Ysn ), σ(Ysn )(dws − dwsn )i I7 = − γ tk−1 Z 2 tk =− µn (t)|Yt¯nn − Zt¯n |2 hDϕ(Ysn ), σ(Ysn )(dws − dwsn )i γ tk−1 Z Z t 4 tk µn (t) hYsn − Zs , σ(Ysn )(dws − dwsn )i − γ tk−1 t¯n

−

4 γ

4 − γ

Z

tk

Z

tk

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i

µn (t)

tk−1

tk−1

µn (t)

Z

t

Z

t

t¯n

hYsn − Zs , (σ(Ysn ) − σ(Zs ))h˙ s i ds

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i

t¯n

hYsn − Zs , ˜b(Ysn ) − b(Zs )i ds

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

4 − γ

2 − γ =:

6 X

Z

tk

Z

tk

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i

µn (t)

tk−1

tk−1

35

Z

t

t¯n

hYsn − Zs , dφns − dψs i

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i Z t µn (t) tr(σσ ∗ (Ysn )) ds × hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i t¯n

I7,i .

i=1

Notice that Z 2 tk (µn (t) − µn (t¯n ))|Yt¯nn − Zt¯n |2 hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i I7,1 = − γ tk−1 Z 2 tk − mn (t¯n )hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i γ tk−1 Z t Z tk 4 µn (s)hDϕ(Ysn ), σ(Ysn )(dws − dwsn )i |Yt¯nn − Zt¯n |2 = 2 γ tk−1 t¯n 4 + 2 γ

Z

tk

tk−1

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i Z t 2 n |Yt¯n − Zt¯n | µn (s)(hDϕ(Ysn ), σ(Ysn )i t¯n

+ hDϕ(Zs ), σ(Zs )i)h˙ s ds

+

4 γ2

4 + 2 γ

Z

tk

Z

tk

tk−1

tk−1

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i |Yt¯nn − Zt¯n |2

|Yt¯nn

2

− Zt¯n |

Z

t

t¯n

Z

µn (s)(hDϕ(Ysn ), ˜b(Ysn )i + hDϕ(Zs ), b(Zs )i) ds

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i t

t¯n

µn (s)(hDϕ(Ysn ), dφns i + hDϕ(Zs ), dψs i)

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i   Z Z t 1 1 4 tk 2 2 ∗ n n 2 n tr(D ϕσσ )(Ys ) − |Dϕσ(Ys )| ds |Y¯ − Zt¯n | µn (s) + 2 γ tk−1 tn 2 γ t¯n × hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i Z 2 tk − mn (t¯n )hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i γ tk−1

36

J. REN AND J. WU

=:

6 X

i I7,1 .

i=1

1 , For the first term I7,1 Z tk Z t 4 1 I7,1 = 2 |Yt¯nn − Zt¯n |2 (µn (s) − µn (t¯n ))hDϕ(Ysn ), σ(Ysn )(dws − dwsn )i γ tk−1 ¯ tn

4 + 2 γ

Z

tk

mn (t¯n )

tk−1

Z

t

t¯n

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i hDϕ(Ysn ), σ(Ysn )(dws − dwsn )i

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i.

However, note that E supt∈[0,T ] |Ytn − Zt |4 < ∞ by Lemma 3.4 and Proposition 3.3, and by Lemma 3.9, Z t Z t k (µn (s) − µn (t¯n ))hDϕ(Ysn ), σ(Ysn )(dws − dwsn )i E ¯ tk−1

tn

× hDϕ(Ytn ), σ(Ytn )(dwt

Z ≤ E

tk

tk−1

t

Z

t¯n

Z + E

tk

Z + E

tk

tk−1

tk−1

3



≤ C∆ + E  Z × E

2 n − dwt )i

(µn (s) − µn (t¯n ))hDϕ(Ysn ), σ(Ysn )(dws − dwsn )i

Z

Z

t t¯n

t t¯n

2 n n × hDϕ(Yt ), σ(Yt ) dwt i

(µn (s) − µn (t¯n ))hDϕ(Ysn ), σ(Ysn ) dws i

(µn (s) − µn (t¯n ))hDϕ(Ysn ), σ(Ysn ) dwsn i

max

t∈[tk−1 ,tk ]

tk

tk−1

2 n n n × hDϕ(Yt ), σ(Yt ) dwt i

Z

t

t¯n

2

× hDϕ(Ytn ), σ(Ytn ) dwtn i

(µn (s) − µn (t¯n ))hDϕ(Ysn ), σ(Ysn ) dws i

hDϕ(Ytn ), σ(Ytn ) dwtn i

4 1/2

,

4 1/2

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES



+ E

Z

max

t∈[tk−1 ,tk ]

 Z × E

tk

tk−1

t t¯n

(µn (s) − µn (t¯n ))hDϕ(Ysn ), σ(Ysn ) dwsn i

hDϕ(Ytn ), σ(Ytn ) dwtn i

4 1/2

4 1/2

≤ C∆3 .

Similar to the term I5,2,1 , 4 γ2

Z

tk

mn (t¯n )

tk−1

4 = 2 γ

+

Z

tk tk−1

4 γ2

4 + 2 γ

4 − 2 γ

−

4 γ2

Z Z Z Z

Z

t

hDϕ(Ysn ), σ(Ysn )(dws − dwsn )i

t¯n

× hDϕ(Ytn ), σ(Ytn ) dwtn i Z t ¯ mn (tn ) hDϕ(Ysn ) − Dϕ(Yt¯nn ), σ(Ysn )(dws − dwsn )i t¯n

tk tk−1

tk

× hDϕ(Ytn ), σ(Ytn ) dwtn i Z t mn (t¯n ) hDϕ(Yt¯nn ), (σ(Ysn ) − σ(Yt¯nn ))(dws − dwsn )i t¯n

× hDϕ(Ytn ), σ(Ytn ) dwtn i

mn (t¯n )

tk−1

tk

mn (t¯n )

tk−1

tk

mn (t¯n )

tk−1

Z

t

t¯n

Z

× hDϕ(Ytn ) − Dϕ(Yt¯nn ), σ(Ytn ) dwtn i t

t¯n

Z

hDϕ(Yt¯nn ), σ(Yt¯nn )(dws − dwsn )i

hDϕ(Yt¯nn ), σ(Yt¯nn )(dws − dwsn )i

× hDϕ(Yt¯nn ), σ(Ytn ) − σ(Yt¯nn )i dwtn t

t¯n

hDϕ(Yt¯nn ), σ(Yt¯nn )(dws − dwsn )i

× hDϕ(Yt¯nn ), σ(Yt¯nn )i dwtn

and Z E

tk

mn (t¯n )

tk−1

Z

t t¯n

hDϕ(Yt¯nn ), σ(Yt¯nn )(dws − dwsn )i × hDϕ(Yt¯nn ), σ(Yt¯nn ) dwtn i

 Z = E E

tk

tk−1

mn (t¯n )

Z

t t¯n



hDϕ(Yt¯nn ), σ(Yt¯nn )(dws − dwsn )i

37

38

J. REN AND J. WU



× hDϕ(Yt¯nn ), σ(Yt¯nn ) dwtn i Ftk−2

≤ Ca(tk−2 )∆.



Summing up we get 1 |EI7,1 | ≤ Ca(tk−2 )∆ + C∆3/2 .

Next, we have by Proposition 3.4 and Lemma 3.4, 2 |EI7,1 |≤

CE(|Ytnk−2

≤ C∆

Z

1/2

− Ztk−2 | |∆wk−1 |)

tk

tk−2

i |EI7,1 | ≤ C∆3/2 ,

2

Z

tk

tk−2

|h˙ s | ds

|h˙ s | ds, i = 3, 5

and 4 | ≤ CE(G1k ), |EI7,1

where t

G1k := |Ytnk−2 − Ztk−2 |2 (|φn |tkk−2 (3.13) ! 2n X θ/2 1 E ∀θ ∈ (0, 1). G k ≤ C∆

∨0

t

+ |ψ|tk−2 k

∨0

)|∆wk−1 |,

k

6 , For the term I7,1 6 I7,1

2 = − mn (tk−2 ) γ

Z

2 = − mn (tk−2 ) γ

Z

tk tk−1 tk tk−1

2 − mn (tk−2 ) γ

Z

tk

2 − mn (tk−2 ) γ

Z

tk

Moreover,  Z E mn (tk−2 )

tk

tk−1

hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i hDϕ(Ytn ) − Dϕ(Yt¯nn ), σ(Ytn )(dwt − dwtn )i

tk−1

tk−1

hDϕ(Yt¯nn ), (σ(Ytn ) − σ(Yt¯nn ))(dwt − dwtn )i hDϕ(Yt¯nn ), σ(Yt¯nn )(dwt − dwtn )i.



hDϕ(Ytn ) − Dϕ(Yt¯nn ), σ(Ytn ) dwtn i

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

 Z ≤ CE mn (tk−2 )

tk

 Z = CE mn (tk−2 )

tk

tk−1

tk−1

|Ytn

39



− Yt¯nn ||dwtn |

Z t σ(Ysn )(dws − dwsn ) ¯ tn

+ tk

Z

t

t¯n

σ(Ysn )h˙ s ds +

Z

t

t¯n

 ˜b(Y n ) ds + φn − φn¯ |dwn | s t t tn

Z t  n n n n n (σ(Y ) − σ(Y ) + σ(Y ))(dws − dw ) |dw | s s¯n s¯n s t tk−1 t¯n   Z t Z tk Z t n n n n n ˙ ˜ b(Ys ) ds + φt¯n − φt |dwt | + E mn (tk−2 ) σ(Ys )hs ds +

 Z ≤ E mn (tk−2 )

tk−1

t¯n

t¯n

Z t Z t  k n n n n max (σ(Ys ) − σ(Ys¯n ))(dws − dws ) |dwt | t∈[tk−1 ,tk ] t¯n tk−1   Z tk Z t n n n + E mn (tk−2 ) σ(Ys¯n )(dws − dws ) |dwt |

 ≤ E mn (tk−2 )

t¯n

tk−1



Z



Z

+ E mn (tk−2 ) + E mn (tk−2 )

tk

tk−1 tk

Z t  Z t ˜b(Y n ) ds |dwn | σ(Y n )h˙ s ds + s s t ¯ ¯ tn

max

tk−1 t∈[tk−1 ,tk ]

tn

|φnt¯n



− φnt ||dwtn |

≤ [E(mn (tk−2 )|∆wk−1 |)2 ]1/2 2 1/2   Z t n n n (σ(Ys ) − σ(Ys¯n ))(dws − dws ) × E max t∈[tk−1 ,tk ] tk−2 tk

Z t  Z t  k−1 n n −1 σ(Y )(dw − dw ) |∆w |∆ dt + s k−1 s¯n s tk−1 tk−1 tk−2   Z tk Z t Z t n n ˜b(Y ) ds |dwn | σ(Ys )h˙ s ds + + E mn (tk−2 ) s t

 Z + E mn (tk−2 )

tk−1



+ E mn (tk−2 ) ≤ C∆

3/2

Z

t¯n

tk

max

tk−1 t∈[tk−1 ,tk ]

t¯n

|φnt¯n



− φnt ||dwtn |

 Z + Ca(tk−2 )∆ + CE |∆wk−1 |

tk

tk−2



|h˙ s | ds + CEG1k ,

40

J. REN AND J. WU

where G1k is defined in (3.13) and  Z tk Z 2n X |h˙ s | ds|∆wk−1 | ≤ C∆θ/2 CE tk−2

k=2

1

0

|h˙ s | ds

∀θ ∈ (0, 1).

Similarly,

(3.14)

 Z E mn (tk−2 )

tk tk−1

hDϕ(Yt¯nn ), σ(Ytn ) − σ(Yt¯nn )i dwtn

≤ Ca(tk−2 )∆ + C∆3/2 + CEG1k ,   Z tk hDϕ(Yt¯nn ), σ(Yt¯nn )i(dwt − dwtn ) E mn (tk−2 )



tk−1

=: EG2k , while n 2 X 1/2  2 2 EGk ≤ C E max n mn (tk ) 1≤k≤2 k=2

Z × E max n 1≤k≤2 

(3.15)

≤ C∆

θ

tk

0

hDϕ(Yt¯nn ), σ(Yt¯nn )i(dwt

∀θ ∈ (0, 1).

2 1/2 n − dwt )

Here the last inequality follows since Z t 2 k n n n E max n hDϕ(Yt¯n ), σ(Yt¯n )i(dwt − dwt ) 1≤k≤2 0 k−1 X = E max n hDϕ(Ytni−1 ), σ(Ytni−1 )i(wti+1 − wti ) 1≤k≤2 i=1

−

k−1 X i=1

hDϕ(Ytni−1 ), σ(Ytni−1 )i(wti

2 − wti−1 )

k−1 X = E max n (hDϕ(Ytni−1 ), σ(Ytni−1 )i − hDϕ(Ytni ), σ(Ytni )i)(wti+1 − wti ) 1≤k≤2 i=1 2 − hDϕ(x), σ(x)iwt1 + hDϕ(Ytnk−1 ), σ(Ytnk−1 )i(wtk − wtk−1 ) ≤C

3 X i=1

6,i J7,1 ,

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

41

where by Lemma 3.7 and the conditions ϕ, σ ∈ Cb2 , Z t 2   k 6,1 n n n n J7,1 := E max n (hDϕ(Yt¯n ), σ(Yt¯n )i − hDϕ(Ytˆn ), σ(Ytˆn )i) dwt ≤ C∆, 1≤k≤2

0

6,2 J7,1 := E(wt21 ) = ∆,   6,3 J7,1 := E max n |hDϕ(Ytnk−1 ), σ(Ytnk−1 )i|2 |∆wk |2 ≤ C∆θ 1≤k≤2

θ ∈ (0, 1).

We then consider I7,2 . Z t Z 4 tk µn (t) hYsn − Zs − (Yt¯nn − Zt¯n ), σ(Ysn )(dws − dwsn )i I7,2 = − γ tk−1 ¯ tn 4 − γ

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i

tk

Z

µn (t)

tk−1

Z

t

t¯n

hYt¯nn − Zt¯n , σ(Ysn )(dws − dwsn )i

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i

=: I7,2,1 + I7,2,2 and I7,2,1

4 =− γ

+

Z

4 γ

(1)

tk

µn (t) tk−1

Z

Z

tk

µn (t)

tk−1

t t¯n

× hDϕ(Ytn ), σ(Ytn ) dwt i

Z

(2)

hYsn − Zs − (Yt¯nn − Zt¯n ), σ(Ysn )(dws − dwsn )i t

t¯n

hYsn − Zs − (Yt¯nn − Zt¯n ), σ(Ysn )(dws − dwsn )i

× hDϕ(Ytn ), σ(Ytn ) dwtn i

=: I7,2,1 + I7,2,1 . (1)

However, note that E(I7,2,1 ) = 0, and by using the Schwartz and BDG inequalities, Lemmas 3.4 and 3.7, (2)

|EI7,2,1 | ≤ C(E|∆wk−1 |2 )1/2 Z t 2 1/2  n n n n × E max hYs − Zs − (Yt¯n − Zt¯n ), σ(Ys )(dws − dws )i t∈[tk−1 ,tk ]

≤ C∆

3/2

.

t¯n

42

J. REN AND J. WU

Now according to Lemma 3.12, 5 E(I7,2,2 (tk ) + I5,2,1 (tk ) + I10 (tk )) ≤ C∆3/2 .

On the other hand, by the assumptions on f and σ, b, as well as Lemma 3.4 and Proposition 3.4, Z tk Z t |EI7,3 | ≤ CE µn (t) |Ysn − Zs |2 |h˙ s | ds t¯n

tk−1

(3.16) ≤ CE



sup

t∈[tk−2 ,tk ]

≤ C∆1/2

Z

tk tk−2

× |Dϕ(Ytn )|kσ(Ytn )k|dwtn |  Z tk 2 n |Yt − Zt | |∆wk−1 | |h˙ s | ds tk−2

|h˙ s | ds

and Z t Z t k 4 µ (t) |Ysn − Zs ||˜b(Ysn ) − b(Zs )| ds E |EI7,4 | ≤ n γ t¯n tk−1 × hDϕ(Ytn ), σ(Ytn )(dwt

tk

Z ≤ C E

µn (t)

tk−1

≤ CE

Z

tk tk−1

+ C∆E

Z

Z

t t¯n

t t¯n

(|Ysn

− Zs |

2



mn (s) ds|dwtn |

tk

tk−2

I7,5

Z



− dwtn )i

+ 1) ds dwtn



|dwtn |

≤ C∆3/2 , Z t Z 4 tk =− µn (t) hYsn − Zs − Yt¯nn + Zt¯n , dφns − dψs i γ tk−1 t¯n 4 − γ

Z

tk

tk−1

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i

µn (t)hYt¯nn − Zt¯n , φnt − φnt¯n − (ψt − ψt¯n )i

=: I7,5,1 + I7,5,2 .

× hDϕ(Ytn ), σ(Ytn )(dwt − dwtn )i

43

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

By using (3.8), |EI7,5,1 | ≤ CE

Z

tk

sup |Ysn − Zs − Ytnk−2 + Ztk−2 |

µn (t)

tk−1

s∈[tk−2 ,t]

t

t

× (|φn |tk−2 + |ψ|tk−2 )|dwtn |

tk

 Z tk−2 n n tk−2 ≤ CE (kY k[tk−2 ,tk ] + kZk[tk−2 ,tk ] )(|φ |tk + |ψ|tk )

tk−1

≤ C∆

3/2



|dwtn |

,

|I7,5,2 | ≤ C|Ytnk−2

− Ztk−2 | Z t  × max t∈[tk−1 ,tk ]

(3.17)

tk−1

× (|φn |ttkk−2





hDϕ(Ytn ), σ(Ytn ) dwt i + |∆wk−1 |

+ |ψ|ttkk−2 ) =: G3k ,

while according to Lemma 3.4 and Proposition 3.4, n

2 X

EG3k

k=1

  ≤ C max n E |Ytnk−2 − Ztk−2 |2 1≤k≤2

Z max × t∈[tk−1 ,tk ] 

× (E(|φn |01 + |ψ|01 )2 )1/2

t

tk−1

2

hDϕ(Ytn ), σ(Ytn ) dwt i

2

+ |∆wk−1 |

1/2

≤ C∆1/2 .

Also by the boundedness of σ we have Z t Z t k γ ∗ n n n n µn (t) tr(σσ )(Ys ) ds × hDϕ(Yt ), σ(Yt )(dwt − dwt )i |EI7,6 | ≤ E 2 t¯n tk−1 ≤ C∆3/2 .

Hence by applying all the above estimates to (3.9), Z tk a(tk ) − a(tk−1 ) ≤ Ca(tk−2 )∆ + C∆3/2 + bk + E mn (t)|h˙ t | dt tk−1

≤ CE(mn (tk−2 ) − mn (tk−1 ) + mn (tk−1 ))∆ + C∆3/2

44

J. REN AND J. WU

+ Cbk + E

Z

tk

tk−1

mn (t)|h˙ t | dt

≤ Ca(tk−1 )∆ + C∆

3/2

Z

+ Cbk + E

tk

tk−1

mn (t)|h˙ t | dt,

where bk := Gk + G1k + G2k + G3k + Ank + C∆1/2

Z

tk

tk−2

|h˙ t | dt,

and hence n

2 X

bk = E

k=1

" 2n  X

Ank + Gk + G1k + G2k + G3k + C∆1/2

k=1

 Z θ/2 ≤ C ∆ + sup 2≤k≤2n

tk tk−2

|h˙ s |2 ds

1/2 

,

Z

tk

tk−2

|h˙ t | dt

#

θ ∈ (0, 1),

and Ank , Gk , G1k , G2k , G3k , are defined in (3.10), (3.11), (3.13), (3.14) and (3.17), respectively. R C

Therefore according to Bihari’s inequality, by denoting hk := e we have

tk tk−1

|h˙ s | ds

a(tk ) ≤ [a(tk−1 )(1 + C∆) + C∆3/2 + bk ]hk

≤ a(tk−1 )hk (1 + C∆) + C∆3/2 hk + bk hk

≤ [(a(tk−2 )(1 + C∆) + C∆3/2 + bk−1 )hk−1 ]hk (1 + C∆) + C∆3/2 hk + bk hk

≤ ···

k−1 X (1 + C∆)i hk · · · hk−i (C∆3/2 + bk−i ) ≤ (1 + C∆) hk · · · h1 a(t0 ) + k

i=0

 Z R CT C 0T |h˙ t | dt θ/2 ≤e e C ∆ + sup 1≤k≤2n

tk

tk−2

|h˙ s |2 ds

1/2 

,

and we obtain the desired result.  Lemma 3.10. ! 1/2   Z tk 2n X 2 1/2 n ˙ |hs | ds . Ai ≤ C ∆ + sup E 2≤k≤2n tk−2 i=1

θ ∈ (0, 1),

,

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

45

Proof. Set Z t n ζt := [(σ(Ysn ) − σ(Yt¯nn ))(dws − dwsn ) + σ(Ysn )h˙ s ds + ˜b(Ysn ) ds]. t¯n

We have for any 2 ≤ k ≤ 2n ,

|E(Ank + Ank+1 )| Z t    k+1 1 n n n n n ¯ µn (tn ) Yt¯n − Zt¯n , (σ(Yt ) − σ(Yt¯n ))w˙ t − (∇σ)σ(Yt¯n ) dt = E 2 tk−1 Z t    k 1 n n n n n = E µn (tk−2 ) Yt¯n − Zt¯n , (σ(Yt ) − σ(Yt¯n ))w˙ t − (∇σ)σ(Yt¯n ) dt 2 tk−1  Z tk+1 (µn (tk−1 ) − µn (tk−2 )) Yt¯nn − Zt¯n , (σ(Ytn ) − σ(Yt¯nn ))w˙ tn +E tk

+E

Z

tk+1

tk

Z ≤ E

tk+1

tk−1

  1 n − (∇σ)σ(Yt¯n ) dt 2  µn (tk−2 ) Yt¯nn − Zt¯n , (σ(Ytn ) − σ(Yt¯nn ))w˙ tn   1 n − (∇σ)σ(Yt¯n ) dt 2

 µn (tk−2 ) Yt¯nn − Zt¯n , (σ(Ytn ) − σ(Yt¯nn ))w˙ tn

  1 − (∇σ)σ(Yt¯nn ) dt + C∆3/2 . 2

Thus by continuing this procedure we get ! 2n X Ani E i=1 Z 1  n 3/2 µn (t1 ) Yt¯nn − Zt¯n , (σ(Ytn ) − σ(Yt¯nn ))w˙ tn ≤ C2 ∆ + E 0

  1 n − (∇σ)σ(Yt¯n ) dt 2

i1/2 h  ≤ C∆1/2 + E µ2n (t1 ) sup |Ytn − Zt |2 0≤t≤1

 Z 1   2 1/2 1 n n n n . × E (σ(Yt ) − σ(Yt¯n ))w˙ t − (∇σ)σ(Yt¯n ) dt 2 0

46

J. REN AND J. WU

Note that σ ∈ Cb2 ,

|σ(Ytn ) − σ(Yt¯nn ) − (∇σ)(Yt¯nn )(Ytn − Yt¯nn )| ≤ C|Ytn − Yt¯nn |2 , Z t Z t n n n n Yt − Yt¯n = σ(Ys )(dws − dws ) + σ(Ysn )h˙ s ds t¯n

+ Then  E sup

Z n

2≤k≤2

≤ CE

0

tk 

0

+ CE



+ CE

sup sup

1≤k≤2n

+ CE



sup

1≤k≤2n

=:

4 X

t

t¯n

˜b(Y n ) ds + φn − φn¯ . s t tn

|Ytn − Yt¯nn |2 |w˙ tn | dt

1≤k≤2n



t¯n

 2  1 n − (∇σ)σ(Yt¯n ) dt 2 2 

(σ(Ytn ) − σ(Yt¯nn ))w˙ tn

1

Z

Z

Z Z Z

tk

∇σ(Yt¯nn )(ζt

0

0

0

+ φnt

tk 

(∇σ)σ(Yt¯nn )

Z

tk 

(∇σ)σ(Yt¯nn )

Z

t t¯n t t¯n

− φnt¯n )w˙ tn dt

2  

− (∇σ)σ(Yt¯nn )

dwsn w˙ tn

 2  1 n − (∇σ)σ(Yt¯n ) dt 2

dt

Tα .

α=1

Note that by (3.8), 2  Z 1 n n 2 n |Yt − Yt¯n | |w˙ t | dt T1 = CE 0

≤ CE

"

n

2 Z X i=1

ti

ti−1

≤ C22n maxn E 1≤i≤2

|Ytn



− Yt¯nn |2 |w˙ tn | dt

!2 #

 sup |Ytn − Ytni−2 ∨0 |4 |∆wi−1 |2 ≤ C∆.

t∈[ti−1 ,ti ]

By Lemma 3.7 and Proposition 3.4, 2   Z tk n n n n ∇σ(Yt¯n )(ζt + φt − φt¯n )w˙ t dt T2 = CE sup 1≤k≤2n

≤ CE

Z

0

1

0

|∇σ(Yt¯nn )ζt w˙ tn | dt

2 

2 

dws w˙ tn

ON APPROXIMATE CONTINUITY AND THE SUPPORT OF RSDES

+ CE



sup

1≤k≤2n

 Z ≤C E

0

+ CE

Z

tk

∇σ(Yt¯nn )(φnt

0

− φnt¯n )w˙ tn dt

47

2 

1/2 1/2  Z 1 1 |w˙ tn |4 dt |ζt |4 dt E 0

"

sup

1≤k≤2n

k Z X i=1

 Z ≤ C∆ ∆ + sup 2≤k≤2n

 Z ≤ C ∆ + sup 1≤k≤2n

ti ti−1

tk

tk−2

tk

tk−2

∇σ(Yt¯nn )(φnt − φnt¯n )w˙ tn dt

|h˙ s |2 ds

|h˙ s |2 ds



∆−1 + CE

!2 #

h 2 i |φn |1 sup |∆wi | 1≤i≤2n



i1/p h  + E sup |∆wk |2p [E(|φn |1 )2q ]1/q 1≤k≤2n

 Z 1−1/p ≤C ∆ + sup 1≤k≤2n

tk

tk−2

|h˙ s |2 ds

Note that T3 = T31 + T32 , where Z tk  Z 1 n T3 := CE sup (∇σ)σ(Yt¯n ) 1≤k≤2n

≤ CE

" 2n X i=1

+2

≤ CE

i=1

∀p, q > 1, 1/p + 1/q = 1.

dws w˙ tn

− (∇σ)σ(Yt¯nn )

X

(∇σ)σ(Ytni−1 ∨0 )(∇σ)σ(Ytnj−1 ∨0 )(|∆wi |2

((∇σ)σ(Ytni−1 ∨0 ))2 (|∆wi |2 − ∆)2

≤ C2n ∆2 ≤ C∆, Z tk Z 2 n T3 := CE sup (∇σ)σ(Yt¯n ) 1≤k≤2n 1

0

t

tˆn

#

dws w˙ tn dt

2

ˆ +∆−t ≤ CE (wtˆn − wt¯n ) dwt ∆ 0  Z 1 2 n 2 |(∇σ)σ(Yt¯n )| |wtˆn − wt¯n | dt ≤ CE Z

0



dt

2

((∇σ)σ(Ytni−1 ∨0 ))2 (|∆wi |2 − ∆)2

i