4 Proof of Theorem 1.2

in PROBABILITY

A MAXIMAL INEQUALITY FOR STOCHASTIC CONVOLUTIONS IN 2-SMOOTH BANACH SPACES

JAN VAN NEERVEN¹

Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands

email: [email protected] JIAHUI ZHU²

Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands

email: [email protected]

SubmittedMay 24, 2011, accepted in final formOctober 24,2011 AMS 2000 Subject classification: Primary 60H05; Secondary 60H15

Keywords: Stochastic convolutions, maximal inequality, 2-smooth Banach spaces, Itô formula.

Abstract

Let(e^tA)t>0be aC₀-contraction semigroup on a 2-smooth Banach spaceE, let(W_t)t>0be a cylin- drical Brownian motion in a Hilbert spaceH, and let(g_t)t>0be a progressively measurable process with values in the spaceγ(H,E)of allγ-radonifying operators fromHtoE. We prove that for all 0<p<∞there exists a constantC, depending only onpandE, such that for allT>0 we have

E sup

06t6T

Z t

0

e^(t−s)Ag_sdW_s

p

6CE Z T

0

kg_tk²_γ(H,E)dt^p₂ .

For p>2 the proof is based on the observation thatψ(x) =kxk^p is Fréchet differentiable and its derivative satisfies the Lipschitz estimatekψ⁰(x)−ψ⁰(y)k 6 C(kxk+kyk)^p−2kx−yk; the extension to 0<p<2 proceeds via Lenglart’s inequality.

1 Introduction

Let (e^tA)t>0 be a C₀-contraction semigroup on a 2-smooth Banach space E and let (W_t)t>0 be a cylindrical Brownian motion in a Hilbert space H. Let (g_t)t>0 be a progressively measurable process with values in the spaceγ(H,E)of allγ-radonifying operators fromH toEsatisfying

Z T

0

kg_tk²_γ(H,E)dt<∞ P-almost surely

1RESEARCH SUPPORTED BY VICI SUBSIDY 639.033.604 OF THE NETHERLANDS ORGANISATION FOR SCIENTIFIC RESEARCH (NWO)

2RESEARCH SUPPORTED BY VICI SUBSIDY 639.033.604 OF THE NETHERLANDS ORGANISATION FOR SCIENTIFIC RESEARCH (NWO)

689

for allT>0. As is well known (see[6,15,16]), under these assumptions the stochastic convolu- tion process

X_t= Z t

0

e^(t−s)Ag_sdWs, t>0,

is well-defined in Eand provides the unique mild solution of the stochastic initial value problem dX_t=AX_tdt+g_tdW_t, X₀=0.

In order to obtain the existence of a continuous version of this process, one usually proves a maximal estimate of the form

E sup

06t6TkX_tk^p6C^pE Z T

0

kg_tk²_γ(H,E)dt^p₂

. (1.1)

The first such estimate was obtained by Kotelenez [11, 12] for C₀-contraction semigroups on Hilbert spaces E and exponent p =2. Tubaro[19]extended this result to exponents p>2 by a different method of proof which applies Itô’s formula to the C²-mapping x 7→ kxk^p. The case p∈(0, 2)was covered subsequently by Ichikawa[10]. A very simple proof, still forC₀-contraction semigroups on Hilbert spaces, which works for allp∈(0,∞), was obtained recently by Hausenblas and Seidler[9]. It is based on the Sz.-Nagy dilation theorem, which is used to reduce the problem to the corresponding problem forC₀-contraction groups. Then, by using the group property, the maximal estimate follows from Burkholder’s inequality. This proof shows, moreover, that the constant C in (1.1) may be taken equal to the constant appearing in Burkholder’s inequality. In particular, this constant depends only onp.

The maximal inequality (1.1) has been extended by Brze´zniak and Peszat [4] toC₀-contraction semigroups on Banach spaces Ewith the property that, for some p∈[2,∞), x 7→ kxk^p is twice continuously Fréchet differentiable and the first and second Fréchet derivatives are bounded by constant multiples of kxk^p−1 and kxk^p−2, respectively. Examples of spaces with this property, which we shall call(C²_p), are the spaces L^q(µ)forq ∈[p,∞). Any(C_p²)space is 2-smooth (the definition is recalled in Section 2), but the converse doesn’t hold:

Example1.1. LetF be a Banach space. The space`<sup>2</sup>(F)is 2-smooth whenever F is 2-smooth[8, Proposition 17]. On the other hand, the norm of`²(F)is twice continuously Fréchet differentiable away from the origin if and only ifF is a Hilbert space[14, Theorem 3.9]. Thus, forq∈(2,∞),

`<sup>2</sup>(`^q)and`²(L^q(0, 1))are examples of 2-smooth Banach spaces which fail property(C_p²) for all p∈[2,∞).

To the best of our knowledge, the general problem of proving the maximal estimate (1.1) forC₀- contraction semigroups on 2-smooth Banach space remains open. The present paper aims to fill this gap:

Theorem 1.2. Let(e^tA)t>0be a C₀-contraction semigroup on a2-smooth Banach space E, let(W_t)t>0

be a cylindrical Brownian motion in a Hilbert space H, and let(g_t)t>0be a progressively measurable process inγ(H,E). If

ZT

0

kg_tk²_γ(H,E)dt<∞ P-almost surely, then the stochastic convolution process X_t = Rt

0e^(t−s)Ag_sdW_s is well-defined and has a continuous version. Moreover, for all0<p<∞there exists a constant C, depending only on p and E, such that

E sup

06t6T

kX_tk^p6C^pE Z T

0

kg_tk²_γ(H,E)dt^p₂ .

Forp>2, the proof of Theorem 1.2 is based on a version of Itô’s formula (Theorem 3.1) which exploits the fact (proved in Lemma 2.1) that in 2-smooth Banach spaces the functionψ(x) =kxk^p is Fréchet differentiable and satisfies the Lipschitz estimate

kψ⁰(x)−ψ⁰(y)k6C(kxk+kyk)^p−2kx−yk.

The extension to exponents 0<p<2 is obtained by applying Lenglart’s inequality (see (4.1)).

We conclude this introduction with a brief discussion of some developments of the inequality (1.1) into different directions in the literature. Seidler[18]has proved the inequality (1.1) with optimal constantC =O(pp)as p→ ∞for positiveC₀-contraction semigroups on the (2-smooth) space E= L^q(µ),q>2. He also proved that the same result holds if the assumption ‘e^tA is a positive contraction semigroup’ is replaced by ‘−Ahas a bounded H^∞-calculus of angle strictly less than

1

2π’. The latter result was subsequently extended by Veraar and Weis[20]to arbitrary UMD spaces Ewith type 2. In the same paper, still under the assumption that−Ahas a boundedH^∞-calculus of angle strictly less than ¹

2π, the following stronger estimate is obtained for UMD spaces Ewith Pisier’s property(α):

E sup

06t6T

kX_tk^p6C^pEkgk^p_γ(L2(0,T;H),E) (1.2)

with a constantC depending only on p andE. If, in addition, E has type 2, then the mapping f⊗(h⊗x)7→(f⊗h)⊗xextends to a continuous embeddingL²(0,T;γ(H,E)),→γ(L²(0,T;H),E) and (1.2) implies (1.1).

Let us finally mention that, for p>2, a weaker version of (1.1) for arbitrary C₀-semigroups on Hilbert spaces has been obtained by Da Prato and Zabczyk[5]. Using the factorisation method they proved that

E sup

06t6T

kX_tk^p6C^pE Z T

0

kg_tk^p_γ(H,E)dt

with a constantC depending on p, E, and T. The proof extendsverbatimtoC₀-semigroups on martingale type 2 spaces. This relates to the above results for 2-smooth spaces through a theorem of Pisier[17, Theorem 3.1], which states that a Banach space has martingale typepif and only if it isp-smooth.

2 The Fréchet derivative of k · k

Let 1<q62. A Banach spaceEisq-smoothif the modulus of smoothness ρk·k(t) =supn

1

2(kx+t yk+kx−t yk)−1 : kxk=kyk=1o satisfiesρk·k(t)6C t^qfor allt>0.

It is known (see[17, Theorem 3.1]) thatEisq-smooth if and only if there exists a constantK>1 such that for allx,y∈E,

kx+yk^q+kx−yk^q62kxk^q+Kkyk^q. (2.1) Lemma 2.1. Let E be a Banach space and let1<q62be given. For p>q setψp(x):=kxk^p.

1. E is q-smooth if and only if the Fréchet derivative ofψqis globally(q−1)-Hölder continuous on E.

2. If E is q-smooth, then for p>q the Fréchet derivative ofψpis locally(q−1)-Hölder continuous on E.

Moreover, for all p>q and x,y∈E we have

kψ⁰p(x)−ψ⁰p(y)k6C(kxk+kyk)^p−qkx−yk^q−1, (2.2) where C depends only on p, q and E.

Proof. If the Fréchet derivative ofψq is(q−1)-Hölder continuous on E, then by the mean value theorem we can find 06θ,ρ61 such that for allx,y∈E,

kx+yk^q+kx−yk^q−2kxk^q= (kx+yk^q− kxk^q) + (kx−yk^q− kxk^q) 6kψ⁰_q(x+θy)−ψ⁰_q(x−ρy)k kyk

6Lk(x+θy)−(x−ρy)k^q−1kyk62^q−1Lkyk^q. Hence the Banach spaceEisq-smooth.

Suppose now that the norm ofEisq-smooth. Then for allx,y∈Ewithkxk,kyk=1 and allt>0 we have

kx+t yk+kx−t yk −2kxk6Kkt yk^q. (2.3) Thus

limt→0

kx+t yk+kx−t yk −2kxk kt yk =0,

which by[7, Lemma I.1.3]means thatk · kis Fréchet differentiable on the unit sphere. Hence, by homogeneity,k · kis Fréchet differentiable onE\{0}. Let us denote by f_xits Fréchet derivative at the pointx6=0.

We begin by showing the(q−1)-Hölder continuity of x 7→f_x on the unit sphere ofE, following the argument of [7, Lemma V.3.5]. We fix x 6= y ∈E such that kxk,kyk= 1 andh∈ E with khk=kx−ykandx−y+h6=0. Since the normk · kis a convex function,

f_y(x−y)6kxk − kyk. Similarly, we have

f_x(h)6kx+hk − kxk, f_y(y−x−h)6k2y−x−hk − kyk. By using above inequalities and the linearity of the function f_x, we have

f_x(h)−f_y(h)6kx+hk − kxk −f_y(h)

=kx+hk − kyk −f_y(x+h−y) +kyk − kxk+f_y(x−y) 6kx+hk − kyk −f_y(x+h−y)

=kx+hk − kyk+f_y(y−x−h) 6kx+hk+k2y−x−hk −2kyk

=

y+kx+h−yk · x+h−y kx+h−yk

+

y− kx+h−yk · x+h−y kx+h−yk

−2kyk 6Kkx+h−yk^q6K(kx−yk+khk)^q=2^qKkx−yk^q,

where we also used (2.3). Since the roles of x and y may be reversed in this inequality, this implies

kf_x−f_yk= sup

khk=kx−yk

|f_x(h)−f_y(h)|

kx−yk 62^qKkx−yk^q−1 This proves the(q−1)-Hölder continuity of the normk · kon the unit sphere.

We proceed with the proof of (2.2); the (q−1)-Hölder continuity of ψq as well as the local (p−1)-Hölder continuity ofψp follow from it. For all x,y∈E with x 6=0 and y6=0 we have ψ⁰p(x) =pkxk^p−1f_x.

It is easy to check that f_x= f ^x

kxk andkf_xk=1. Following once more the argument of[7, Lemma V.3.5], this gives

kψ⁰p(x)−ψ⁰p(y)k=p

kxk^p−1f_x− kyk^p−1f_y

6p

kxk^p−1(f ^x

kxk−f ^y

kyk) +p

(kxk^p−1− kyk^p−1)f ^y

kyk

6p2^qKkxk^p−1

x kxk− y

kyk

q−1+p

kxk^p−1− kyk^p−1

6p2^qKkxk^p−qkyk^1−q

xkyk −ykxk

q−1

+p

kxk^p−1− kyk^p−1

=p2^qKkxk^p−qkyk^1−q

kyk(x−y) +y(kyk − kxk)

q−1+p

kxk^p−1− kyk^p−1

6p2^qKkxk^p−qkyk^1−q(2kykkx−yk)^q−1+p

kxk^p−1− kyk^p−1

=p2^2q−1Kkxk^p−qkx−yk^q−1+p

kxk^p−1− kyk^p−1 .

(2.4) Ifq6p62, then by the inequality|t^r−s^r|6|t−s|^r, valid for 0<r61 ands,t∈[0,∞), we have

kxk^p−1− kyk^p−1 6

kxk − kyk

p−1

6kx−yk^p−16(kxk+kyk)^p−qkx−yk^q−1. Ifp>2, by applying the mean value theorem, for someθ∈[0, 1]we have

kxk^p−1− kyk^p−1

= (p−1)

kθx+ (1−θ)yk^p−2f_{θx+(1−θ)y}(x−y) 6(p−1)(kxk+kyk)^p−2kx−yk

6(p−1)(kxk+kyk)^p−2(kxk+kyk)^2−qkx−yk^q−1

= (p−1)(kxk+kyk)^p−qkx−yk^q−1. Also, sinceψ⁰_p(0) =0, for y6=0 we have

kψ⁰p(0)−ψ⁰p(y)k=pkyk^p−1=pkyk^p−1

y kyk

p−1

6pkyk^p−1

y kyk

q−1

=pkyk^p−qkyk^q−1.

The above lemma will be combined with the next one, which gives a first order Taylor formula with a remainder term involving the first derivative only.

Lemma 2.2. Let E and F be Banach spaces, let 0 < α 6 1, and let ψ : E → F be a Fréchet differentiable function whose Fréchet derivative ψ⁰ :E → L(E,F)is locally α-Hölder continuous.

Then for all x,y∈E we have

ψ(y) =ψ(x) +ψ⁰(x)(y−x) +R(x,y), where

R(x,y) = Z1

0

(ψ⁰(x+r(y−x))(y−x)−ψ⁰(x)(y−x))dr. (2.5) Proof. Pickw∈Esuch thatkwk61 and consider the function f :R→F by

f(θ):=ψ(x+θw).

For all x^∗∈F^∗,〈f⁰,x^∗〉is locallyα-Hölder continuous. To see this, note that for|θ1|,|θ2|6Rand kxk6Rwe havekx+θ1wk,kx+θ2wk62R, so by assumption there exists a constantC_2R such that

|〈f⁰(θ1)−f⁰(θ2),x^∗〉|=|〈ψ⁰(x+θ1w)w,x^∗〉 − 〈ψ⁰(x+θ2w)w,x^∗〉|

6kψ⁰(x+θ1w)−ψ⁰(x+θ2w)k kx^∗k6C_2R|θ1−θ2|^αkx^∗k. Applying Taylor’s formula and[1, Lemma 1, Theorem 3]to the function〈f,x^∗〉we obtain

〈f(t)−f(0),x^∗〉=t〈f⁰(0),x^∗〉+〈R_f(0,t),x^∗〉, where R_f(0,t) = R1

0 t(f⁰(r t)− f⁰(0))dr. Now let x,y ∈ E be given and set t =ky−xk and w=_k^y_y⁻₋^x_xk. With these choices we obtain

〈ψ(y),x^∗〉 − 〈ψ(x),x^∗〉 − 〈ψ⁰(x)(y−x),x^∗〉=〈ψ(x+t w),x^∗〉 − 〈ψ(x),x^∗〉 −t〈ψ(⁰x)w,x^∗〉

=〈f(t)−f(0)−t f⁰(0),x^∗〉

= Z1

0

t〈f⁰(r t)−f⁰(0),x^∗〉dr

= Z1

0

〈ψ⁰(x+r(y−x))(y−x)−ψ⁰(x)(y−x),x^∗〉dr.

Sincex^∗∈F^∗was arbitrary, this proves the lemma.

3 An Itô formula for k · k

From now on we shall always assume that Eis a 2-smooth Banach space. We fixT >0 and let (Ω,F,P)be a probability space with a filtration(Ft)_t∈[0,T]. LetH be a real Hilbert space, and denote by γ(H,E)the Banach space of allγ-radonifying operators from H to E. We denote by M([0,T];γ(H,E))the space of all progressively measurable processesξ:[0,T]×Ω→γ(H,E) such that

ZT

0

kξtk²_γ(H,E)dt<∞ P-almost surely.

The space of all suchξwhich satisfy E

Z T

0

kξtk²_γ(H,E)dt^p₂

<∞ is denoted byM^p([0,T];γ(H,E)), 0<p<∞.

On (Ω,F,P), let (W_t)t∈[0,T] be an (Ft)t∈[0,T]-cylindrical Brownian motion in H. For adapted simple processesξ∈M([0,T];γ(H,E))of the form

ξt=

n−1

X

i=0

1_(ti,ti+1](t)⊗A_i,

whereΠ ={0= t₀ < t₁ < · · ·< t_n = T} is a partition of the interval [0,T]and the random variablesA_iareFt_i-measurable and take values in the space of all finite rank operators fromHto E, we define the random variableI(ξ)∈L⁰(Ω,FT;E)by

I(ξ):=

n−1

X

i=0

A_i(W_ti+1−W_ti) where(h⊗x)W_t := (W_th)⊗x. It is well known that

EkI(ξ)k²6C²E Z T

0

kξtk²_γ(H,E)dt,

whereC depends onp andEonly. It follows that I has a unique extension to a bounded linear operator M²([0,T];γ(H,E)) to L²(Ω,FT;E). By a standard localisation argument, I extends continuous linear operator fromM([0,T];γ(H,E))toL⁰(Ω,FT;E). In what follows we write

Zt

0

ξsdWs:=I(1_(0,t]ξ), t∈[0,T]. This stochastic integral has the following properties:

1. For all ξ∈ M([0,T];γ(H,E)) the process t →Rt

0ξsdW_s is an E-valued continuous local martingale, which is a martingale ifξ∈M²([0,T];γ(H,E)).

2. For allξ∈M([0,T];γ(H,E))and stopping timesτwith values in[0,T], Zτ

0

ξtdW_t= Z T

0

1_[0,τ](t)ξtdW_t P-almost surely. (3.1)

3. For allξ∈M²([0,T];γ(H,E))and 06u<t6T, E

Z t

u

ξsdWs

2

|Fu

6CE

Z t

u

kξsk²_γ(H,E)ds|Fu

. (3.2)

4. (Burkholder’s inequality[2,6]) For all 0<p<∞there exists a constantC, depending only onpandE, such that for allξ∈M^p([0,T];γ(H,E))andt∈[0,T],

E sup

s∈[0,t]

Zs

0

ξudW_u

p

6CE Z t

0

kξsk²_γ(H,E)ds^p₂

. (3.3)

An excellent survey of the theory of stochastic integration in 2-smooth Banach spaces with com- plete proofs is given in Ondreját’s thesis[16], where also further references to the literature can be found.

In what follows we fix p>2 and setψ(x):=ψp(x) =kxk^p. Since we assume thatEis 2-smooth, this function is Fréchet differentiable. Following the notation of Lemma 2.2 we set

R_ψ(x,y):= Z1

0

(ψ⁰(x+r(y−x))(y−x)−ψ⁰(x)(y−x))dr.

We have the following version of Itô’s formula.

Theorem 3.1(Itô formula). Let E be a2-smooth Banach space and let26p<∞. Let(a_t)t∈[0,T]

be an E-valued progressively measurable process such that

E Z T

0

ka_tkdt^p

<∞

and let(g_t)_t∈[0,T]be a process in M^p([0,T];γ(H,E)). Fix x∈E and let(X_t)_t∈[0,T]be given by X_t=x+

Zt

0

a_sds+ Z t

0

g_sdW_s.

The process s 7→ψ⁰(X_s)g_s is progressively measurable and belongs to M¹([0,T];H), and for all t∈[0,T]we have

ψ(X_t) =ψ(x) + Z t

0

ψ⁰(X_s)(a_s)ds+ Z t

0

ψ⁰(X_s)(g_s)dWs+ lim

n→∞

m(n)−1

X

i=0

R_ψ(X_tⁿ

i∧t,X_tⁿ

i+1∧t) (3.4) with convergence in probability, for any sequence of partitionsΠn={0=t₀ⁿ<t₁ⁿ<· · ·<t_m(n)ⁿ =T} whose meshes kΠnk := max₀₆i6m(n)−1|t_iⁿ₊₁−t_iⁿ| tend to 0 as n → ∞. Moreover, there exists a constant C and, for each" >0, a constant C_", both independent of a and g, such that

Elim inf

n→∞

m(n)−1

X

i=0

|R_ψ(X_tⁿ

i∧t,X_tⁿ

i+1∧t)|6"CE sup

s∈[0,t]kX_sk^p+C_"E Z t

0

kg_sk²_γ(H,E)ds₂^p

. (3.5)

The proof shows that we may takeC_"=C⁰("^1−2p+1)for some constantC⁰ independent ofa, g, and".

Before we start the proof of the theorem we state some lemmas. The first is an immediate conse- quence of Burkholder’s inequality (3.3).

Lemma 3.2. Under the assumptions of Theorem 3.1 we have

E sup

06t6TkX_tk^p6CE Z T

0

ka_skds^p +CE

ZT

0

kg_sk²_γ(H,E)ds^p₂ .

Lemma 3.3. Under the assumptions of Theorem 3.1, the process t 7→ψ⁰(X_t)(g_t)is progressively measurable and belongs to M¹([0,T];H).

Proof. By the identitykψ⁰(x)k=pkxk^p−1and Hölder’s inequality,

E ZT

0

kψ⁰(X_t)(g_t)k²_Hdt¹₂ 6E

ZT

0

kψ⁰(X_t)k²kg_tk²_γ(H,E)dt¹₂

6E sup

t∈[0,T]kX_tk^p−1ZT 0

kg_tk²_γ(H,E)ds¹₂

6C E sup

t∈[0,T]kX_tk^pp−1_p E

Z T

0

kg_tk²_γ(H,E)ds^p₂¹_p ,

and the right-hand side is finite by the previous lemma. The progressively measurability is clear.

This lemma implies that the stochastic integral in (3.4) is well-defined.

Lemma 3.4. Let06u6 t 6 T be arbitrary and fixed. Under the assumptions of Theorem 3.1, the process s 7→ψ⁰(X_u)(g_s) is progressively measurable and belongs to M¹([0,T];H). Moreover, P-almost surely,

ψ⁰(X_u) Zt

u

g_sdW_s= Z t

u

ψ⁰(X_u)(g_s)dW_s. Proof. By similar estimates as in the previous lemma,

E Zt

u

kψ⁰(X_u)(g_s)k²Hds¹₂

6C(EkX_uk^p)^p−1p E

Zt

u

kg_sk²_γ(H,E)ds^p₂¹_p .

The progressively measurability is again clear. To prove the identity we first assume that g is a simple adapted process of the form

g_s=

n−1X

i=0

1_(ti,ti+1](s)A_i,

whereΠ = {u = t₀ < t₁ < · · · < t_n = t} is a partition of the interval[0,T] and the random variables areFt_i-measurable and take values in the space of all finite rank operators fromHtoE.

Then,

ψ⁰(X_u) Z t

u

g_sdW_s=ψ⁰(X_u)ⁿX⁻¹

i=0

A_i(W_ti+1−W_ti)

=

n−1

X

i=0

ψ⁰(X_u)(A_i(W_ti+1−W_ti)) = Zt

u

ψ⁰(X_u)(g_s)dW_s.

For general progressively measurable g ∈ L^p(Ω;L²([0,T];γ(H,E))), the identity follows by a routine approximation argument.

Proof of Theorem 3.1. The proof of the theorem proceeds in two steps. All constants occurring in the proof may depend on Eandp, even where this is not indicated explicitly, but not on T. The numerical value of the constants may change from line to line.

Step 1– Applying Lemma 2.2 to the functionψ(x) =kxk^p and the processX, we have, for every t∈[0,T],

ψ(X_t)−ψ(x) =

m(n)−1

X

i=0

ψ(X_tⁿ

i+1∧t)−ψ(X_tⁿ

i∧t)

=

m(n)−1

X

i=0

ψ⁰(X_tn i∧t)(X_tn

i+1∧t−X_tn i∧t) +

m(n)−1

X

i=0

R_ψ(X_tn i∧t,X_tn

i+1∧t). We shall prove the identity (3.4) by showing that

n→∞lim

m(n)−1

X

i=0

ψ⁰(X_tn i∧t)(X_tn

i+1∧t−X_tn i) =

Z t

0

ψ⁰(X_s)(a_s)ds+ Zt

0

ψ⁰(X_s)(g_s)dWs

with convergence in probability. In view of the definition ofX_t, it is enough to show that

nlim→∞

m(n)−1

X

i=0

ψ⁰(X_tn

i∧t)Z tⁿ_i+1∧t tⁿ_i∧t

a_sds

− Zt

0

ψ⁰(X_s)(a_s)ds

=0 P-almost surely and

nlim→∞

m(n)−1

X

i=0

ψ⁰(X_tn

i∧t)Z tⁿ_i+1∧t t_iⁿ∧t

g_sdW_s

− Z t

0

ψ⁰(X_s)(g_s)dW_s=0 in probability. (3.6)

By (2.2),P-almost surely we have lim sup

n→∞

m(n)−1

X

i=0

ψ⁰(X_tn

i∧t)Z tⁿ_i+1∧t tⁿ_i∧t

a_s

− Z t

0

ψ⁰(X_s)(a_s)ds

6lim sup

n→∞

m(n)−1

X

i=0

Z tⁿ_i+1∧t t_iⁿ∧t

(ψ⁰(X_tn

i∧t)−ψ⁰(X_s))(a_s)ds

6C sup

s∈[0,T]kX_sk^p−2×lim sup

n→∞

m(n)−1

X

i=0

Zt_i+1ⁿ ∧t tⁿ_i∧t

kX_tⁿ

i∧t−X_sk ka_skds 6C sup

s∈[0,T]kX_sk^p−2×lim sup

n→∞

sup

06i6m(n)−1 sup

s∈[tⁿ_i∧t,tⁿ_i+1∧t]kX_tn

i∧t−X_sk

×^mX⁽ⁿ⁾⁻¹

i=0

Z tⁿ_i+1∧t t_iⁿ∧t

ka_skds

=0,

where we used the continuity of the processX in the last line.

Next, by Lemma 3.4 and the inequalities (3.2) and (2.2),

m(n)−1

X

i=0

ψ⁰(X_tⁿ

i∧t)Z^t_i+1ⁿ ∧t tⁿ_i∧t

g_sdW_s

− Z ^t

0

ψ⁰(X_s)(g_s)dW_s

=

m(n)−1

X

i=0

Z tⁿ_i+1∧t tⁿ_i∧t

ψ⁰(X_tn

i∧t)(g_s)dW_s− Z t

0

ψ⁰(X_s)(g_s)dW_s

= Zt

0 m(n)−1

X

i=0

1₍t_iⁿ,tⁿ_i+1](s)(ψ⁰(X_tⁿ

i∧t)−ψ⁰(X_s))(g_s)dWs.

The localized stochastic integral being continuous from M([0,t];γ(H,E)))into L⁰(Ω,Ft;E), in order to prove that the right-hand side converges to 0 in probability it suffices to prove that

n→∞lim s7→

m(n)−1

X

i=0

1₍tⁿ_i,tⁿ_i+1](s)(ψ⁰(X_tn

i∧t)−ψ⁰(X_s))(g_s)

L²([0,t];H)=0 in probability.

For this, in turn, it suffices to observe thatP-almost surely

nlim→∞

m(n)−1

X

i=0

1_(tⁿ

i,tⁿ_i+1](s)(ψ⁰(X_tn

i∧t)−ψ⁰(X_s))

L^∞([0,t];E^∗)

= lim

n→∞ sup

06i6n−1

sup

s∈[tⁿ_i∧t,tⁿ_i+1∧t]kψ⁰(X_tn

i∧t)−ψ⁰(X_s)k=0 by the path continuity ofX.

Step 2– In this step we prove the estimate (3.5). By (2.2), for allx,y∈Eandr∈[0, 1]we have

|ψ⁰(x+r(y−x))−ψ⁰(x)|6(kxk^p−2kx−yk+kx−yk^p−1). Combining this with (2.5) we obtain

|R_ψ(X_tn i∧t,X_tn

i+1∧t)|6CkX_tn

i∧tk^p−2kX_tn

i+1∧t−X_tn

i∧tk²+CkX_tn

i+1∧t−X_tn

i∧tk^p. (3.7) We shall estimate the two terms on the right hand of (3.7) side separately.

For the first term, using the inequality|a+b|²62|a|²+2|b|²we obtain

m(n)−1

X

i=0

kX_tn

i∧tk^p−2kX_tn

i+1∧t−X_tn i∧tk² 62

m(n)−1

X

i=0

kX_tn i∧tk^p−2

Zt_i+1ⁿ ∧t tⁿ_i∧t

a_sds

2+2

m(n)−1

X

i=0

kX_tn i∧tk^p−2

Zt_i+1ⁿ ∧t tⁿ_i∧t

g_sdW_s

2

=:I₁ⁿ+I₂ⁿ. For the first term we have

I₁ⁿ62C sup

s∈[0,t]kX_sk^p−2×sup

i

Z tⁿ_i+1∧t tⁿ_i∧t

a_sds ×

m(n)−1

X

i=0

Z tⁿ_i+1∧t t_iⁿ∧t

a_sds

62C sup

s∈[0,t]kX_sk^p−2×sup

i

Z tⁿ_i+1∧t tⁿ_i∧t

a_sds ×

Z t

0

ka_skds.