Under Boundedness Assumption of Z

3.4. HIGHER ORDER DIFFERENTIABILITY 49 Remark 3.4.3. The key point of the proof of Theorem 3.4.1 is seen in the estimation of (3.17) and (3.19). The degree of the integrand in H ^p-norm is two and we have no tools to estimate integrals in which the degree of the integrands are greater than two; for example,E[(∫_T

0 ∇^kZ_s³ds)^p],E[(∫_T

0 ∇^kZ_s⁴ds)^p] and so on. Thus, if (3.19) contains derivatives of Z more than two, we fail to estimate E[(∫T

0 ∥∇A_s∥ds)^p]. (A4)-3) assures that there appears at most two derivatives of Z in (3.19); r ≤2.

50 CHAPTER 3. MALLIAVIN DIRREFENTIABILITY OF SOLUTIONS We introduce some examples on Theorem 3.4.4.

Example 3.4.5. Let (Wt)0≤t≤T be a one-dimensional Brownian motion and a function g: [0, T]× R ∋ (t, x) 7→ g(t, x) ∈ R belong to C^1,2([0, T]×R). Set Y_t =g(t, W_t) and Z_t= ^∂g_∂x(t, W_t). Then (Y, Z) is a solution to the BSDE;

Y_t =g(T, W_T)−

∫ _T

t

(∂g

∂t +1 2

∂²g

∂x² )

(s, W_s)ds−

∫ _T

t

Z_sdW_s, and it holds

D_uZ_t= ∂²g

∂x²(t, W_t)1_[0,t](u).

We now can construct some examples satisfying the assumptions (A4) or (A5) by above formulae.

(1) Let g(t, x) = tsinx. Then, (Y, Z), defined as above, is a unique solution to the BSDE;

Yt=TsinWT +

∫ T t

(

−sinWt+ Y_t 2

) ds−

∫ T t

ZsdWs, which satisfies (A5) and sup_u,t,ω|D_uZ_t(ω)|<∞ as well as (A4).

(2) Let g(t, x) = xarctan(2x)− ¹₄log(1 + 4x²). Then, (Y, Z), defined as above, is a unique solution to the BSDE;

Y_t =W_T arctan(2W_T)− 1

4log(1 + 4W_T²)−

∫ _T

t

cos²Z_sds−

∫ _T

t

Z_sdW_s. (A4)-3) is not satisfied but (A5) and sup_u,t,ω|D_uZ_t(ω)|<∞are satisfied since

D_uZ_t = 4

1 + 4W_t²1_[0,t](u).

First, we introduce the following lemma given by extracting the argument of the result of Zhen et al. [26, Proposition 2].

Lemma 3.4.6. Let (Y, Z) be an L² solution to the BSDE (3.1) and suppose y·f(t, y, z)≤y·a_t+b_t|y|²+c_t|y||z|, (t, y, z)∈[0, T]×R^d×R^n×d, where(a_t)_0≤≤T is aR^d-valued progressively measurable process satisfyingE[∫T

0 |a_s|²ds]<

∞, and (bt)0≤t≤T and (ct)0≤t≤T are real progressively measurable processes satis-fying sup_t,ω(|b_t(ω)|+|c_t(ω)|)<∞. Then, there exists a γ >0 such that

|Y_t|²e^γt+ 1 2E

[∫ T t

e^γs|Z_s|²ds Ft

]

≤E [

|ξ|²e^γT +

∫ T t

e^γs|a_s|²ds Ft

]

, 0≤t ≤T.

3.4. HIGHER ORDER DIFFERENTIABILITY 51 Proof. Applying the Itˆo formula to the function [0, T]×R^d∋(t, y)7→ |y|²e^γt ∈R with y=Y_t and (3.1) yields that for any γ >0,

|Y_t|²e^γt+

∫ T t

e^γs(

γ|Y_s|²+|Z_s|²) ds

=|ξ|²e^γT +

∫ T t

2e^γsY_s·f(s, Y_s, Z_s)ds−2

∫ T t

e^γsY_s·Z_sdW_s. By the identities 2ab≤a²+b² and 2ab≤2a²+¹₂b², we get

2Y_s·f(s, Y_s, Z_s)≤2Y_s·a_s+ 2b_s|Y_s|²+ 2c_s|Y_s||Z_s|

≤2|Ys||as|+ 2|bs||Ys|²+ 2|cs||Ys||Zs|

≤ |Y_s|²+|a_s|²+ 2∥b∥_∞|Y_s|² + 2∥c∥²_∞|Y_s|²+ 1 2|Z_s|², where ∥·∥_∞ represents the supremum with respect to (s, ω) ∈ [0, T]×Ω. Then, we obtain

|Y_t|²e^γt+

∫ T t

e^γs {

(γ−2∥b∥_∞−2∥c∥²_∞−1)|Y_s|²+1 2|Z_s|²

} ds

≤ |ξ|²e^γT +

∫ T t

e^γs|a_s|²ds−2

∫ T t

e^γsY_s·Z_sdW_s. By choosing γ ≥2∥b∥_∞+ 2∥c∥²_∞+ 1 and taking the conditional expectation, we get

|Y_t|²e^γt+ 1 2E

[∫ T t

e^γs|Z_s|²ds Ft

]

≤E [

|ξ|²e^γT +

∫ T t

e^γs|a_s|²ds Ft

] .

Proof of Theorem 3.4.4. For simplicity of notation, we give the proof in the case d = n = 1. In the proof, notation C represents a positive constant which may change from place to place.

Let (Y, Z)∈S^∞(R)×H ^∞(R) be a unique solution to the BSDE (3.1).

Step 1: We show that for any p≥2,

i) (Y, Z)∈L^a1,p(R)×L^a1,p(R) and (∇Y,∇Z)∈S^p(H)×H^p(H), ii) (D_·Y, D_·Z)∈Src²(R,P¯)×H²(R,P¯) and for almost all 0≤u, t ≤T,

|DuYt|+E [∫ T

t

|DuZs|²ds Ft

]

+∥∇Yt∥_H +|Zt| ≤C, iii) (D_·Y, D_·Z)∈S_rc^p(R,P¯)×H^p(R,P¯).

52 CHAPTER 3. MALLIAVIN DIRREFENTIABILITY OF SOLUTIONS Letp≥2. (A5) implies (A3)^′. Hence, by Corollary 3.1.3, i) holds.

We show ii). (∇Y,∇Z)∈S^p(H)×H^p(H) solves (3.2). (D_·Y, D_·Z) belongs to Src²(R,P¯)×H²(R,P¯) and solves

D_uY_t=D_uξ−1_(t,T](u)Z_u +

∫ T t

{D_uf(s, Y_s, Z_s) +∂_yf(s, Y_s, Z_s)D_uY_s+∂_zf(s, Y_s, Z_s)D_uZ_s}ds

−

∫ T t

D_uZ_sdW_s, 0≤t ≤T, a.e.u∈[0, T]. (3.20) Takeu∈[0, T] satisfying (3.20). Then, (D_uY, D_uZ)∈Src²(R)×H²(R) solves D_uY_t=D_uξ

+

∫ _T

t

{D_uf(s, Y_s, Z_s) +∂_yf(s, Y_s, Z_s)D_uY_s+∂_zf(s, Y_s, Z_s)D_uZ_s}ds

−

∫ _T

t

D_uZ_sdW_s, u≤t≤T.

Lemma 3.4.6 yields that there exists a γ >0 such that

|D_uY_t|²e^γt+ 1 2E

[∫ T t

e^γs|D_uZ_s|²ds Ft

]

≤E [

|D_uξ|²e^γT +

∫ _T

t

e^γs|D_uf(s, Y_s, Z_s)|ds Ft

]

, u≤t≤T.

By (A5)-1),6), we obtain

|D_uY_t|²+E [∫ T

t

|D_uZ_s|²ds Ft

]

≤C, u≤t ≤T.

If 0 ≤t < u, then D_uY_t=D_uZ_t= 0 and E

[∫ T t

|DuZs|²ds Ft

]

=E [∫ T

u

|DuZs|²ds Ft

]

=E [

E [∫ _T

u

|D_uZ_s|²ds Fu

]Ft

] . Thus, we get

|D_uY_t|²+E [∫ T

t

|D_uZ_s|²ds Ft

]

≤C, 0≤t≤T. (3.21)

3.4. HIGHER ORDER DIFFERENTIABILITY 53 By taking integrals with respect to u, we obtain

∥∇Y_t∥²_H ≤C, 0≤t≤T.

Since D_tY_t = Z_t for almost all t ∈ [0, T] and by (3.21), |Z_t| ≤ C for almost all t ∈[0, T]. Thus, we get for almost all u, t∈[0, T],

|D_uY_t|+E [∫ T

t

|D_uZ_s|²ds Ft

]

+∥∇Y_t∥_H +|Z_t| ≤C.

We will show iii). By ii), (D_·Y, D_·Z) ∈ Src²(R,P¯)×H ²(R,P¯) is a unique solution to the BSDE (3.20). Namely, putting ¯Y_t¹(u) = D_uY_t − 1_[0,t](u)Z_u, ( ¯Y¹(·), D_·Z)∈S²(R,P¯)×H ²(R,P¯) is a unique solution to the BSDE;

Y¯_t¹(u) = D_uξ−Z_u+

∫ _T

t

{D_uf(s, Y_s, Z_s) +∂_yf(s, Y_s, Z_s)1_[0,s](u)Z_u

+∂yf(s, Ys, Zs) ¯Y_s¹(u) +∂zf(s, Ys, Zs)DuZs}ds

−

∫ T t

D_uZ_sdW_s, 0≤t ≤T, a.e.u∈[0, T].

Let p≥2. By ii) and (A5)-3),6), we get E

[∫ _T

0

(∫ _T

0

|D_uf(s, Y_s, Z_s) +∂_yf(s, Y_s, Z_s)1_[0,s](u)Z_u|ds )p

du ]

<∞, E

[∫ T 0

sup

0≤t≤T

|1_[0,t](u)Z_u|^pdu ]

<∞.

Thus, we obtain (D_·Y, D_·Z)∈S_rc^p(R,P¯)×H^p(R,P¯).

Step 2: We show the following Claims 1-4 for k ≥2 by induction:

Claim 1 Letp≥2. Then, (∇^k−1Y,∇^k−1Z)∈L^a_1,p(H^⊗(k−1))×L^a_1,p(H^⊗(k−1)) and (∇^kY,∇^kZ)∈S^p(H^⊗k)×H ^p(H^⊗k) is a unique solution to the BSDE;

∇^kY_t=∇^kξ−

k−1

∑

i=0

∇ⁱ (∫ _·

·∧t

∇^k−1−iZ_sds )

+

∫ _T

t

{A^k_s +B_s^k∇^kY_s+ Γ^k_s∇^kZ_s} ds−

∫ _T

t

∇^kZ_sdW_s, 0≤t≤T, where B_t^k =∂_yf(t, Y_t, Z_t), Γ^k_t =∂_zf(t, Y_t, Z_t) and A^k is defined inductively as

A¹_t =∇f(t, Y_t, Z_t),

A^k_t =∇A^k_t⁻¹+^∇B_t^k−1∇^k−1Y_t+^∇Γ^k_t⁻¹∇^k−1Z_t, k ≥2.

54 CHAPTER 3. MALLIAVIN DIRREFENTIABILITY OF SOLUTIONS Moreover, it holds that

A^k_t =∇^kf(t, Y_t, Z_t)

+∑

1,k

(∇^α1Y_t⊗ · · · ⊗ ∇^αm−1−rY_t

⊗ ∇^β1Z_t⊗ · · · ⊗ ∇^βrZ_t⊗ ∇^γ∂_y^m−1−r∂_z^rf(t, Y_t, Z_t))^Σ

+∑

2,k

∂_y^m−r∂_z^rf(t, Y_t, Z_t)

×(∇^α1Y_t⊗ · · · ⊗ ∇^αm−rY_t⊗ ∇^β1Z_t⊗ · · · ⊗ ∇^βrZ_t)^Σ, (3.22) where the notations of summation represent

∑

1,k

:=

∑k m=2

m∑−1 r=0

∑

α∈N^mND^−1−r

β∈N^rND

γ∈N

|α|+|β|+γ=k

, ∑

2,k

:=

∑k m=2

∑m r=0

∑

α∈N^mND^−r

β∈N^rND

|α|+|β|=k

,

N^kND and the superscript Σ represent the same as in the proof of Theorem 3.4.1.

Claim 2 The following holds;

E [∫ T

0

(∫ _T

0

∥K˜_uA^k_s∥H^⊗(k−1)ds )2

du ]

<∞.

In addition, (D_·∇^k−1Y, D_·∇^k−1Z)∈S_rc²(H^⊗(k−1),P¯)×H ²(H^⊗(k−1),P¯) and D_t∇^k−1Y_t=∇^k−1Z_t for almost all t∈[0, T].

Claim 3 For almost all 0≤t ≤T, u∈[0, T]^k and v ∈[0, T]^k−1,

|D^k_uY_t|+E [∫ T

t

|D^k_uZ_s|²ds Ft

]

+|D^k_v⁻¹Z_t|+∇^kY_t

H^⊗k +∇^k−1Z_t

H^⊗(k−1) ≤C.

Claim 4 (D_·∇^k−1Y, D_·∇^k−1Z)∈Src^∞(H^⊗(k−1),P¯)×H^∞(H^⊗(k−1),P¯).

We show the case whenk = 2. Let p≥2. By Corollary 3.3.2, (∇²Y,∇²Z)∈ S^p(H^⊗2)×H ^p(H^⊗2) solves (3.11). Then, Claim 1 holds. As in the proof of Theorem 3.4.1, Claim 2 holds. We will show Claim 3 and 4.

We now prove Claim 3. For a.e.(u, v) ∈ [0, T]², (D²_u,vY, D²_u,vZ) ∈ S²(R)× H ²(R) solves

D²_u,vY_t=D²_u,vξ+

∫ T t

{K˜_u,v² A²_s+∂_yf(s, Y_s, Z_s)D²_u,vY_s+∂_zf(s, Y_s, Z_s)D_u,v² Z_s}ds

3.4. HIGHER ORDER DIFFERENTIABILITY 55

−

∫ T t

D_u,v² ZsdWs, u∨v ≤t≤T, where

K˜_u,v² A²_s =D_u,v² f(s, Y_s, Z_s) +D_uY_sD_v∂_yf(s, Y_s, Z_s) +D_uZ_sD_v∂_zf(s, Y_s, Z_s) +D_u∂_yf(s, Y_s, Z_s)D_vY_s+D_u∂_zf(s, Y_s, Z_s)D_vZ_s

+∂_y²f(s, Y_s, Z_s)D_uY_sD_vY_s+∂_y∂_zf(s, Y_s, Z_s)D_uY_sD_vZ_s +∂_z∂_yf(s, Y_s, Z_s)D_uZ_sD_vY_s+∂_z²f(s, Y_s, Z_s)D_uZ_sD_vZ_s.

Fix a.e.(u, v)∈[0, T]² as above. By Lemma 3.4.6, there exists aγ2 >0 such that

|D²_u,vY_t|²e^γ2t+1 2E

[∫ T t

e^γ2s|D²_u,vZ_s|²ds Ft

]

≤E [

|D_u,v² ξ|²e^γ2T +

∫ T t

e^γ2s|K˜_u,v² A²_s|²ds Ft

]

, u∨v ≤t≤T. (3.23) Since sup_u,t,ω|D_uZ_t(ω)| <∞, by (A5)-3),5),6) and Step 1-ii), we see for almost all s∈[0, T],

|K˜_u,v² A²_s|² ≤C.

Hence, we get

E [∫ _T

t

|K˜_u,v² A²_s|²ds Ft

]

≤C.

By Step 1-ii), (A5)-1) and (3.23), we see

|D_u,v² Y_t|+E [∫ T

t

|D_u,v² Z_s|²ds Ft

]

≤C, u∨v ≤t ≤T.

Thus, in the same manner as Step 1-ii), we obtain for almost all u, v, t∈[0, T],

|D_u,v² Yt|+E [∫ T

t

|D_u,v² Zs|²ds Ft

]

≤C.

And then, for almost all v, t ∈[0, T], we get

|D_vZ_t|=|D_vD_tY_t| ≤C.

Integration of |D_u,v² Y_t|² and |D_vZ_t|² with respect tou and v yield also that ∇²Y_t

H^⊗2 +∥∇Z_t∥_H ≤C.

56 CHAPTER 3. MALLIAVIN DIRREFENTIABILITY OF SOLUTIONS Claim 3 is proved.

We now prove Claim 4. (D_·∇Y, D_·∇Z)∈Src²(H,P¯)×H ²(H,P¯) is a unique solution to the BSDE;

D_u∇Y_t=D_u∇ξ−D_u (∫ _·

·∧t

Z_sds )

−1_(t,T](u)∇Z_u +

∫ T t

{K˜_uA²_s+∂_yf(s, Y_s, Z_s)D_u∇Y_s+∂_zf(s, Y_s, Z_s)D_u∇Z_s}ds

−

∫ T t

D_u∇Z_sdW_s, 0≤t ≤T, du⊗dP-a.e.

Namely, putting ¯Y_t²(u) =D_u∇Y_t−D_u(∫_·∧t

0 Z_sds)−1_[0,t](u)∇Z_u, ( ¯Y²(·), D_u∇Z)∈ S²(H,P¯)×H ²(H,P¯) is a unique solution to the BSDE;

Y¯_t²(u) = D_u∇ξ−D_u (∫ _·

0

Z_sds )

− ∇Z_u +

∫ _T

t

{

K˜_uA²_s+∂_yf(s, Y_S, Z_s) (

D_u

(∫ _·∧s

0

Z_rdr )

+1_[0,s](u)∇Z_u )

+∂_yf(s, Y_s, Z_s) ¯Y_s²(u) +∂_zf(s, Y_s, Z_s)D_u∇Z_s }

ds

−

∫ _T

t

D_u∇Z_sdW_s, 0≤t≤T, du⊗dP-a.e.

Let p ≥ 2. Since ∥D_u(∫_·∧s

0 Z_rdr)∥²H = ∫s

0 |D_uZ_r|²dr and by Claim 3 and (A5)-3),5),6), we see

E [∫ _T

0

(∫ _T

0

K˜_uA²_s+∂_yf(s, Y_S, Z_s) (

D_u

(∫ _·∧s

0

Z_rdr )

+1_[0,s](u)∇Z_u) H

ds )p

du ]

≤C (

1 +E [∫ T

0

{ sup

0≤t≤T|D_uY_t|^p+ (∫ T

0

|D_uZ_s|²ds )^p₂}

du ])

<∞, E

[∫ T 0

sup

0≤t≤T

D_u

(∫ _·∧t 0

Z_sds )

+1_[0,t](u)∇Z_u ^p

H

du ]

<∞.

Thus, we obtain (D_·∇Y, D_·∇Z)∈Src^p(H,P¯)×H ^p(H,P¯). The proofs of Claims 1-4 for k = 2 completes.

Next, assumek >2 and Claims 1-4 for 2,3, . . . , k−1 hold.

We will show Claim 1. Letp≥2. By the inductive assumption, (∇^k−1Y,∇^k−1Z)∈ S^2p(H^⊗(k−1))×H ^2p(H^⊗(k−1)) is a unique solution to the BSDE;

∇^k−1Y_t =∇^k−1ξ−

k−2

∑

i=0

∇ⁱ (∫ _·

·∧t

∇^k−2−iZ_sds )

3.4. HIGHER ORDER DIFFERENTIABILITY 57 +

∫ T t

{A^k_s⁻¹+B_s^k−1∇^k−1Ys+ Γ^k_s⁻¹∇^k−1Zs

}ds−

∫ T t

∇^k−1ZsdWs, 0≤t≤T, (3.24)

whereB^k−1_t =∂_yf(t, Y_t, Z_t), Γ^k−1_t =∂_zf(t, Y_t, Z_t), A^k_t⁻¹ =∇^kf(t, Y_t, Z_t)

+ ∑

1,k−1

(∇^α1Y_t⊗ · · · ⊗ ∇^αm−1−rY_t

⊗ ∇^β1Z_t⊗ · · · ⊗ ∇^βrZ_t⊗ ∇^γ∂_y^m−1−r∂_z^rf(t, Y_t, Z_t))^Σ

+ ∑

2,k−1

∂_y^m−r∂_z^rf(t, Y_t, Z_t)

×(∇^α1Y_t⊗ · · · ⊗ ∇^αm−rY_t⊗ ∇^β1Z_t⊗ · · · ⊗ ∇^βrZ_t)^Σ. We show (∇^k−1Y,∇^k−1Z)∈L^a1,p(H^⊗(k−1))×L^a1,p(H^⊗(k−1)) by applying Theorem 3.2.5. By (3.24) and (A5)-5), we see (A2)-3) is satisfied . The correspondence to (3.6) is as follows;

ξ =∇^k−1ξ, ζ_t=

k−2

∑

i=0

∇ⁱ

(∫ _·∧t 0

∇^k−2−iZ_sds )

, which satisfy (A2)-1),4). We see that for any F, G∈H^⊗(k−1),

∇B_t^k−1F =∇∂_yf(t, Y_t, Z_t)⊗F +∂_y²f(t, Y_t, Z_t)∇Y_t⊗F +∂_z∂_yf(t, Y_t, Z_t)∇Z_t⊗F, B˜^k_t⁻¹ =∂_yf(t, Y_t, Z_t),

∇Γ^k_t⁻¹G=∇∂_zf(t, Y_t, Z_t)⊗G+∂_y∂_zf(t, Y_t, Z_t)∇Y_t⊗G+∂_z²f(t, Y_t, Z_t)∇Z_t⊗G, Γ˜^k_t⁻¹ =∂_zf(t, Y_t, Z_t).

Hence, (A2)-6) is satisfied.

By messy but not difficult calculation, we obtain that∇A^k_t⁻¹+^∇B_t^k−1∇^k−1Y_t+

∇Γ^k_t⁻¹∇^k−1Z_t is equal to the right-hand side of (3.22). Then by (A5)-3),5),6), we get

E

[(∫ _T

0

∇A^k_s⁻¹

H^⊗kds )p]

≤C









1 + ∑

m,r∈Z+

1≤m+r≤2

∑

α∈(Z⁺∩[1,k−1])^m+r m+r≤|α|≤k

E

[(∫ T 0

∏m j=1

∥∇^αjY_s∥_H^⊗αj

∏r j^′=1

∥∇^αj′+mZ_s∥_H^⊗αj′+mds )p]

58 CHAPTER 3. MALLIAVIN DIRREFENTIABILITY OF SOLUTIONS

+ ∑

m,r∈Z⁺ 3≤m+r≤k

∑

α∈(Z+∩[1,k−2])^m+r m+r≤|α|≤k

E

[(∫ T 0

∏m j=1

∥∇^αjY_s∥_H^⊗αj

∏r j^′=1

∥∇^αj′+mZ_s∥_H^⊗αj′+mds )p]







 , (3.25)

where the products in (3.25) are defined to take 1 when m = 0 or r= 0. By the H¨older inequality, for each term of the first summation in (3.25), we see

E

[(∫ T 0

∏m j=1

∥∇^αjY_t∥_H^⊗αj

∏r j^′=1

∥∇^αj′+mZ_s∥_H^⊗αj′+mds )p]

≤ {

E [

sup

0≤t≤T

∏m j=1

∥∇^αjY_t∥^2p_H⊗αj

]}¹₂ 



E



(∫ T 0

∏r j^′=1

∥∇^αj′+mZ_s∥_H^⊗αj′+mds )2p







1 2

≤(1 +T^p2)

∏m j=1

∥∇^αjY∥^p_S²pj(H^⊗αj)

∏r j^′=1

∥∇^αj′+mZ∥^p_H^2qj′(H^⊗αj′+m)

<∞,

where 1/p = ∑_m

j=11/p_j = ∑_r

j^′=11/q_j′. By the inductive assumption Claim 3, all ∥∇^αjY_s∥_H^⊗αj and ∥∇^αj′+mZ_s∥_H^⊗αj′+m in the second summation in (3.25) are bounded; because for each 1 ≤ i ≤ m +r, α_i ≤ k−2. Therefore, the second summation in (3.25) is bounded. Thus, we get

E

[(∫ T 0

∇A^k_s⁻¹

H^⊗kds )p]

<∞. (A2)-5) is satisfied.

Now, we see that (A2)-2),3) are satisfied because the properties corresponding to them are shown in previous k on (A2)-7) and Claim 2.

From the above results, applying Theorem 3.2.5 yields that (∇^k−1Y,∇^k−1Z)∈ L^a1,p(H^⊗(k−1))×L^a1,p(H^⊗(k−1)) and that (∇^kY,∇^kZ)∈S^p(H^⊗k)×H ^p(H^⊗k) is a unique solution to the BSDE;

∇^kY_t=∇^kξ−

k−1

∑

i=0

∇ⁱ (∫ _·

·∧t

∇^k−1−iZ_sds )

+

∫ T t

{∇A^k−1_s +^∇B^k−1∇^k−1Y_s+^∇Γ^k−1∇^k−1Z_s+ ˜B_s^k−1∇^kY_s+ ˜Γ^k−1_s ∇^kZ_s }

ds

−

∫ T t

∇^kZ_sdW_s, 0≤t≤T.

3.4. HIGHER ORDER DIFFERENTIABILITY 59 DefineA^k_t =∇A^k_t⁻¹+^∇B_t^k−1∇^k−1Y_t+^∇Γ^k_t⁻¹∇^k−1Z_t. As mentioned above,A^k_t is written as (3.22). Claim 1 is proved.

We show Claim 2. By (3.22) and (A5)-3),5),6), we get E

[∫ T 0

(∫ _T

0

∥K˜_uA^k_s∥H^⊗(k−1)ds )2

du ]

≤C {

1 +∑

1.1

E [∫ T

0

(I₁^m,r,α(u))²du ]

+∑

1.2

E [∫ T

0

(I₁^m,r,α(u))²du ]

+∑

2.1

E [∫ T

0

(I₂^m,r,α(u))²du ]

+∑

2.2

E [∫ T

0

(I₂^m,r,α(u))²du ]}

, (3.26)

where I₁^m,r,α(u) =

∫ _T

0

D_u∇^α1−1Y_s

H^⊗(α1−1)

∏m j=2

∥∇^αjY_s∥_H^⊗αj

∏r j^′=1

∥∇^αj′+mZ_s∥_H^⊗αj′+mds,

∑

1.1

= ∑

m,r∈Z+

m≥1 1≤m+r≤2

∑

α∈(Z+∩[1,k−1])^m+r m+r≤|α|≤k

, ∑

1.2

= ∑

m,r∈Z+

m≥1 3≤m+r≤k

∑

α∈(Z+∩[1,k−2])^m+r m+r≤|α|≤k

,

I₂^m,r,α(u) =

∫ _T

0

D_u∇^α1+m−1Z_s

H^{⊗(α1+m−1)}

∏m j=1

∥∇^αjY_s∥_H^⊗αj

∏r j^′=2

∥∇^αj′+mZ_s∥_H^⊗αj′+mds,

∑

2.1

= ∑

m,r∈Z⁺ r≥1 1≤m+r≤2

∑

α∈(Z+∩[1,k−1])^m+r m+r≤|α|≤k

, ∑

2.2

= ∑

m,r∈Z⁺ r≥1 3≤m+r≤k

∑

α∈(Z+∩[1,k−2])^m+r m+r≤|α|≤k

,

defining a product ∏_b

j=ax_j = 1 if b < a.

If m, r ∈ Z+, m ≥1, 1 ≤ m+r ≤ 2 and α ∈(Z+∩[1, k−1])^m+r, m+r ≤

|α| ≤k, then by the Schwarz inequality and the inductive assumption (1), E

[∫ _T

0

(I₁^m,r,α(u))²du ]

≤E [∫ _T

0

du

∫ _T

0

D_u∇^α1−1Y_s²

H^⊗(α1−1)ds

×

∫ T 0

∏m j=2

∥∇^αjY_s∥²_H^⊗αj

∏r j^′=1

∥∇^αj′+mZ_s∥²_H^⊗αj′+mds ]

≤(T +T²)E [ _m

∏

j=1

sup

0≤t≤T∥∇^αjY_s∥²_H^⊗αj

∫ _T

0

∏r j^′=1

∥∇^αj′+mZ_s∥²_H^⊗αj′+mds ]

≤(T +T²) {

E [ _m

∏

j=1

sup

0≤t≤T

∥∇^αjY_s∥⁴_H^⊗αj ]}¹₂

60 CHAPTER 3. MALLIAVIN DIRREFENTIABILITY OF SOLUTIONS

×



E



(∫ T 0

∏r j^′=1

∥∇^αj′+mZs∥²_H^⊗αj′+mds )2







1 2

≤(T +T²)

∏m j=1

∥∇^αjY∥²_S²pj(H^⊗αj)

∏r j^′=1

∥∇^αj′+mZ∥²_H4(H^⊗αj′+m)

<∞,

where, to see in the fourth inequality above, r ≤1 is used, and 1/2 =∑_m

j=11/p_j. Hence, we get

∑

1.1

E [∫ T

0

(I₁^m,r,α(u))²du ]

<∞.

Ifm, r ∈Z+,r≥1, 1≤m+r≤2 andα ∈(Z+∩[1, k−1])^m+r,m+r≤ |α| ≤k, then by the inductive assumption Claim 1,

E [∫ T

0

(I₂^m,r,α(u))²du ]

≤ E [∫ T

0

du

∫ T 0

Du∇^α1+m−1Zs²

H^{⊗(α1+m−1)}ds

×

∫ _T

0

∏m j=1

∥∇^αjY_s∥²_H^⊗αj

∏r j^′=2

∥∇^αj′+mZ_s∥²_H^⊗αj′+mds ]

≤ E [∫ _T

0

∥∇^α1+mZ_s∥²_H^⊗α1+mds

×

∫ T 0

∏m j=1

∥∇^αjYs∥²_H^⊗αj

∏r j^′=2

∥∇^αj′+mZs∥²_H^⊗αj′+mds ]

≤ (1 +T) {

E

[(∫ T 0

∥∇^α1+mZ_s∥²_H^⊗α1+mds

)2]}¹

2

×



E



∏^m

j=1

sup

0≤t≤T∥∇^αjY_s∥⁴_H^⊗αj (∫ T

0

∏r j^′=2

∥∇^αj′+mZ_s∥²_H^⊗αj′+mds )2







1 2

≤ (1 +T)

∏m j=1

∥∇^αjY∥²_S⁴_(H^⊗αj)

∏r j^′=1

∥∇^αj′+mZ∥_H4(H^⊗αj′+m)

<∞. Thus, we get

∑

2.1

E [∫ T

0

(I₂^m,r,α(u))²du ]

<∞.

3.4. HIGHER ORDER DIFFERENTIABILITY 61 If α_i ≤k −2 for 1 ≤ i ≤m+r, then by the inductive assumption Claim 3,

∥∇^αjY_s∥_H^⊗αj and ∥∇^αj′+mZ_s∥_H^⊗αj′+m are bounded. Then, we see

∑

1.2

E [∫ _T

0

(I₁^m,r,α(u))²du ]

+∑

2.2

E [∫ _T

0

(I₂^m,r,α(u))²du ]

<∞. Hence, we obtain

E [∫ T

0

(∫ T 0

∥K˜_uA^k_s∥ds )²

du ]

<∞,

which implies (A2)^′-8). Thus, by Theorem 3.3.1, we get (D_·∇^k−1Y, D_·∇^k−1Z) ∈ Src²(H^⊗(k−1),P¯)×H ²(H^⊗(k−1),P¯) andD_t∇^k−1Y_t=∑_k−1

i=1 D_t∇ⁱ⁻¹(∫_·∧t

0 ∇^k−1−iZ_sds)+

∇^k−1Z_t. Since for 1 ≤i≤k−1, D_u∇ⁱ⁻¹

(∫ _·∧t

0

∇^k−1−iZ_sds) ²

H^⊗(k−1)

=







0, t ≤u,

∫ _t

u

D_u∇^k−2Z_s²

H^⊗(k−2)ds, u < t, we obtain D_t∇^k−1Y_t=∇^k−1Z_t for almost all t∈[0, T]. Claim 2 is proved.

We prove Claim 3. For a.e.u= (u₁, . . . , u_k)∈[0, T]^k, (D^k_uY, D_u^kZ)∈Src²(R)× H ²(R) solves

D^k_uY_t=D^k_uξ+

∫ T t

{K˜_u^kA^k_s+∂_yf(s, Y_s, Z_s)D_u^kY_s+∂_zf(s, Y_s, Z_s)D_u^kZ_s}ds

−

∫ T t

D_u^kZ_sdW_s, u¯≤t≤T, where ¯u= max{u₁, . . . , u_k}. By Lemma 3.4.6, there exists a γ_k >0 such that

|D^k_uY_t|²e^γkt+ 1 2E

[∫ _T

t

e^γks|D_u^kZ_s|²ds Ft

]

≤E [

|D^k_uξ|²e^γkT +

∫ T t

e^γks|K˜_u^kA^k_s|²ds Ft

]

, u¯≤t≤T. (3.27) By (3.22), we see

|K˜_u^kA^k_s|² ≤C {

1 +∑

1,k

∑

σ∈Sk

J1,m,r,α,β,σ

u (s) +∑

2,k

∑

σ∈Sk

J2,m,r,α,β,σ

u (s)

}

, (3.28) where S_k represents the symmetric group of degree k and

J1,m,r,α,β,σ

u (s) =

m∏−1−r j=1

|D_u^ασ,αj

j Ys|²

∏r j^′=1

|D^βj′

u^σ,β_j′ Zs|²|D^γ_u^σ,γ∂_y^m−1−r∂_z^rf(t, Yt, Zt)|²,

62 CHAPTER 3. MALLIAVIN DIRREFENTIABILITY OF SOLUTIONS J2,m,r,α,β,σ

u (s) =

m∏−r j=1

|D_u^ασ,α^j j

Y_s|²

∏r j^′=1

|D^βj′

u^σ,β_j′ Z_s|², α₀ = 1, β₀ = 1,

u^σ,α_j = (u_σ(^∑j

i=1α_i−1), u_σ(^∑j

i=1α_i−1+1), . . . , u_σ(^∑j

i=1α_i−1+αj−1)), u^σ,β_j′ = (u

σ(|α|+∑j′

i=1β_i−1), u

σ(|α|+∑j′

i=1β_i−1+1), . . . , u

σ(|α|+∑j′

i=1β_i−1+βj′−1)), u^σ,γ = (u_{σ(|α|+|β|+1)}, u_{σ(|α|+|β|+2)}, . . . , u_{σ(|α|+|β|+γ)}).

We divide the first summation of (3.28) into three ones;

∑

1,k

∑

σ∈Sk

J_u¹(s) = ∑

1,k r=0

∑

σ∈Sk

J1,m,r,α,β,σ

u (s) +∑

1,k r=1

∑

σ∈Sk

J1,m,r,α,β,σ

u (s) +∑

1,k r≥2

∑

σ∈Sk

J1,m,r,α,β,σ

u (s).

Ifr = 0, derivatives ofZ do not appear inJ1,m,r,α,β,σ

u (s) and eachα_j ≤k−1. By the inductive assumption and (A5)-6), we get

∑

1,k r=0

∑

σ∈Sk

E [∫ _T

t

J1,m,r,α,β,σ

u (s)ds

Ft

]

≤C.

Ifr= 1, β ∈Nandβ₁ ≤k−1. By the inductive assumption and (A5)-6), we get

∑

1,k r=1

∑

σ∈Sk

E [∫ T

t

J1,m,r,α,β,σ

u (s)ds

Ft

]

≤C∑

1,k r=0

∑

σ∈Sk

E [∫ T

t

|D^β1

u^σ,β₁ Z_s|²ds Ft

]

≤C.

If r≥2, each βj^′ ≤k−2. By the inductive assumption and (A5)-6), we get

∑

1,k r≥2

∑

σ∈S_k

E [∫ _T

t

J1,m,r,α,β,σ

u (s)ds

Ft

]

≤C.

In the same manner as above, we can obtain

∑

2,k

∑

σ∈Sk

E [∫ T

t

J2,m,r,α,β,σ

u (s)ds

Ft

]

≤C.

Thus by (3.28), we get E

[∫ T t

|K˜_u^kA^k_s|²ds Ft

]

≤C.

3.4. HIGHER ORDER DIFFERENTIABILITY 63 Hence by (3.27) and (A5)-1), we obtain

|D_u^kY_t|²+E [∫ T

t

|D_u^kZ_s|²ds Ft

]

≤C, u¯≤t ≤T.

In the same manner as Step 1-ii), we see

|D^k_uY_t|²+E [∫ T

t

|D^k_uZ_s|²ds Ft

]

≤C, 0≤t≤T.

And then, for almost all (v, t)∈[0, T]^k−1×[0, T], we get

|D_v^k−1Z_t|=|D^k_v⁻¹D_tY_t| ≤C.

Integration of |D_u^kY_t|² and |D^k_v⁻¹Z_t|² with respect to u and v yield also that ∇^kY_t²

H^⊗k+∇^k−1Z_t²

H^⊗(k−1) ≤C.

Claim 3 is now proved.

We show Claim 4. (D_·∇^k−1Y, D_·∇^k−1Z)∈S_rc²(H^⊗(k−1),P¯)×H ²(H^⊗(k−1),P¯) is a unique solution to the BSDE;

Du∇^k−1Yt=Du∇^k−1ξ−ζT(u) +ζt(u) +

∫ T t

{K˜_uA^k_s+∂_yf(s, Y_s, Z_s)D_u∇^k−1Y_s+∂_zf(s, Y_s, Z_s)D_u∇^k−1Z_s}ds

−

∫ T t

D_u∇^k−1Z_sdW_s, 0≤t ≤T, du⊗dP-a.e., whereζt(u) = ∑k−1

i=1 Du∇ⁱ⁻¹(∫_·∧t

0 ∇^k−1−iZsds)+1[0,t](u)∇^k−1Zu. Namely, putting Y¯_t^k(u) =D_u∇^k−1Y_t−ζ_t(u), ( ¯Y²(·), D_u∇Z)∈S²(H^⊗(k−1),P¯)×H ²(H^⊗(k−1),P¯) is a unique solution to the BSDE;

Y¯_t^k(u) = D_u∇^k−1ξ−ζ_T(u) +

∫ _T

t

{K˜_uA^k_s+∂_yf(s, Y_S, Z_s)ζ_s(u)

+∂_yf(s, Y_s, Z_s) ¯Y_s^k(u) +∂_zf(s, Y_s, Z_s)D_u∇^k−1Z_s}ds

−

∫ _T

t

D_u∇^k−1Z_sdW_s, 0≤t≤T, du⊗dP-a.e.

Let p≥2. By Claim 2, we get D_u∇^k−2Z_s =D_uD_s∇^k−2Y_s and by Claim 3, E

[∫ T 0

sup

0≤t≤T∥ζt(u)∥^p_H⊗(k−1)du ]

64 CHAPTER 3. MALLIAVIN DIRREFENTIABILITY OF SOLUTIONS

≤CE [∫ T

0

{(∫ T 0

∥D_u∇^k−2Z_s∥²_H⊗(k−2)ds )^p₂

+∥∇^k−1Z_u∥^p_H⊗(k−1)

} du

]

<∞.

Thus in the same manner as (3.26) and (A5)-3)-6), we get E

[∫ T 0

(∫ T 0

∥K˜_uA^k_s+∂_yf(s, Y_S, Z_s)ζ_s(u)∥H^⊗(k−1)ds )p

du ]

≤C (

1 +

k−2

∑

α=0

E [∫ _T

0

{ sup

0≤t≤T∥D_u∇^αY_t∥^2p_H⊗α + (∫ _T

0

∥D_u∇^αZ_s∥²_H⊗αds )p}

du ])

<∞. Hence, we obtain (D_·∇^k−1Y, D_·∇^k−1Z)∈Src^p(H^⊗(k−1),P¯)×H ^p(H^⊗(k−1),P¯).

Acknowledgements

I would like to express my sincere gratitude to my supervisor, Professor Setsuo Taniguchi, for his patience and persistent guidance on me throughout my Ph.D period of time. I am profoundly grateful to him for all of the help, encouragement, and suggestions. Furthermore, his rigorous and uncompromised attitudes toward mathematics told me a lot of importance in studying mathematical problems.

Without his valuable suggestions and contributions, it is not possible for me to achieve these results.

65

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