Regularity of solutions - FBSDEs with quadratic growth

Appendix 2.A Auxiliary result for the one-dimensional BSDE

3.4 FBSDEs with quadratic growth

3.4.2 Regularity of solutions

8k₂²T²e^2k¹^T k₄²+k₇²

+ 8k₅²T²

kδyk²_S∞

+ 24√

3k²₆L²₄T^1−ε

T^ε+ 2 +εL²₄kz·W˜ⁱk²

BMO( ˜Pⁱ)

+εL²₄k¯z·W˜ⁱk²

BMO( ˜Pⁱ)

kδz·W˜ⁱk²

BMO( ˜Pⁱ).

With the strictly positive constantsc1, c2depending only onk3andC2from Lemma A.1.3,

kδYk²_S∞+c1kδZ·Wk²_BMO(P₎

≤m⁰

8k₂²T²e^2k¹^T k₄²+k²₇

+ 8k₅²T²

kδyk²_S∞

+ 24

√

3k²₆L²₄T^1−εc2m⁰ T^ε+ 2 + 2εL²₄c2C₂²

kδz·Wk²_BMO(P₎. LettingT be small enough so that





 m⁰

1 +_c¹

8k₂²T²e^2k¹^T k₄²+k²₇

+ 8k₅²T²

≤ ¹₂ 24√

3k²₆L²₄T^1−εc₂m⁰ 1 +_c¹

T^ε+ 2 + 2εL²₄c₂C₂²

≤ ¹₂, (3.4.3) it follows thatΨdefines a contraction mapping. Then, there exists a fixed point (Y, Z)∈ B. Hence there exists a constantC_k,λwhich depends only onk_i, λ_isuch that whenT ≤Ck,λ, FBSDE (3.2.2) admits a unique solution(X, Y, Z)such that (X, Y, Z ·W) belongs toS²(R^m)× S^∞(R^m

0)×BMO andkYk_S∞(R^m⁰) ≤ C₁, kZ·Wk_BMO ≤C₂.

3.4.2 Regularity of solutions

For any initial valuex ∈ R^m, we denote by(X^x, Y^x, Z^x) the unique solution of the FBSDE (3.2.2). The following two results provide regularity of the solution upon the parameterx.

Theorem 3.4.1 (Continuity). Assume (B1) - (B4). With the same constant C_k,λ as in Theorem 3.2.2, ifT ≤C_k,λ, the functionx7→(X^x, Y^x, Z^x)is continuous.

Proof. LetT ≤ C_k,λ and(X^x, Y^x, Z^x)be the solution of the FBSDE (3.2.2) for anyx∈R. Notice thatX^x−X^x⁰is bounded. In fact, using the Lipschitz continuity condition onb, we have

|X_t^x−X_t^x⁰| ≤ |x−x⁰|+k1 t

|X_u^x−X_u^x⁰|du+k2 t

|Y_u^x−Y_u^x⁰|du

≤ |x−x⁰|+k₂TkY^x−Y^x⁰k_S^∞+k₁

|X_u^x−X_u^x⁰|du

≤

|x−x⁰|+k2TkY^x−Y^x⁰k_S^∞ e^k¹^t, by Gronwall’s lemma. Thus

kX^x−X^x⁰k_S^∞ ≤

|x−x⁰|+k₂TkY^x−Y^x⁰k_S^∞

e^k¹^T. (3.4.4) On the other hand, arguing such as in the proof of Theorem 3.2.2, we have, for eachi= 1, . . . , m⁰,

Y_t^i,x−Y_t^i,x⁰ +

Z_u^i,x−Z_u^i,x⁰dW˜_uⁱ

=hⁱ(X_T^x)−hⁱ(X_T^x⁰) +

lⁱ_u(X_u^x, Y_u^x, Z_u^x)−lⁱ_u(X_u^x⁰, Y_u^x⁰, Z_u^x⁰)du

whereW˜ⁱ=W−R.

0ηⁱ_sdswith|η_sⁱ| ≤k3(1+|Z_s^i,x|+|Z_s^i,x⁰)is a Brownian motion under the equivalent measureP˜ⁱ = E(ηⁱ·W)T ·P. Hence, similar to Theorem 3.2.2, with the same constantsc₁, c₂andC₂,

kY^x−Y^x⁰k²_∞+c₁k(Z^x−Z^x⁰)·Wk²_BMO

≤m⁰

16k²₂T²e^2k¹^T k²₄+k²₇

+ 8k²₅T²

kY^x−Y^x⁰k²_∞ + 16m⁰e^2k¹^T k²₄+k²₇

|x−x⁰|² + 24

√

3k₆²L²₄T^1−εc2m⁰ T^ε+ 2 + 2εL²₄c2C₂²

k(Z^x−Z^x⁰)·Wk²_BMO. Therefore, it follows from (3.4.3) that

kY^x−Y^x⁰k²_S∞ ≤ 16m⁰e^2k¹^T k₄²+k₇² 1−m⁰ 16k²₂T²e^2k¹^T k²₄+k²₇

+ 8k²₅T²|x−x⁰|². (3.4.5) and

c1kZ^x−Z^x⁰k²_BMO ≤32m⁰e^2k¹^T k₄²+k²₇

|x−x⁰|². (3.4.6) Combining with (3.4.4)

kX^x−X^x⁰k_S^∞

≤ 1 +k2T s

16m⁰e^2k¹^T k₄²+k₇² 1−m⁰ 16k²₂T²e^2k¹^T k²₄+k²₇

+ 8k²₅T²

e^k¹^T|x−x⁰|.

(3.4.7)

This proves continuity of the solution.

(B5) The functionsb;h;f andlare continuously differentiable.

(B6) The functionsh⁰;∂xb;∂yb;∂xl;∂yl;∂zlandf⁰are Lipschitz continuous in all variables with Lipschitz constantK.

Theorem 3.4.2 (Differentiability). Assume (B1) - (B6). With the same constant C_k,λas in Theorem 3.2.2, ifT ≤C_k,λ, the functionx7→(X^x, Y^x, Z^x)is differen-tiable.

Proof. Let T ≤ C_k,λ, x, x⁰ ∈ R^m andλ, λ⁰ > 0. Let e_j = (0, . . . ,1, . . . ,0) be the unit vector in R^m the jth component of which is 1 and all the others 0.

Given (X^x+λeⁱ, Y^x+λe^j, Z^x+λe^j), (X^x⁰^+λ⁰^e^j, Y^x⁰^+λ⁰^e^j, Z^x⁰^+λ⁰^e^j), (X^x, Y^x, Z^x) and(X^x⁰, Y^x⁰, Z^x⁰)solutions of the FBSDE (3.2.2), we define the processesN^x,λ :=

(X^x+λe^j −X^x)/λ;N^x⁰^,λ⁰ := (X^x⁰^+λ⁰^e^j −X^x⁰)/λ⁰;U^x,λ:= (Y^x+λe^j −Y^x)/λ;

U^x⁰^,λ⁰ := (Y^x⁰^+λ⁰^e^j−Y^x⁰)/λ⁰andV^x,λ:= (Z^x+λe^j−Z^x)/λ;V^x⁰^,λ⁰ := (Z^x⁰^+λ⁰^e^j− Z^x⁰)/λ⁰. Furthermore, for θ ∈ [0,1], λ > 0, x ∈ R^m, we define the processes Λ^x,λ := X^x+θλN^x,λ,Γ^x,λ := Y^x +θλU^x,λ and∆^x,λ := Z^x+θλV^x,λ. Let N^i,x,λ,U^i,x,λ, V^i,x,λ,Λ^i,x,λ,Γ^i,x,λand∆^i,x,λbe theith component ofN^x,λ, U^x,λ, V^x,λ,Λ^x,λ,Γ^x,λ and∆^x,λ, respectively for each i= 1, . . . , m⁰. Let us first show that there exists a constantCindependent ofxandλsuch that

kN^λk²_S∞+kU^λk²_S∞+kV^λ·Wk²_BMO≤C. (3.4.8) Since

N_t^x,λ=ei+

0 1

∂xbu(X_u^x+θ(X_u^x+λeⁱ−X_u^x), Y_u^x+θ(Y_u^x+λeⁱ−Y_u^x))N_u^x,λdθ du

0 1

∂ybu(X_u^x+θ(X_u^x+λeⁱ−X_u^x), Y_u^x+θ(Y_u^x+λeⁱ−Y_u^x))U_u^x,λdθ du,

and∂xband∂ybare bounded, it follows from Gronwall’s inequality that

|N_t^x,λ| ≤e^k¹^t

1 +k₂TkU^x,λk_S^∞

. (3.4.9)

We have U_t^i,x,λ=

∂_xhⁱ(Λ^x,λ_T )N_T^x,λdθ+

t 1

∂_zf_uⁱ(∆^i,x,λ_u )V_u^i,x,λ

+∂_xlⁱ_u(Λ^x,λ_u ,Γ^x,λ_u ,∆^x,λ_u )N_u^x,λ+∂_ylⁱ_u(Λ^x,λ_u ,Γ^x,λ_u ,∆^x,λ_u )U_u^x,λ +∂zlⁱ_u(Λ^x,λ_u ,Γ^x,λ_u ,∆^x,λ_u )V_u^x,λdθ du−

V_u^i,x,λdWu.

Hence, similar to the proof of Theorem 3.2.2, we have U_t^i,x,λ+

V_u^i,x,λdW˜_uⁱ

∂_xhⁱ(Λ^x,λ_T )N_T^x,λdθ+

t 1

∂_xlⁱ_u(Λ^x,λ_u ,Γ^x,λ_u ,∆^x,λ_u )N_u^x,λ

+∂_ylⁱ_u(Λ^x,λ_u ,Γ^x,λ_u ,∆^x,λ_u )U_u^x,λ+∂_zl_uⁱ(Λ^x,λ_u ,Γ^x,λ_u ,∆^x,λ_u )V_u^x,λdθ du, whereW˜ⁱ =W −R.

0ζ_sⁱdswith|ζ_sⁱ| ≤k3(1 + 2|(1−θ¯s)Zs^i,x+ ¯θsZs^i,x+λe^j|)for some predictable process θ¯s ∈ [0,1] is a Brownian motion under the equivalent measureP˜ⁱ =E(ζⁱ·W)_T ·P. Therefore similar to Theorem 3.2.2, with the same constantsc1, c2andC2,

kU^x,λk²_∞+c₁kV^x,λ·Wk²_BMO

≤m⁰

16k₂²T²e^2k¹^T k₄²+k²₇

+ 8k₅²T²

kU^x,λk²_∞+ 16m⁰e^2k¹^T k₄²+k₇² + 24√

3k²₆L²₄T^1−εc₂m⁰ T^ε+ 2 + 2εL²₄c₂C₂²

kV^x,λ·Wk²_BMO. Therefore, it follows from (3.4.3) that

kU^x,λk²_S∞ ≤ 16m⁰e^2k¹^T k₄²+k²₇ 1−m⁰ 16k²₂T²e^2k¹^T k²₄+k₇²

+ 8k²₅T². (3.4.10) and

c₁kV^x,λk²_BMO≤32m⁰e^2k¹^T k²₄+k²₇

. (3.4.11)

Combining with (3.4.9), kN^x,λk_S^∞ ≤ 1 +k2T

s 16m⁰e^2k¹^T k²₄+k²₇ 1−m⁰ 16k₂²T²e^2k¹^T k₄²+k₇²

+ 8k₅²T²

! e^k¹^T. (3.4.12) Now, estimating the difference gives

|N_t^x,λ−N_t^x⁰^,λ⁰|=

0 1

∂xbu(Λ^x,λ_u ,Γ^x,λ_u )N_u^x,λ+∂ybu(Λ^x,λ_u ,Γ^x,λ_u )U_u^x,λ

−∂_xb_u(Λ^x_u⁰^,λ⁰,Γ^x_u⁰^,λ⁰)N_u^x⁰^,λ⁰ −∂_yb_u(Λ^x_u⁰^,λ⁰,Γ^x_u⁰^,λ⁰)U_u^x⁰^,λ⁰dθdu

≤ Zt

|∂_xb_u(Λ^x,λ_u ,Γ^x,λ_u )||N_u^x,λ−N_u^x⁰^,λ⁰|

+|∂_xb_u(Λ^x,λ_u ,Γ^x,λ_u )−∂_xb_u(Λ^x_u⁰^,λ⁰,Γ^x_u⁰^,λ⁰)||N_u^x⁰^,λ⁰| +|∂_yb_u(Λ^x,λ_u ,Γ^x,λ_u )||U_u^x,λ−U_u^x⁰^,λ⁰|

+|∂_yb_u(Λ^x,λ_u ,Γ^x,λ_u )−∂_yb_u(Λ^x_u⁰^,λ⁰,Γ^x_u⁰^,λ⁰)||U_u^x⁰^,λ⁰|dθdu.

(3.4.13)

Then, using (B1) and (B6) and applying Gronwall’s lemma, we have kN^x,λ−N^x⁰^,λ⁰k_S^∞

≤e^k¹^T k2TkU^x,λ−U^x⁰^,λ⁰kS^∞+K(kN^x⁰^,λ⁰kS^∞+kU^x⁰^,λ⁰kS^∞)

2 kX^x−X^x⁰kS^∞

+K(kN^x⁰^,λ⁰k_S^∞+kU^x⁰^,λ⁰k_S^∞)

2 kX^x+λe^j −X^x⁰^+λ⁰^e^jk_S^∞ +K(kN^x⁰^,λ⁰kS^∞+kU^x⁰^,λ⁰kS^∞)

2 kY^x−Y^x⁰k_S^∞ +K(kN^x⁰^,λ⁰k_S^∞ +kU^x⁰^,λ⁰k_S^∞)

2 kY^x+λe^j−Y^x⁰^+λ⁰^e^jkS^∞

! . On the other hand,

U_t^i,x,λ−U_t^i,x⁰^,λ⁰+

(V_u^i,x,λ−V_u^i,x⁰^,λ⁰)dW˜_uⁱ

∂_xhⁱ(Λ^x,λ_T )N_T^x,λ−∂_xhⁱ(XΛ^x_T⁰^,λ⁰)N_T^x⁰^,λ⁰dθ

t 1

(∂zf_uⁱ(∆^i,x,λ_u )−∂zf_uⁱ(∆^i,x_u ⁰^,λ⁰))V_u^i,x⁰^,λ⁰dθ du

t 1

∂_xl_uⁱ(Λ^x,λ_u ,Γ^x,λ_u ,∆^x,λ_u )N_u^x,λ+∂_yl_uⁱ(Λ^x,λ_u ,Γ^x,λ_u ,∆^x,λ_u )U_u^x,λ +∂_zl_uⁱ(Λ^x,λ_u ,Γ^x,λ_u ,∆^x,λ_u )V_u^x,λ−∂_xlⁱ_u(Λ^x_u⁰^,λ⁰,Γ^x_u⁰^,λ⁰,∆^x_u⁰^,λ⁰)N_u^x⁰^,λ⁰

−∂_ylⁱ_u(Λ^x_u⁰^,λ⁰,Γ^x_u⁰^,λ⁰,∆^x_u⁰^,λ⁰)U_u^x⁰^,λ⁰−∂_zlⁱ_u(Λ^x_u⁰^,λ⁰,Γ^x_u⁰^,λ⁰,∆^x_u⁰^,λ⁰)V_u^x⁰^,λ⁰dθ du, where W˜ⁱ = W −R_.

0ζ_sⁱds is defined as above. Rearranging the terms on the right hand side such as in (3.4.13) using successively (B3), (B4), (B6) and using Cauchy-Schwarz’ inequality, similar to Theorem 3.2.2, with the same constants c1, c2, C2, we have

kU^x,λ−U^x⁰^,λ⁰k²_∞+c₁k(V^x,λ−V^x⁰^,λ⁰)·Wk²_BMO

≤m⁰

16k₂²T²e^2k¹^T k₄²+k₇²

+ 16k₅²T²

kU^x,λ−U^x⁰^,λ⁰k²_∞+I₁+I₂ + 24√

3k²₆L²₄T^1−εc2m⁰ T^ε+ 2 + 2εL²₄c2C₂²

k(V^x,λ−V^x⁰^,λ⁰)·Wk²_BMO, where

I1 = 4K²m⁰e^2k¹^T(k₄²+k²₇)

kN^x⁰^,λ⁰kS^∞+kU^x⁰^,λ⁰kS^∞

kX^x−X^x⁰kS^∞

+kX^x+λe^j−X^x⁰^+λ⁰^e^jk_S^∞+kY^x−Y^x⁰k_S^∞+kY^x+λe^j−Y^x⁰^+λ⁰^e^jk_S^∞2

I2 = 24m⁰K²

kN_T^x⁰^,λ⁰k²_S∞

kX_T^x−X_T^x⁰k_S^∞+kX_T^x+λe^j−X^x

0+λ⁰ej

T k_S^∞2

4c²₂L²₄kV^x⁰^,λ⁰·Wk²_BMO+ 2T c₂

kN_T^x⁰^,λ⁰k_S^∞+kU_T^x⁰^,λ⁰k_S^∞2

k(Z^x−Z^x⁰)·Wk²_BMO+k(Z^x+λe^j−Z^x⁰^+λ⁰^e^j)·Wk²_BMO +

T c2kV^x⁰^,λ⁰·Wk²_BMO+T²

kN_T^x⁰^,λ⁰k_S^∞+kU_T^x⁰^,λ⁰k_S^∞2

kX^x−X^x⁰k_S^∞+kX^x+λe^j−X^x⁰^+λ⁰^e^jk_S^∞ +kY^x−Y^x⁰kS^∞+kY^x+λe^j−Y^x⁰^+λ⁰^e^jkS^∞

2 .

Hence, it follows from the Equations (3.4.3), (3.4.5), (3.4.6), (3.4.7) and (3.4.8) that there exists a constantC >˜ 0which does not depend onx, x⁰ andλ, λ⁰ such that

kN^x,λ−N^x⁰^,λ⁰k²_S∞+kU^x,λ−U^x⁰^,λ⁰k²_S∞+k(V^x,λ−V^x⁰^,λ⁰)·Wk²_BMO

≤C˜ |x−x⁰|+|λ−λ⁰| .

This proves the differentiability ofx7→(X^x, Y^x, Z^x).

3.A Multidimensional BSDEs with terminal condition of bounded Malliavin derivative

In this section, we extend the existence result of Cheridito and Nam [16] to the multidimensional case where thei^thcomponent of the generator depends only on (y, zⁱ). For simplicity, we prove the crucial boundedness ofZ in this setting and leave out the existence since it follows as in [16, Theorem 2.2]. We consider the BSDE

Y_t=ξ+

g_u(Y_u, Z_u)du−

Z_udW_u. (3.A.1)

We make the following assumptions:

(D1) g : Ω×[0, T]×R^m

0 ×R^m

0×d → R^m

0 is a continuous and measurable function such thatg_tⁱ(y, z) = gⁱ_t(y, zⁱ),i = 1, . . . , m⁰ and there exists a constantB ∈R+and a nondecreasing functionρ:R+→R+such that

gt(y, z)−gt(y⁰, z⁰) ≤B

y−y⁰

+ρ |z| ∨ z⁰

z−z⁰ for allt∈[0, T],y, y⁰ ∈R^m

0 andz, z⁰∈R^m

0×d.

(D2) g·(0,0)∈ H⁴ and there exist Borel-measurable functionsqij : [0, T]→

We first prove a useful lemma under the following stronger conditions:

(D1’) gis continuously differentiable in(y, z)is such thatg_tⁱ(y, z) =g_tⁱ(y, zⁱ),

Lemma 3.A.1. If (D1’), (D2’) and (D3) hold, then the BSDE (3.A.1)admits a unique solution(Y, Z)∈ S⁴(R^m [1,∞). It follows from [28, Theorem 5.1 and Proposition 5.3] that the BSDE (3.A.1) has a unique solution(Y, Z)∈ S⁴(R^m

and for each fixedr, denoting(U_t^j,r, V_t^j,r) = (D^jrYt, Dr^jZt), then(U^j,r, V^j,r) is

Using the conditions (D1’), we have U_t^ij,r=D_r^jξⁱ+

It is easy to see that the unique solution of the above ODE is given by u^j_t =

Theorem 3.A.2. If (D1) - (D3) hold, then the BSDE(3.A.1)has a unique solution as in the proof of [16, Theorem 2.2] and in combination with [28, Proposition 5.1]

the result follows.

3.B Multidimensional BSDEs with superquadratic growth

In this section, we will drop the assumption that thei^thcomponent of the generator depends only on(y, zⁱ). We obtain solvability for small time horizon. Under an ad-ditional condition on the growth functionρ, the existence result hold for arbitrarily large time horizon. We consider the BSDE

Yt=ξ+

We make the following assumptions:

(H1) g : Ω×[0, T]×R^m

0 ×R^m

0×d → R^m

0 is a continuous and measurable function such that there exists a constantB ∈ R+ and a nondecreasing functionρ:R+→R+such that

(H3) The terminal conditionξ ∈ D^1,2(R^m⁰)and there exist constantsAij ≥0 such that

D_t^jξⁱ

≤Aij for alli= 1, . . . , m⁰;j= 1, . . . , d.

We first prove a useful lemma under the following stronger conditions:

(H1’) gis continuously differentiable in(y, z)is such that there exist constants B ∈R+, ρ∈R+such that

|∂_ygt(y, z)| ≤B, |∂_zgt(y, z)| ≤ρ, for allt∈[0, T],y, y⁰ ∈R^m

0 andz, z⁰∈R^m

0×d. (H2’) Condition (D2) holds for all(y, z)∈R^m

0 ×R^m

0×d.

Lemma 3.B.1. If (H1’), (H2’) and (H3) hold, then the BSDE (3.B.1) admits a unique solution(Y, Z)∈ S⁴(R^m

0)× H⁴(R^m

0×d), and

|Z_t^j|²≤

m⁰

i=1



A²_ij +

q²_ij(s)e⁻(^2B+ρ²⁺¹)^(T^−s)ds



e(^2B+ρ²⁺¹)^(T^−t), P⊗dt-a.e.

Proof. By [16, Lemma 2.5], condition (H3) implies E|ξ|^p < +∞, for all p ∈ [1,∞). It follows from [28, Theorem 5.1 and Proposition 5.3] that the BSDE (3.B.1) has a unique solution(Y, Z)∈ S⁴(R^m

0)×H⁴(R^m

0×d). Moreover,(Y, Z)∈ L^1,2_a (R^m⁰^+m⁰^×d)fori= 1, . . . , m⁰; j= 1, . . . , d,

(D^j_rY_tⁱ, D^j_rZ_tⁱ) = (U_t^ij,r, V_t^ij,r) P⊗dt⊗dr-a.e. andZ_t^ij =U_t^ij,t P⊗dt-a.e., where

U_t^ij,r= 0, V_t^ij,r= 0, for 0≤t < r ≤T,

and for each fixedr, denoting(U_t^j,r, V_t^j,r) = (D^jrYt, Dr^jZt), then(U^j,r, V^j,r) is the unique solution inS²(R^m

0)× H²(R^m

0×d)of the BSDE

U_t^j,r =D_r^jξ+

∂ygs(Ys, Zs)U_s^j,r+∂zgs(Ys, Zs)V_s^j,r+D_r^jgs(Ys, Zs)ds

−

V_s^j,rdWs.

Applying Itô’s formula to|U_t^j,r|²yields

Taking conditional expectation with respect toF_tandP, using condition (H3)

|U_t^j,r|²≤E Gronwall’s inequality implies that

|U_t^j,r|² ≤

where the constant λ ∈ R+ is chosen such that R

R^m⁰^+m0×dβ(x)dx = 1. Set βⁿ(x) :=n^m⁰^+m⁰^×dβ(nx),n∈N\ {0}, and define

g_tⁿ(ω, x) :=

R^m⁰^+m0×d

gt(ω, x⁰)βⁿ(x−x⁰)dx⁰.

Then allgⁿsatisfy (H1’) and (H2’). Therefore by Lemma 3.B.1 there exist unique solutions (Yⁿ, Zⁿ) ∈ S⁴(R^m

0) × H⁴(R^m

0×d) to the BSDEs corresponding to (gⁿ, ξ), and

|Z_t^n,j|² ≤

m⁰

i=1



A²_ij+

q_ij²(s)e⁻(^2B+ρ²^(Q)+1)^(T^−s)ds



e(^2B+ρ²^(Q)+1)^(T^−t)

≤

m⁰

i=1



A²_ij+

q_ij²(s)ds



e(^2B+ρ²^(Q)+1)^T. SinceT ≤ _2B+ρ^{log 2}2(Q)+1, we obtain

|Z_t^n,j|² ≤2





m⁰

i=1

A²_ij +

m⁰

i=1 T

q²_ij(s)ds



.

Thus, following the same procedure as in the proof of [16, Theorem 2.2] and in combination with [28, Proposition 5.1] the result follows.

Theorem 3.B.3. If (H1), (H3) hold andρis such thatP∞ n=0

log 2

2B+ρ²(2ⁿQ)+1 > T, (H2) holds for all(y, z) ∈ R^m

0 ×R^m

0×d such that|z| ≤ 2^NQwhere N is the smallest integer such thatPN

n=0

log 2

2B+ρ²(2ⁿQ)+1 ≥T. Then the BSDE(3.B.1)has a unique solution inS⁴(R^m⁰)× H^∞(R^m⁰^×d)and

|Z_t| ≤2^NQ, P⊗dt-a.e.

Proof. From Theorem 3.B.2, the BSDE (3.B.1) has a unique solution inS⁴(R^m

0)×

H^∞(R^m⁰^×d)and|Z_t| ≤Qon[T −_2B+ρ^{log 2}2(Q)+1, T]. By Lemma 3.B.1, we have

|D_r^jY_T₋ log 2 2B+ρ2(Q)+1

|²≤

m⁰

i=1

2|A_ij|²+

m⁰

i=1 T

2|q_ij(t)|²dt.

By similar arguments as in Lemma 3.B.1 and Theorem 3.B.2, the BSDE (3.B.1) has a unique solution inS⁴(R^m⁰)×H^∞(R^m⁰^×d)on[T−_2B+ρ^{log 2}2(Q)+1−_2B+ρ^{log 2}2(2Q)+1, T−

log 2

2B+ρ²(Q)+1]with terminal conditionY_T₋ ^{log 2}

2B+ρ2(Q)+1, and

|D^j_rY_T₋ log 2

2B+ρ2(Q)+1−2B+ρ2(2Q)+1^{log 2} |²≤

m⁰

i=1

2²|A_ij|²+

m⁰

i=1 T

(2²+ 2)|q_ij(t)|²dt,

|Z_t| ≤2Q, t∈[T − log 2

2B+ρ²(Q) + 1− log 2

2B+ρ²(2Q) + 1, T − log 2

2B+ρ²(Q) + 1].

By recurrence, form≥2, the BSDE (3.B.1) has a unique solution inS⁴(R^m

0)× H^∞(R^m⁰^×d)on[T−Pm

n=0

log 2

2B+ρ²(2ⁿQ)+1, T−Pm−1 n=0

log 2

2B+ρ²(2ⁿQ)+1]with termi-nal conditionY_T₋^P^m−1

n=0

log 2

2B+ρ2(2nQ)+1, and

|D_r^jY_T₋^Pm n=0

log 2 2B+ρ2(2nQ)+1

|²≤

m⁰

i=1

2^m+1|A_ij|²+

m⁰

i=1 T

(

m+1

k=1

2^k)|q_ij(t)|²dt,

|Z_t| ≤2^mQ, t∈[T−

n=0

log 2

2B+ρ²(2ⁿQ) + 1, T −

m−1

n=0

log 2

2B+ρ²(2ⁿQ) + 1].

Hence the existence follows from a pasting argument. The uniqueness follows from a similar argument as in the proof of [16, Theorem 2.2].

Remark 3.B.4. Ifρ(x) = p

log(1 +x)for x ≥ 0, then P∞ n=0

log 2

2B+ρ²(2ⁿQ)+1 =

∞. Indeed, sinceρ(2ⁿQ)≤p

log(2ⁿ(1 +Q)), we have

∞

n=0

log 2

2B+ρ²(2ⁿQ) + 1 ≥

∞

n=0

log 2

2B+ log(2ⁿ(1 +Q)) + 1

∞

n=0

log 2

2B+ log(1 +Q) +nlog 2 + 1 =∞.

BSDEs on Finite and Infinite Horizon with Time-delayed Generators

4.1 Introduction

In Delong and Imkeller [24, 25], the theory of backward stochastic differential equations (BSDEs) was extended to BSDEs with time delay generators (delay BS-DEs). These are non-Markovian BSDEs in which the generator at each positive time t may depend on the past values of the solutions. This class of equations turned out to have natural applications in pricing and hedging of insurance con-tracts, see Delong [23].

The existence result of Delong and Imkeller [24], proved for standard Lips-chitz generators and small time horizon, has been refined by dos Reis et al. [26]

who derived additional properties of delay BSDEs such as path regularity and ex-istence of decoupled systems. Furthermore, exex-istence of delay BSDE constrained above a given continuous barrier has been established by Zhou and Ren [77] in a similar setup. More recently, Briand and Elie [13] proposed a framework in which quadratic BSDEs with sufficiently small time delay in the value process can be solved.

In addition to the inherent non-Markovian structure of delay BSDEs, the dif-ficulty in studying these equations comes from that the inter-temporal changes of the value and control processes always depend on their entire past, hence making it hard to obtain boundedness of solutions or even BMO-martingale properties of the stochastic integral of the control process. This suggests that delay BSDEs can ac-tually be solved forward and backward in time and in this regard, share similarities with forward backward stochastic differential equations (FBSDEs), see Section 4.4 for a more detailed discussion.

The object of the present chapter is to study delay BSDEs in the case where the past values of the solutions are weighted with respect to some scaling function.

In economic applications, these weighting functions can be viewed as representing the perception of the past of an agent. For multidimentional BSDEs with possibly infinite time horizon, we derive existence, uniqueness and stability of delay BSDE in this weighting-function setting. In particular, we show that when the delay van-ishes, the solutions of the delay BSDEs converge to the solution of the BSDE with no delay, hence recovering a result obtained by Briand and Elie [13] for different types of delay. Moreover, we prove that in our setting existence and uniqueness also hold in the case of reflexion on a càdlàg barrier. We observe a link between delay BSDEs and coupled FBSDE and, based on the findings in chapter 3, we de-rive existence of delay quadratic BSDEs in the case where only the value process is subjected to delay. We refer to Briand and Elie [13] for a similar result, again for a different type of delay and in the one-dimensional case.

In the next section, we specify our probabilistic structure and the form of the equation, then present existence, uniqueness and stability results. Sections 4.3 and 4.4 are dedicated to the study of reflected delay BSDEs and quadratic and superquadratic BSDEs with delay in value process, respectively.

4.2 BSDEs with time delayed generators

We work on a filtered probability space(Ω,F,(F_t)_t∈[0,T_], P) with T ∈ (0,∞].

We assume that the filtration is generated by ad-dimensional Brownian motion W and it is complete and right continuous. Let us also assume that F = F_T. We endow Ω×[0, T] with the predictable σ-algebra and R^k with its Borel σ-algebra. Unless otherwise stated, all equalities and inequalities between random variables and stochastic processes will be understood in theP-a.s. and P ⊗ dt-a.e. sense, respectively. Forp ∈ [1,∞) andm ∈ N, we denote by S^p(R^m) the space of predictable and continuous processesXvalued inR^msuch thatkXk^p_Sp:=

E[(sup_t∈[0,T_]|X_t|)^p] < ∞and byH^p(R^m)the space of predictable processesZ valued inR^m×d such thatkZk^p_Hp := E[(R_T

0 |Z_u|² du)^p/2] < ∞. For a suitable integrandZ, we denote byZ ·W the stochastic integral(R_t

0ZudWu)_t∈[0,T_]ofZ with respect toW. From Protter [68],Z·W defines a continuous martingale for everyZ ∈ H^p(R^m). Processes(φt)t∈[0,T]will always be extended to[−T,0)by settingφt= 0fort∈[−T,0). We equipRwith theσ-algebraB(R)consisting of Borel sets of the usual real line with possible addition of the points−∞,+∞, see Bogachev [11].

Letξ be anF_T-measurable terminal condition and ganR^m-valued function.

Given two measuresα1 andα2 on[−∞,∞], and two weighting functionsu, v : [0, T]→R, we study the existence of the BSDE

Yt=ξ+

g(s,Γ(s))ds−

ZsdWs, t∈[0, T], (4.2.1)

where

Γ(s) :=





−T

u(s+r)Ys+rα1(dr),

−T

v(s+r)Zs+rα2(dr)



. (4.2.2) Example 4.2.1. 1. BSDE with infinite horizon: Ifu =v = 1andα₁ =α₂ =δ₀ the Dirac measure at0, then Equation (4.2.1) reduces to the classical BSDE with infinite time horizon and stantard Lipschitz generator.

2.Pricing of insurance contracts:Let us consider the pricing problem of an insur-ance contractξwritten on a weather derivative. It is well know, see for instance [3]

that such contracts can be priced by investing in a highly correlated, but tradable derivative. In the Merton model, assuming that the latter asset has dynamics

dSt=St(µtdt+σtdWt),

then the insurer chooses a numberπ_tof shares ofSto buy at timetand fixes a cost ctto be paid by the client. Hence, he seeks to find the priceV0such that

dV_t=c_tdt+π_tσ_t(dW_t+θ_tdt)

withθ_t=σ⁰_t(σ_tσ_t)⁻¹µ_t. It is natural to demand the costc_tat timetto depend on the past values of the insurance premiumV_t, for instance to account for historical weather data. A possible cost criteria is

c_t:=M_t Z0

−T

cos(2π

P (t+s))V_t+sds

whereP accounts for the weather periodicity andM is a scaling parameter. Thus, the insurance premium satisfies the delay BSDE

Vt=ξ+





−T

Mucos(2π

P (u+s))Vu+sds+Zuσuθu



 du−

ZudWu. ♦

4.2.1 Existence

Our existence result for the BSDE (4.2.1) is obtained under the following assump-tions:

(A1) α₁, α₂are two deterministic, finite valued measures supported on[−T,0].

(A2) u, v : [0, T]→ Rare Borel measurable functions such thatu ∈ L¹(dt) andv∈L²(dt).

(A3) g: Ω×[0, T]×R^m×R^m×d→R^mis measurable, such thatRT

0 g(s,0,0)ds∈ L²(R^m)and satisfies the standard Lipschitz condition: there exists a con-stantK >0such that

|g(t, y, z)−g(t, y⁰, z⁰)| ≤K(|y−y⁰|+|z−z⁰|) for everyy, y⁰ ∈R^mandz, z⁰∈R^m×d.

(A4) ξ ∈L²(R^m)and isF_T-measurable.

Theorem 4.2.2. Assume (A1)-(A4). If

(K²α²₁([−T,0])kuk²_L1(dt)≤ ₂₅¹ ,

K²α²₂([−T,0])kvk²_L2(dt)≤ ₂₅¹, (4.2.3) then BSDE(4.2.1)admits a unique solution(Y, Z)∈ S²(R^m)× H²(R^m×d).

For the proof we need the following lemma ona prioriestimates of solutions of (4.2.1).

Lemma 4.2.3 (A priori estimation). Assume (A1)-(A3). For everyξ,ξ¯∈L²(R^m), (y, z),(¯y,z)¯ ∈ S²(R^m)× H²(R^m×d)and(Y, Z),( ¯Y ,Z)¯ ∈ S²(R^m)× H²(R^m×d) satisfying

(Yt=ξ+RT

t g(s, γ(s))ds−RT

t ZsdWs

Y¯_t= ¯ξ+RT

t g(s,γ(s))ds¯ −RT

t Z¯_sdW_s, t∈[0, T] with







γ(s) = R0

−T u(s+r)y_s+rα₁(dr),R0

−Tv(s+r)z_s+rα₂(dr)

¯ γ(s) =

−T u(s+r)¯ys+rα1(dr),R0

−Tv(s+r)¯zs+rα2(dr)

. Then, one has

kY −Y¯k²_S2(R^m)+kZ−Zk¯ ²_H2(R^m×d)

≤20K²α²₁([−T,0])kuk²_L1(dt)ky−yk¯ ²_S2(R^m)

+ 10 ξ−ξ¯

L²(R^m)+ 20K²α²₂([−T,0])kvk²_L2(dt)kz−zk¯ ²_H2(R^m×d). Proof. Let(y, z)∈ S²(R^m)× H²(R^m×d), by assumptions (A1) and (A3), using

2ab≤a²+b² and [26, Lemma 1.1], we have and taking conditional expectation with respect toF_tyields

Y_t−Y¯_t=E

By Doob’s maximal inequality and2ab≤a²+b², we obtain side, taking square and expectation to both sides and2ab≤a²+b², we have E

≤2K²α²₁([−T,0])kuk²_L1(dt)ky−yk¯ ²_S2+ 2K²α²₂([−T,0])kvk²_L2(dt)kz−zk¯ ²_H2. Hence,

kY −Y¯k²_S2(R^m)+kZ−Zk¯ ²_H2(R^m×d)≤20K²α²₁([−T,0])kuk²_L1(dt)ky−yk¯ ²_S2(R^m)

10E

|ξ−ξ|¯²

+ 20K²α²₂([−T,0])kvk²_L2(dt)kz−zk¯ ²_H2(R^m×d).

This concludes the proof.

Proof ( of Theorem 4.2.2). Let(y, z)∈ S²(R^m)× H²(R^m×d)and define the pro-cess γ(s) :=

−Tu(s+r)y_s+rα₁(dr),R0

−T v(s+r)z_s+rα₂(dr)

. Similar to Lemma 4.2.3, it follows from (A1)-(A4) that



ξ+

g(s, γ(s))ds





<∞.

According to the martingale representation theorem, there exists a unique Z ∈ H²(R^m×d)such that for allt∈[0, T],



ξ+

g(s, γ(s))ds

F_t



=E



ξ+

g(s, γ(s))ds



+

ZsdWs. Putting

Y_t:=E



ξ+

g(s, γ(s))ds F_t



, 0≤t≤T, the pair(Y, Z)belongs toS²(R^m)× H²(R^m×d)and satisfies

Y_t=ξ+

g(s, γ(s))ds−

Z_sdW_s, 0≤t≤T.

Thus we have constructed a mappingΦfromS²(R^m)× H²(R^m×d)to itself such thatΦ(y, z) = (Y, Z). Let(y, z),(¯y,z)¯ ∈ S²(R^m)× H²(R^m×d), and(Y, Z) = Φ(y, z),( ¯Y ,Z) = Φ(¯¯ y,z). By Lemma 4.2.3, we have¯

kY −Y¯k²_S2(R^m)+kZ−Zk¯ ²_H2(R^m×d)≤10K²α²₁([−T,0])kuk²_L1(dt)ky−yk¯ ²_S2(R^m)

+ 10K²α²₂([−T,0])kvk²_L2(dt)kz−zk¯ ²_H₂₍

R^m×d)

so that if condition (4.2.3) is satisfied,Φis a contraction mapping which therefore admits a unique fixed point on the Banach space S²(R^m) × H²(R^m×d). This

completes the proof.

4.2.2 Stability

In this subsection, we study stability of the BSDE (4.2.1) with respect to the delay measures. In particular, in Corollary 4.2.5 below we give conditions under which a sequence of solutions of BSDEs with time delayed generator converges to the solution of a standard BSDE with no delay. Given two measuresαandβ, we write α≤βifα(A)≤β(A)for every measurable setA.

Theorem 4.2.4. Assume (A2)-(A4). Fori = 1,2andn∈ N, letαⁿ_i, αi be mea-sures satisfying (A1); with αⁿ_i satisfying(4.2.3)in Theorem 4.2.2 and such that αⁿ_i([−T,0]) converges toα_i([−T,0]). Ifαⁿ₁ ≤ α₁ (orα₁ ≤ αⁿ₁) and αⁿ₂ ≤ α₂ (orα2 ≤ αⁿ₂), thenkYⁿ−Yk_S2(R^m) → 0andkZⁿ−Zk_H2(R^m×d) → 0, where (Yⁿ, Zⁿ) and(Y, Z) are solutions of the BSDE (4.2.1)with delay given by the measures(αⁿ₁, αⁿ₂)and(α₂, α₂), respectively.

Proof. From Theorem 4.2.2, for everyn, there exists a unique solution(Yⁿ, Zⁿ) to the BSDE (4.2.1) with delay given by the measures(αⁿ₁, αⁿ₂). Sinceαⁿ_i, i = 1,2satisfy (4.2.3) in Theorem 4.2.2 andα_iⁿ([−T,0])converges toα_i([−T,0]), it follows thatαisatisfy (4.2.3) and by Theorem 4.2.2 there exists a unique solution (Y, Z)to the BSDE with delay given by(α1, α2). Using it follows similar to the proof of Lemma 4.2.3 that

+ 2K²E i= 1,2, define positive measures satisfying (A1). Therefore,

E Using [26, Lemma 1.1], we obtain

E Similarly, for the control processes we have

Hence

kYⁿ−Yk²_S2(R^m)+kZⁿ−Zk²_H2(R^m×d)

≤20K²(αⁿ₁([−T,0]))²kuk²_L1(dt)kYⁿ−Yk²_S2(R^m)

+ 20K²((αⁿ₁ −α1)([−T,0]))²kuk²_L1(dt)kYk²_S2(R^m)

+ 20K²(αⁿ₂([−T,0]))²kvk²_L2(dt)kZⁿ−Zk²_H2(R^m×d)

+ 20K²((αⁿ₂ −α2)([−T,0]))²kvk²_L2(dt)kZk²_H2(R^m×d)

≤ 4

5kYⁿ−Yk²_S2(R^m)+4

5kZⁿ−Zk²_H2(R^m×d)

+ 20K²((αⁿ₁ −α₁)([−T,0]))²kuk²_L1(dt)kYk²_S2(R^m)

+ 20K²((αⁿ₂ −α₂)([−T,0]))²kvk²_L2(dt)kZk²_H₂₍

R^m×d).

Therefore, the result follows from the convergence ofαⁿ_i([−T,0]),i= 1,2.

The following is a direct consequence of the above stability result. We denote by δ₀the Dirac measure at0.

Corollary 4.2.5. Assume (A2)-(A4). Fori= 1,2andn∈Nletαⁿ_i be measures satisfying (A1) and(4.2.3)in Theorem 4.2.2 and such thatαⁿ_i([−T,0])converges to1. Ifδ0 ≤α₁ⁿ(orαⁿ₁ ≤δ0) andδ0 ≤αⁿ₂ (orαⁿ₂ ≤δ0), thenkYⁿ−Yk_S2(R^m)→ 0andkZⁿ−Zk_H2(R^m×d)→0, where(Yⁿ, Zⁿ)is the solution of the BSDE(4.2.1) with delay given by(αⁿ₁, αⁿ₂)and(Y, Z)is the solution of BSDE without delay.

We conclude this section with the following counterexample which shows that the conditionα1 ≤ αⁿ₁ (orαⁿ₁ ≤ α1) andα2 ≤ α₂ⁿ(or α₂ⁿ ≤ α2) is needed in the above theorem.

Example 4.2.6. Assume thatm = d = 1. We denote by δ₀ andδ−1 the Dirac measures at 0 and−1, respectively. It is clear that δ0([−1,0]) = δ−1([−1,0]).

Consider the delay BSDEs Yt= 1 +

1/5





−1

Ys+r+Zs+r



δ0(dr)ds−

ZsdWs (4.2.5) and

Y¯t= 1 +

1/5





−1

Y¯s+r+ ¯Zs+r



δ−1(dr)ds−

Z¯sdWs. (4.2.6)

Since BSDE (4.2.6) takes the formY¯_t= 1−R1

t Z¯_udW_s, it follows thatY¯_t= 1for allt ∈ [0,1]. On the other hand, (4.2.5) is the standard BSDE without delay, its solution can be written asYt=E[H₁^t| F_t], where the deflator(H_s^t)s≥tat timetis given bydH_s^t=−^H₅^s^t(ds+dWs). Thus,Yt=exp(−1/5(1−t))and fort∈[0,1),

Y_t<Y¯_t. ♦

4.3 Reflected BSDEs with time-delayed generators

The probabilistic setting and the notation of the previous section carries over to the present one. In particular, we fix a time horizonT ∈(0,∞]and we assumem= 1.

Forp∈[1,∞), we further introduce the spaceM^p(R)of adapted càdlàg processes Xvalued inRsuch thatkXk^p_Mp :=E[(sup_t∈[0,T_]|X_t|)^p]<∞and byA^p(R), we denote the subspace of elements ofM^p(R)which are increasing processes starting at0. Let (S_t)_t∈[0,T] be a càdlàg adapted real-valued process. In this section, we study existence of solutions(Y, Z, K)of BSDEs reflected on the càdlàg barrierS and with time-delayed generators. That is, processes satisfying

Y_t =ξ+ ZT

g(s,Γ(s))ds+K_T −K_t− ZT

Z_sdW_s, t∈[0, T](4.3.1)

Y ≥S (4.3.2)

0 (Yt−−St−)dKt= 0 (4.3.3)

withΓdefined by (4.2.2). Consider the condition (A5) E

sup_0≤t≤T(S_t⁺)²

<∞andS_T ≤ξ.

Theorem 4.3.1. Assume (A1)-(A5). If

(K²α²₁([−T,0])kuk²_L1(dt)≤ ₃₆¹,

K²α²₂([−T,0])kvk²_L2(dt) ≤ ₃₆¹, (4.3.4) then RBSDE (4.3.1)admits a unique solution (Y, Z, K) ∈ M²(R)× H²(R^d)× A²(R)satisfying

Y_t= ess sup

τ∈Tt





g(s,Γ(s))ds+S_τ1_{{τ <T}_}+ξ1_{τ=T_}

F_t



,

whereT is the set of all stopping times taking values in[0, T]andT_t={τ ∈ T : τ ≥t}.

Proof. For any given(y, z) ∈ M²(R)× H²(R^d), similar to the proof of Lemma 4.2.3, we have



ξ+

g(s, γ(s))ds





<∞

withγdefined as in Lemma 4.2.3. Hence, from [50, Theorem 3.3] forT <∞and [1, Theorem 3.1] forT =∞the reflected BSDE

Y_t=ξ+

g(s, γ(s))ds+K_T −K_t−

Z_sdW_s

with barrierSadmits a unique solution(Y, Z, K)such that(Y, Z)∈ B, the space of processes (Y, Z) ∈ M²(R)× H²(R^d) such that Y ≥ S, andK ∈ A²(R).

Moreover,Y admits the representation Y_t= ess sup Doob’s maximal inequality implies that

Since(Y, K)and( ¯Y ,K)¯ satisfy (4.3.2) and (4.3.3), we have

|Y_t−Y¯t|²+

|Z_s−Z¯s|²ds≤2

(Ys−Y¯s)(g(s, γ(s))−g(s,γ(s)))ds¯

−2

(Y_s−Y¯_s)(Z_s−Z¯_s)dW_s. Hence





|Z_s−Z¯s|²ds





≤E

sup

0≤t≤T

|Y_t−Y¯t|²

# +E











|g(s, γ(s))−g(s,γ¯(s))|ds





2



. In view of the proof of Lemma 4.2.3, we deduce

kY −Y¯k²_M2(R)+kZ−Z¯k²_H2(R^d)≤9E











|g(s, γ(s))−g(s,¯γ(s))|ds





2





≤18K²α²₁([−T,0])kuk²_L1(dt)ky−yk¯ ²_M2(R)

+ 18K²α²₂([−T,0])kvk²_L2(dt)kz−zk¯ ²_H2(R^d).

By condition (4.3.4),Φis a contraction mapping and therefore it admits a unique fixed point which combined with the associated processK is the unique solution

of the RBSDE (4.3.1).

4.4 Quadratic and superquadratic BSDEs with delay in value process

In this section, we study quadratic and superquadratic BSDEs with delay in value process through the connection between BSDEs with time-delayed generators and FBSDEs. We work in the probabilistic setting and with the notation of Section 4.2.

Standard methods to solve BSDEs with quadratic growth in the control variable often rely either on boundedness of the control process, see for instance [69] and [16], or on BMO estimates for the stochastic integral of the control process, see for instance [73]. However, as shown in [24], solutions of BSDEs with time-delayed generators do not, in general, satisfy boundedness and BMO properties so that new methods are required to solve quadratic BSDE with time-delayed generators.

Recently, [13] obtained existence and uniqueness of solution for a quadratic BSDE

with delay only in the value process. We show below that using FBSDE theory, it is possible to generalize their results to multidimension and considering a different kind of delay. Moreover, our argument allows to solve equations with generators of superquadratic growth.

Let α₁ be the uniform measure on [−T,0], α₂ the Dirac measure at 0. Put u(s) = v(s) = 1, fors ∈ [0, T]. We are considering the following BSDE with time delay only in the value process:

Y_t=ξ+

g(s,

Y_rdr, Z_s)ds−

Z_sdW_s, t∈[0, T]. (4.4.1)

We denote byD^1,2 the space of all Malliavin differentiable random variables and forξ ∈ D^1,2 denote byD_tξ its Malliavin derivative. We refer to Nualart [61] for a thorough treatment of the theory of Malliavin calculus, whereas the definition and properties of the BMO-space and norm can be found in [47]. We make the following assumptions:

(B1) g : [0, T]×R^m ×R^m×d → R^m is a continuous function such that gⁱ(y, z) =gⁱ(y, zⁱ)and there exists a constantK >0as well as a nonde-creasing functionρ:R+→R+such that

|g(s, y, z)−g(s, y⁰, z⁰)| ≤K|y−y⁰|+ρ(|z| ∨ |z⁰|)|z−z⁰|,

|g(s, y, z)−g(s, y⁰, z)−g(s, y, z⁰) +g(s, y⁰, z⁰)| ≤K(|y−y⁰|+|z−z⁰|) for alls∈[0, T],y, y⁰∈R^mandz, z⁰ ∈R^m×d.

(B2) ξ is F_T-measurable such that ξ ∈ D^1,2(R^m) and there exist constants A_ij ≥0such that

|D^j_tξⁱ| ≤Aij, i= 1, . . . , m; j = 1, . . . , d, for allt∈[0, T].

(B3) g: Ω×[0, T]×R^m×R^m×dis measurable,g(s, y, z) =f(s, z)+l(s, y, z) where f and lare measurable functions with fⁱ(s, z) = fⁱ(s, zⁱ), i = 1, . . . , mand there exists a constantK ≥0such that

|f(s, z)−f(s, z⁰)| ≤K(1 +|z|+|z⁰|)|z−z⁰|,

|l(s, y, z)−l(s, y⁰, z⁰)| ≤K|y−y⁰|+K(1 +|z|+|z⁰|)|z−z⁰|,

|f(s, z)| ≤K(1 +|z|²),

|l(s, y, z)| ≤K(1 +|z|¹⁺),

for some0≤ <1and for alls∈[0, T],y, y⁰ ∈R^mandz, z⁰∈R^m×d.

(B4) ξ is F_T-measurable such that there exist a constant K ≥ 0 such that

|ξ| ≤K.

(B5) g : Ω×[0, T]×R×R^d → Ris progressively measurable, continuous process for any choice of the spatial variables and for each fixed(s, ω)∈ [0, T]×Ω, g(s, ω,·) is continuous. g is increasing in y and for some constantK ≥0such that

|g(s, y, z)| ≤K(1 +|z|), for alls∈[0, T],y∈Randz∈R^d.

(B6) ξisF_T-measurable such thatξ ∈L².

(B7) g : Ω×[0, T]×R×R^d → Ris progressively measurable, continuous process for any choice of the spatial variables and for each fixed(s, ω)∈ [0, T]×Ω, g(s, ω,·) is continuous. g is increasing in y and for some constantK ≥0such that

|g(s, y, z)| ≤K(1 +|z|²), for alls∈[0, T],y∈Randz∈R^d.

Proposition 4.4.1. AssumeT ∈(0,∞).

1. If (B1)-(B2) are satisfied, then there exists a constant C ≥ 0 such that for sufficiently small T, BSDE (4.4.1) admits a unique solution (Y, Z) ∈ S²(R^m)× H²(R^m×d)such that|Z| ≤C.

2. If (B3)-(B4) are satisfied, then there exist constantsC₁, C₂ ≥ 0such that for sufficiently small T, BSDE (4.4.1) admits a unique solution (Y, Z) ∈ S²(R^m)× H²(R^m×d)such that|Y| ≤C1andkZ·dWk_BMO ≤C2. 3. Ifm=d= 1and (B5)-(B6) are satisfied, then BSDE(4.4.1)admits at least

a solution(Y, Z)∈ S²(R)× H²(R^d).

4. If m = d = 1 and (B4) and (B7) are satisfied, then BSDE(4.4.1)admits at least a solution(Y, Z) ∈ S²(R)× H²(R^d)such thatY is bounded and Z·W is a BMO martingale.

Proof. Define the function b : R^m → R^m by setting for y ∈ R^m, bⁱ(y) = yⁱ, i= 1, . . . , m. Fort∈[0, T], put

Xt=

b(Ys)ds.

Thus BSDE (4.4.1) can be written as the coupled FBSDE (X_t=Rt

0b(Y_s)ds, Yt=ξ+RT

t g(s, Xs, Zs)ds−RT

t ZsdWs

(4.4.2) so that 1. and 2. follow from chapter 3, and 3. and 4. from [5].

The above theorem provides an explanation why it is not enough to solve a time-delayed BSDE backward in time, one actually needs to consider both the forward and backward parts of the solution due to the delay.

Appendix

A.1 BMO martingales

We recall some results and properties of BMO martingales, for a thorough treat-ment, we refer to Kazamaki [47]. For any uniformly integrable martingaleM with M0= 0andp∈[1,∞), define

kMk_{BM O}_p := sup

τ∈T

kE[hMi_T − hMi_τ|^p²F_τ]¹^pk_∞.

We will useBM Op(P)when it is necessary to indicate the underlying probability measure, and just writeBM O whenp = 2. We recall the following results from the literature.

Lemma A.1.1. LetM be a BMO martingale. Then we have:

(1) The stochastic exponentialE(M)is uniformly integrable.

(2) There exists a numberr >1such thatE(M)_T ∈L^r. This property follows from the Reverse Hölder inequality. The maximalr with this property can be expressed explicitly in terms of theBM O norm of M. There exists as well an upper bound forkE(M)_Tk^r_Lr depending only onT, rand theBM O norm ofM.

(3) For probability measuresP andQ satisfyingdQ = E(M)_TdP for M ∈ BM O(P), the processMˆ =M − hMiis aBM O(Q)martingale.

(4) Energy inequalities imply the inclusionBM O ⊂ H^p for allp ≥ 1. More precisely, for M = R

αdW with BM O norm C, the following estimate

Im Dokument Essays on Multidimensional BSDEs and FBSDEs (Seite 61-0)