FBSDE with time delayed generators: Lp-solutions, differentiability, representation formulas and path regularity

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FBSDE with time delayed generators: Lp-solutions, differentiability, representation formulas and path

regularity

Gonçalo dos Reis, Anthony Réveillac, Jianing Zhang

To cite this version:

Gonçalo dos Reis, Anthony Réveillac, Jianing Zhang. FBSDE with time delayed generators: Lp-

solutions, differentiability, representation formulas and path regularity. 2010. �hal-00508892�

FBSDE with time delayed generators: Lp-solutions,

differentiability, representation formulas and path regularity ^∗

Gon¸ calo dos Reis CMAP Ecole Polytechnique ´

Route de Saclay 91128 Palaiseau Cedex [email protected]

Anthony R´ eveillac Institut f¨ ur Mathematik Humboldt-Universit¨ at zu Berlin

Unter den Linden 6 10099 Berlin [email protected]

Jianing Zhang Institut f¨ ur Mathematik Humboldt-Universit¨ at zu Berlin

Unter den Linden 6 10099 Berlin [email protected]

August 6, 2010

Abstract

We extend the work of Delong and Imkeller (2010a,b) concerning Backward stochastic dif- ferential equations with time delayed generators (delay BSDE). We provide sharper a priori estimates and show that the solution of a delay BSDE is in L

^p

. We introduce decoupled systems of SDE and delay BSDE (which we term delay FBSDE) and give sufficient condi- tions for the variational differentiability of their solutions. We connect these derivatives to the Malliavin derivatives of such delay FBSDE via the usual representation formulas which in turn give access to several path regularity results. In particular we prove an extension of the L

²

-path regularity result for delay FBSDE.

2010 AMS subject classifications: Primary: 60H10; Secondary: 60H30, 60H07, 60G17 Key words and phrases: Backward stochastic differential equation, BSDE, delay, time de- layed generators, Lp-solutions, differentiability, calculus of variations, Malliavin Calculus, path regularity.

Introduction

The theory of nonlinear backward stochastic differential equations (BSDEs) was introduced by Pardoux and Peng (1990) with its main motivations being mathematical finance (see El Karoui et al. (1997)) and stochastic control theory (see Yong and Zhou (1999)). In the last twenty years much effort has been given to this type of equations and nowadays many classes of BSDEs and results on them are available. Due to tractability, common results are achieved within a Marko- vian framework. Under certain conditions the BSDE’s solution exhibits a Markov structure and hence can be interpreted as an instantaneous transformation of the underlying Markov process that spans the stochastic basis of the underlying probability space. This in turn yields access to the theory of partial differential equations via the non-linear Feynman-Kac formula.

∗

Gon¸ calo dos Reis kindly acknowledges financial support by the DFG Research Center MATHEON to visit

the Humboldt-Universit¨ at zu Berlin as well of the Chair Financial risks of the Risk Foundation sponsored by

Soci´ et´ e G´ en´ erale. Anthony R´ eveillac is grateful to DFG Research Center MATHEON, project E2 for financial

support. Jianing Zhang acknowledges financial support by the DFG IRTG 1339 SMCP.

Moving away from the Markovian setting, Delong and Imkeller (2010a,b) introduce a new class of BSDE labeled backward stochastic differential equations with time delayed generators (delay BSDEs). The dynamics of these BSDEs are governed by

Y

t

= ξ + Z

T

t

f (s, Y (s), Z (s))ds − Z

T

t

Z

s

dW

s

, t ∈ [0, T ],

where the generator f at time s ∈ [0, T ] is allowed to depend on the past values of the solution (Y, Z) over the time interval [0, s] and ξ is a measurable random variable. In these two works the authors answered thoroughly several fundamental questions: existence and uniqueness of a square integrable solution, comparison principles, existence of a measure solution, BMO martin- gale properties for the control component Z of the solution, Malliavin differentiability for delay BSDEs driven by a Wiener process and a generalized Poisson martingale. To the best of our knowledge the only existence and uniqueness results for this class of BSDEs follow from those two works. As pointed out by Delong (2010), delay BSDEs appear naturally in finance and insurance related problems of pricing and hedging of contracts. In the same work the author analyses a vast scope of contracts to which this class of BSDEs can be applied to.

Paying consideration to and seeking reference from the state of the art of BSDEs with non- time delayed generators, the next step concerning delay BSDEs is to obtain a feasible numerical scheme. Here, the main obstacle is the presence of the control process Z in the generator.

This process is usually obtained via the predictable representation property of the underlying stochastic basis, and initially all one knows about Z is that it is a square integrable process.

To steer in the direction of a numerical scheme a deeper analysis on the fine properties of the solution of such equations is required. As for numerics for Lipschitz continuous BSDEs (see for example Bouchard and Touzi (2004) or Bender and Denk (2007)) one is usually forced to gather several results concerning the path regularity properties of the solution process before being able to give proper convergence results. Such path properties include not only sample path continuity but also estimations on the time increments of the components of the solution by the size of the time increment. For the purpose of establishing such path properties we first need to prove several auxiliary results.

Our agenda consists of refining and extending the existence and uniqueness results obtained in

Delong and Imkeller (2010a,b) and then steer into the direction of the smoothness properties of

the solution of delay BSDEs. We start by improving the original results of Delong and Imkeller

(2010a) concerning their a priori estimates by providing sharper versions of them. In Lemma

2.1 from Delong and Imkeller (2010a), a priori estimates are given expressing the difference (in

norm) of solutions of two delay BSDE as the difference of the respective terminal conditions

and generators. These a priori estimates fall short of the usual a priori estimates one expects

to see due to the presence of the solutions of both delay BSDE on the right hand side of the

estimate. We establish an a priori estimate in the classical form where the right hand side of

the estimate contains the difference of generators evaluated at their zero spatial state and hence

is independent of the BSDEs’ solutions. Within the topic of a priori estimates we extend the

results of Delong and Imkeller (2010a) in another direction. We show that given extra regularity

of the terminal condition and the generator, the solution will inherit this regularity. This allows

us to state moment and a priori estimates in general L

^p

-spaces and not solely in L

²

. The proof

of these estimates relies on techniques from Delong and Imkeller (2010a) and on computations

carried out for non-time delayed BSDEs in the spirit of Wang et al. (2007). The usual techniques

to obtain higher order moment estimates fail in the setting of delay BSDEs, the reason for this

can be seen in (10) below - usually the dynamics of Y

t

is given by sums of integrals over the

interval [t, T ] but for delay BSDEs we see from (10) that the dynamics of Y

t

depends also on a

integral over the whole interval [0, T ]. These estimates pave the way to a result of existence and uniqueness of solutions to delay BSDE with Lipschitz continuous generators in general L

^p

spaces for p ≥ 2. Inevitably, in analogy to Delong and Imkeller (2010a,b) a compatibility condition on the Lipschitz constant and terminal time is required to obtain existence of solutions (see our Theorem 2.8).

A customary field of application of BSDEs consists in coupling them with SDEs, giving rise (in our case) to systems of delay forward-backward SDEs (delay FBSDEs). We show that when coupling a delay BSDE with a forward diffusion assuming appropriate regularity conditions, we obtain smoothness properties of the solution in terms of the involved parameters, in particular with respect to the initial condition of the forward diffusion. Combining this with the Malliavin differentiability proved in Delong and Imkeller (2010b) enables us to derive the usual represen- tation formulas for FBSDE which display the relationship between the Malliavin derivatives of the solution process and their variational (classical) derivatives. It is somewhat surprising that such a relationship still holds for they are usually consequences of the BSDE’s Markov property which due to path dependency clearly fails to materialize in the context of delay FBSDE.

With this collection of results we are finally able to address the path regularity issue of delay BSDE. Using the techniques employed in Imkeller and Dos Reis (2010a,b), we establish path continuity for the components of the solution of delay FBSDE and we give a result that bounds the norm of the increments in time of Y and Z by the size of the time increment. We expect that these results will open the door to the derivation of concrete numerical schemes and their convergence rate and intend to tackle these problems in our future research.

The paper is organized as follows: in Section 1 we fix notations and elaborate on the type of time-delayed BSDEs that we consider. In Section 2 we refine and extend the a priori estimates obtained in Delong and Imkeller (2010a) and then use them to establish existence and uniqueness of solutions in general L

^p

spaces. In Section 3 we introduce the delay FBSDE framework and use results from the previous sections to obtain the differentiability of the solution process with respect to the initial state of a forward diffusion. The representation formulas and the path regularity results are presented in Section 4.

1 Preliminaries

Let (Ω, F, P ) be a probability space equipped with a standard d-dimensional Brownian motion W . For a fixed real number T > 0 we consider the filtration F := (F

_t

)

t≥0

generated by W and augmented by all P -null sets. The filtered probability space (Ω, F, F , P ) satisfies the usual conditions. Depending on whether we work on R

^d

or R

^d×m

, the Euclidean norm respectively the Hilbert-Schmidt operator norm is denoted by | · |. Furthermore, ∇ denotes the canonical gradient differential operator and for a function h(x, y) : R

^m

× R

^d

→ R

ⁿ

, we write ∇

_x

h or ∇

_y

h for the derivatives with respect to x and y. We work with the following topological vector spaces:

• For p ≥ 2, let L

^p

( R

^m

) be the space of F

_T

-measurable random variables ξ : Ω → R

^m

normed by kξk

_L^p

:= E

|ξ|

^p

1/p

.

• For β ≥ 0 and p ≥ 1, H

^p_β

( R

^d×m

) denotes the space of all predictable process ϕ with values in R

^d×m

such that the norm kϕk

_H^p

β

:= E h R

T

0

e

^βs

|ϕ

_s

|

²

ds

p/2

i

1/p

< ∞.

• For β ≥ 0 and p ≥ 2, S

_β^p

( R

^d×m

) denotes the space of all predictable processes η with values in R

^d×m

such that the norm kηk

_S^p

β

:= E h

sup

_0≤t≤T

e

^βt

|η

_t

|

²

p/2

i

1/p

< ∞.

We omit referencing the range space if no ambiguity arises. It is fairly easy to see that for any β, β ¯ ≥ 0 the norms on H

_β^p

, H

^p_¯

β

and S

_β^p

, S

^p_¯

β

are equivalent.

Some notation

We introduce a notational convention which will be used throughout the text: for an arbitrarily given integrable function f : [0, T ] → R

^m

, trivially extended to [−T, 0) via f (t) 1

[−T ,0)

(t) = 0, and a given deterministic measure α supported on [−T, 0] which is not necessarily atomless, we denote

(f · α)(t) :=

Z

0

−T

f (t + v)α(dv), t ∈ [0, T ], (f

^p

· α)(t) :=

Z

0

−T

|f(t + v)|

^p

α(dv), t ∈ [0, T ], p ≥ 2.

Similarly, for a given process (ϕ

t

)

t∈[0,T]

, extended to [−T, 0) by imposing ϕ

t

= 0 on [−T, 0), we denote

(ϕ · α)(t) :=

Z

0

−T

ϕ

t+v

α(dv), t ∈ [0, T ], (1)

and

(ϕ

^p

· α)(t) :=

Z

0

−T

|ϕ

_t+v

|

^p

α(dv), t ∈ [0, T ], p ≥ 2. (2) We now give a lemma concerning the change of integration order for (1) and (2), which will become useful in the sequel.

Lemma 1.1. Let ϕ be a process and α a non-random finite measure supported on [−T, 0). Then we have the following change of integration order: for every k ≥ 1

Z

_T

t

(ϕ

^k

· α)(s)ds = Z

_T

0

α [r − T, (r − t) ∧ 0)

|ϕ

_r

|

^k

dr, ∀t ∈ [0, T ], P − a.s.

Moreover, if we have for p ≥ 1 that ϕ ∈ H

^p₀

, then we also have that k(ϕ · α)k

^p_Hp

β

≤ M

p

kϕk

^p_Hp 0

, where M

p

= (e

^βT

)

^p/2

α([−T, 0))

p

.

Proof. Let t in [0, T ] and k ∈ [1, +∞). We have that Z

T

t

(ϕ

^k

· α)(s)ds = Z

T

t

Z

0

−T

|ϕ

_s+v

|

^k

α(dv)ds = Z

0

−T

Z

T t

|ϕ

_s+v

|

^k

ds α(dv)

= Z

0

−T

Z

T+v (t+v)∨0

|ϕ

_r

|

^k

dr α(dv) = Z

T

0

Z

(r−t)∧0 (r−T)

|ϕ

_r

|

^k

α(dv) dr

= Z

T

0

α [r − T, (r − t) ∧ 0)

|ϕ

_r

|

^k

dr.

The second claim follows by applying Jensen’s inequality and changing the integration order as done above, i.e. for any β ≥ 0 and p ≥ 1 we have

E Z

T

0

e

^βs

|(ϕ · α)(s)|

²

ds

p/2

≤ e

^βT

α([−T, 0))

p/2

E Z

T

0

(|ϕ|

²

· α)(s)ds

p/2

≤ M

p

E Z

T

0

|ϕ

_s

|

²

ds

p/2

= M

p

kϕk

^p_Hp 0

,

which concludes the proof.

2 General results on BSDE with time delayed generators

In this section we give a brief recapitulation of BSDE with time delayed generators and discuss the setting they are studied under. We then establish convenient a priori estimates on the difference of two solutions to such equations which will play a central role in proving existence and uniqueness of solutions in the more general H

^p

-spaces.

2.1 BSDEs with time delayed generators

Let us start with a recap on BSDE with time delayed generators. For two non-random finite measures α

Y

, α

Z

supported on [−T, 0), we define

α := α

Y

([−T, 0)) ∨ α

Z

([−T, 0)). (3)

Given p ≥ 2, we assume that the following holds:

(H1) ξ is an F

_T

-measurable random variable which belongs to L

^p

( R

^m

);

(H2) the generator f : Ω × [0, T ] × R

^m

× R

^m×d

→ R

^m

is measurable, F -adapted and satisfies the following Lipschitz condition: there exists a constant K > 0 such that

f (t, y, z) − f (t, y

⁰

, z

⁰

)

2

≤ K |y − y

⁰

|

²

+ (|z − z

⁰

|

²

holds for dP ⊗ dt-almost all (ω, t) ∈ Ω × [0, T ] and for every (y, z), (y

⁰

z

⁰

) ∈ R

^m

× R

^d×m

; (H3) E

h R

T

0

|f (s, 0, 0)|

²

ds

p/2

i

< ∞;

(H4) f (t, ·, ·) = 0 if t < 0.

Following the notation from equation (1), we write (Y · α

Y

)(t) =

Z

0

−T

Y

_t+v

α

Y

(dv) and (Z · α

Z

)(t) = Z

0

−T

Z

_t+v

α

Z

(dv), 0 ≤ t ≤ T,

for some processes (Y

_t

)

_t∈[0,T]

and (Z

_t

)

_t∈[0,T]

satisfying appropriate integrability conditions. As- sumption (H2) and Jensen’s inequality then imply

(H2’)

f t, (Y · α

Y

)(t), (Z · α

Z

)(t)

− f t, (Y

⁰

· α

Y

)(t), (Z

⁰

· α

Z

)(t)

2

≤ K

(Y − Y

⁰

) · α

Y

(t)

2

+

(Z − Z

⁰

) · α

Z

(t)

2

≤ L

(Y − Y

⁰

)

²

· α

Y

(t) + (Z − Z

⁰

)

²

· α

Z

(t) ,

where L = Kα with the real number α given by (3). The focus of our study are BSDE with time delayed generators which are of the type

Y

_t

= ξ + Z

T

t

f s, Γ(s) ds −

Z

T t

Z

_s

dW

_s

, 0 ≤ t ≤ T, (4) where Γ abbreviates

Γ(t) :=

Z

0

−T

Y

t+v

α

Y

(dv), Z

0

−T

Z

t+v

α

Z

(dv)

=

(Y · α

Y

)(t), (Z · α

Z

)(t)

, 0 ≤ t ≤ T. (5) Using a fixed point argument, Delong and Imkeller (2010a) have shown that a BSDE of the type (4)-(5) admits a unique solution if the parameters of the equation (4) are sufficiently small, i.e.

if the Lipschitz constant K > 0 or the terminal time T > 0 satisfy a smallness condition. The

following L

²

existence and uniqueness result is a straightforward modification of Theorem 2.1

from Delong and Imkeller (2010a).

Theorem 2.1. Let p = 2 and assume that (H1)-(H4) are satisfied. Assume that the non-negative constants T , L = Kα, β are such that

(8T + 1 β )L

Z

0

−T

e

^−βu

ρ(du) max{1, T } < 1.

where ρ ∈ {α

Y

, α

Z

}. Then the delay BSDE (4)-(5) has a unique solution (Y, Z) ∈ S

_β²

( R

^m

) × H

_β²

( R

^d×m

).

Remark 2.2. In Delong and Imkeller (2010a), this result is proved for the one-dimensional case d = m = 1. It is clear that by the nature of the fixed point argument, the proof is insensitive to dimension of the equation.

Remark 2.3. Given that a compatibility condition is necessary in order to establish existence and uniqueness of solutions and moreover that we will be giving an extended version of it, all the proofs in this section are given with extra detail in order to better control the constants involved in each result.

2.2 A “sharper” a priori estimate

In Delong and Imkeller (2010a), the authors provide a priori estimates for the time delayed BSDE (4) (see their Lemma 2.1), which estimates the norms of the difference between two BSDE solutions in terms of the terminal condition and the generator. It is worth mentioning that Lemma 2.1 from Delong and Imkeller (2010a) establishes a priori estimates whose right hand side again depends on the solution of the delay BSDE. in the context of Delong and Imkeller (2010a), such a result suffices to establish existence and uniqueness of solutions in S

_β²

× H

²_β

, but the situation becomes more intricate when the same issues are considered on S

_β^p

× H

^p_β

for p > 2.

In this section we refine the estimates from Delong and Imkeller (2010a) by providing a right hand side which only depends on the problem’s data (i.e. the differences between the terminal conditions and the corresponding states of the generators at y = 0 and z = 0).

As a starting observation, we have that if (4) admits a solution (Y, Z) in H

^p_β

( R

^m

) × H

^p_β

( R

^m×d

), then we also have that Y ∈ S

_β^p

(R

^m

).

Lemma 2.4. Let β ≥ 0, p ≥ 2 and assume that (H1)-(H4) hold. If the delay BSDE (4) admits a solution (Y, Z) ∈ H

^p_β

( R

^m

) × H

^p_β

( R

^m×d

), then we also have that Y ∈ S

_β^p

( R

^m

).

Proof. Throughout let t ∈ [0, T ] and p ≥ 2. Since all β -norms are equivalent, it suffices to show the result for β = 0. We drop the β-subscripts in the following. The pair (Y, Z) satisfies

Y

t

= ξ + Z

T

t

f s, (Y · α

Y

)(s), (Z · α

Z

)(s) ds −

Z

T t

Z

s

dW

s

, hence we have

sup

0≤t≤T

|Y

_t

| ≤ |ξ| + Z

T

0

f s, (Y · α

Y

)(s), (Z · α

Z

)(s)

ds + sup

0≤t≤T

Z

T t

Z

_s

dW

_s

.

Combining the fact of Z ∈ H

^p

with the inequalities by Young, Doob and Burkholder-Davis- Gundy (BDG), we obtain

E h

sup

0≤t≤T

Z

T t

Z

s

dW

s

2

p/2

i

≤ 2

^p/2

E h

Z

T

0

Z

_s

dW

_s

2

+ sup

0≤t≤T

Z

t 0

Z

_s

dW

_s

2

p/2

i

≤ 2

^p/2

E h

2 sup

0≤t≤T

Z

t 0

Z

s

dW

s

2

p/2

i

= 2

^p

E h

sup

0≤t≤T

Z

t 0

Z

s

dW

s

p

i

≤ 2

^p

C

_p

E h Z

T

0

|Z

_s

|

²

ds

p/2

i

< ∞.

Next observe that by the Lipschitz property of the generator f (notice that (H2) implies (H2’)), it follows that

Z

T 0

f s, (Y · α

Y

)(s), (Z · α

Z

)(s)

2

ds

p/2

≤ 2

^p/2

Z

T

0

f (s, 0, 0)

2

ds + Z

T

0

f s, (Y · α

Y

)(s), (Z · α

Z

)(s)

− f (s, 0, 0)

2

ds

p/2

≤ 2

^p/2

2

^p/2−1

Z

T

0

f(s, 0, 0)

2

ds

p/2

+ L

Z

T 0

(|Y |

²

· α

Y

)(s) + (|Z |

²

· α

Z

)(s)

ds

p/2

. The second term in the bracket can be further estimated by

L

Z

T 0

(|Y |

²

· α

Y

)(s) + (|Z |

²

· α

Z

)(s) ds

^p/2

≤ 2

^p/2−1

L

^p/2

Z

T 0

(|Y |

²

· α

Y

)(s)ds

p/2

+ Z

T

0

(|Z|

²

· α

Z

)(s)ds

p/2

≤ 2

^p/2−1

L

^p/2

α

^p/2

Z

T 0

|Y

_s

|

²

ds

p/2

+ Z

T

0

|Z

_s

|

²

ds

p/2

,

where the last line follows from Lemma 1.1. This estimate together with (H3) yields E

h Z

T 0

f s, (Y · α

Y

)(s), (Z · α

Z

)(s)

2

ds

p/2

i

< ∞.

Using hypothesis (H1), i.e. that ξ is in ∈ L

^p

, we can conclude that Y ∈ S

^p

must hold.

Let us define the maximum of the weighted measure of [−T, 0) via

˜ α :=

Z

0

−T

e

^−βs

α

Y

(ds) ∨ Z

0

−T

e

^−βs

α

Z

(ds), β ≥ 0. (6)

The next results establish a priori estimates for the solutions of two time-delayed BSDEs as given by (4). We distinguish between the cases p = 2 and p > 2, and we shall start with the case p = 2 in the following

Proposition 2.5 (A priori estimates for p = 2). Let p = 2 and β, γ > 0. Consider i ∈ {1, 2}

and let (Y

ⁱ

, Z

ⁱ

) ∈ S

_β²

× H

²_β

be the solution of the delay BSDE (4) with terminal condition ξ

ⁱ

and generator f

ⁱ

satisfying (H1)-(H4). Denote by L > 0 the Lipschitz constant of f

¹

as given in (H2’) and set δY = Y

¹

− Y

²

, δZ = Z

¹

− Z

²

and δ

₂

f

_t

= f

¹

t, (Y

²

· α

Y

)(t), (Z

²

· α

Y

)(t)

− f

²

t, (Y

²

· α

Y

)(t), (Z

²

· α

Y

)(t)

for t ∈ [0, T ]. Assume that β, γ > 0 satisfy γ > αL ˜ and β − γ − αL ˜

γ > 0. (7)

Then there exists a constant C

₂

= C

₂

(β, γ, α, L) ˜ > 0 which depends on β, γ, α, L ˜ such that the following a priori estimate holds:

kδY k

²_S2

β

+ kδY k

²_H2

β

+ kδZk

²_H2 β

≤ C

₂

n E

h

e

^βT

|δY

_T

|

²

i + E

h Z

T 0

e

^βs

|δ

₂

f

_s

|

²

ds io

. (8)

Proof. Throughout let t ∈ [0, T ], i ∈ {1, 2} and define Γ

ⁱ

as in (5) for the pair (Y

ⁱ

, Z

ⁱ

). An application of Itˆ o’s formula to the semimartingale e

^βt

|δY

_t

|

²

for β > 0 yields

e

^βt

|δY

_t

|

²

+ Z

T

t

βe

^βs

|δY

_s

|

²

ds + Z

T

t

e

^βs

|δZ

_s

|

²

ds

= e

^βT

|δY

_T

|

²

+ Z

T

t

2e

^βs

δY

_s

, f

¹

(s, Γ

¹

(s)) − f

²

(s, Γ

²

(s)) ds −

Z

T t

2e

^βs

hδY

_s

, δZ

_s

dW

_s

i

≤ e

^βT

|δY

_T

|

²

+ Z

T

t

γe

^βs

|δY

_s

|

²

ds + Z

T

t

e

^βs

γ

f

¹

(s, Γ

¹

(s)) − f

¹

(s, Γ

²

(s))

2

ds + 2

Z

T t

e

^βs

δY

_s

, δ

₂

f

_s

ds −

Z

T t

2e

^βs

hδY

_s

, δZ

_s

dW

_s

i.

where the last inequality results from Young’s inequality for some γ > 0. Reorganizing and taking condition (H2’) for the generator f

¹

into account, we get

e

^βt

|δY

_t

|

²

+ Z

T

t

(β − γ)e

^βs

|δY

_s

|

²

ds + Z

T

t

e

^βs

|δZ

_s

|

²

ds

≤ e

^βT

|δY

_T

|

²

+ Z

T

t

e

^βs

γ L h

(|δY |

²

· α

Y

)(s) + (|δZ|

²

· α

Z

)(s) i ds + 2

Z

T t

e

^βs

δY

s

, δ

2

f

s

ds − Z

T

t

2e

^βs

hδY

_s

, δZ

s

idW

_s

.

By a change of integration order argument similar to that in the proof of Lemma 1.1 we obtain for j ∈ {

Y

,

Z

} and φ

^Y

= δY , φ

^Z

= δZ

Z

T t

e

^βs

(|φ

^j

|

²

· α

j

)(s)ds

= Z

T

t

Z

0

−T

e

^β(s+v)

e

^−βv

1

{s+v≥0}

|φ

^j_s+v

|

²

α

_j

(dv)ds

= Z

₀

−T

Z

_T+v

(t+v)∨0

e

^βr

e

^−βv

1

{r≥0}

|φ

^j_r

|

²

dr α

_j

(dv) = Z

_T

0

Z

(r−t)∧0 r−T

e

^βr

e

^−βv

|φ

^j_r

|

²

α

_j

(dv) dr

= Z

T

0

e

^βr

|φ

^j_r

|

²

Z

0

−T

e

^−βv

α

j

(dv) dr ≤

Z

T 0

˜

αe

^βr

|φ

^j_r

|

²

dr, (9)

with ˜ α given by (6). Continuing the inequality from above we get e

^βt

|δY

_t

|

²

+

Z

T t

(β − γ )e

^βs

|δY

_s

|

²

ds + Z

T

t

e

^βs

|δZ

_s

|

²

ds ≤ e

^βT

|δY

_T

|

²

+ 2 Z

T

t

e

^βs

δY

s

, δ

2

f

s

ds +

Z

T 0

˜ αL

γ e

^βs

|δY

_s

|

²

+ |δZ

_s

|

²

ds −

Z

T t

2e

^βs

hδY

_s

, δZ

_s

dW

_s

i. (10) Putting t = 0 and taking expectations yields

β − γ − αL ˜ γ

E

h Z

_T

0

e

^βs

|δY

_s

|

²

ds i

+ 1 − αL ˜ γ

E

h Z

_T

0

e

^βs

|δZ

_s

|

²

ds i

≤ E h

e

^βT

|δY

_T

|

²

i + 2E

h Z

T 0

e

^βs

δY

s

, δ

2

f

s

ds i

≤ E h

e

^βT

|δY

_T

|

²

i + 2 E

h sup

0≤t≤T

e

^β2t

|δY

_t

| Z

T

0

e

^β2s

|δ

₂

f

_s

|ds i

≤ E h

e

^βT

|δY

_T

|

²

i + γ

⁰

E

h sup

0≤t≤T

e

^βt

|δY

_t

|

²

i + 1

γ

⁰

E h Z

T

0

e

^β2s

|δ

₂

f

_s

|ds

2

i

where we have used Young’s inequality with some γ

⁰

> 0 to be specified later. From the last expression we deduce

kδY k

²_H2 β

+ kδZk

²_H2 β

≤ C n E

h

e

^βT

|δY

_T

|

²

i

+ γ

⁰

kδY k

²_S2 β

+ 1 γ

⁰

E

h Z

T 0

e

^β2s

|δ

₂

f

_s

|ds

2

o

, (11) where C > 0 is a constant depending β, γ, α, L. In order to obtain the ˜ S

_β²

-estimate for δY , we observe that we have

δY

_t

≤ δY

_T

+ Z

T

t

f

¹

s, Γ

¹

(s)

− f

¹

s, Γ

²

(s) ds +

Z

T t

δ

₂

f

_s

ds −

Z

T t

δZ

_s

dW

_s

.

Multiplying by the monotone increasing function e

^β2t

and taking the conditional expectation with respect to F

_t

, we get

e

^β2t

δY

_t

≤ E

e

^β2t

|δY

_T

| + e

^β2t

Z

T

t

f

¹

s, Γ

¹

(s)

− f

¹

s, Γ

²

(s)

ds + e

^β2t

Z

T

t

δ

₂

f

_s

ds

F

_t

≤ E

e

^β2T

|δY

_T

| + Z

T

t

e

^β2s

f

¹

s, Γ

¹

(s)

− f

¹

s, Γ

²

(s) ds +

Z

t 0

e

^β2s

f

¹

s, Γ

¹

(s)

− f

¹

s, Γ

²

(s) ds +

Z

T t

e

^β2s

δ

2

f

s

ds +

Z

t 0

e

^β2s

δ

2

f

s

ds

F

_t

= E

e

^β2T

|δY

_T

| + Z

T

0

e

^β2s

f

¹

s, Γ

¹

(s)

− f

¹

s, Γ

²

(s) ds +

Z

T 0

e

^β2s

δ

2

f

s

ds F

_t

. Using Doob’s inequality, we obtain

kδY k

²_S2 β

≤ 4 E h

E h

e

^β2T

|δY

_T

| + Z

T

0

e

^β2s

f

¹

s, Γ

¹

(s)

− f

¹

s, Γ

²

(s) ds +

Z

T 0

e

^β2s

δ

2

f

s

ds

F

_T

i

2

i

≤ 12 E h

e

^βT

|δY

_T

|

²

+ T Z

T

0

e

^βs

f

¹

s, Γ

¹

(s)

− f

¹

s, Γ

²

(s)

2

ds + Z

T

0

e

^β2s

δ

₂

f

_s

ds

2

i

,

where the last line follows by Jensen’s inequality. Since f

¹

satisfies (H2’), an application of Lemma 1.1 yields

kδY k

²_S2 β

≤ 12 n

E h

e

^βT

|δY

_T

|

²

i

+ ˜ αT L

kδY k

²_H2

β

+ kδZ k

²_H2 β

+ E

h Z

T 0

e

^β2s

δ

2

f

s

ds

2

io . Hence, plugging into (11) we find

1 − 12Cγ

⁰

αT L ˜ E

h sup

0≤t≤T

e

^βt

|δY

_t

|

²

i

≤ 12 n

1 + C αT L ˜ E

h

e

^βT

|δY

_T

|

²

i

+ 1 + Cγ

⁰⁻¹

αT L ˜ E

h Z

T 0

e

^β2s

δ

2

f

s

ds

2

io . Choosing γ

⁰

small enough so that (1 − 12Cγ

⁰

αT L) ˜ > 0 is satisfied, we conclude that estimate (8) holds for a constant C

2

= C

2

(β, γ, α, L). ˜

The proof for the case p > 2 is more involved and utilizes techniques from the proof of Proposition 2.5. The main reason for the proof to be more involved can be seen in (10). Usually the dynamics of Y

_t

is described by integrals over the interval [t, T ] but for delay BSDEs we see from (10) that the dynamics of Y

_t

depends also on a integral over the whole interval [0, T ].

Proposition 2.6. Let p > 2 and β, γ > 0. Consider i ∈ {1, 2} and denote by (Y

ⁱ

, Z

ⁱ

) ∈ S

_β^p

×H

_β^p

the solution of the delay BSDE (4) with terminal condition ξ

ⁱ

and generator f

ⁱ

satisfying (H1)- (H4). Denote by L > 0 the Lipschitz constant of f

¹

in (H2’) and set δY = Y

¹

−Y

²

, δZ = Z

¹

−Z

²

and δ

2

f

t

= f

¹

t, Y

²

(t), Z

²

(t)

− f

²

t, Y

²

(t), Z

²

(t)

. Assume that β, γ > 0 satisfy (7). Then there exist constants γ

2

, γ

3

> 0 such that for L and T small enough (i.e. chosen such that the constants D

_i

, i ∈ {1, . . . , 5}, specified in the proof are positive) there exists a constant C

_p

= C

_p

(β, γ, γ

₂

, γ

₃

, α, L, T ˜ ) > 0 satisfying the a priori estimate

kδY k

^p_Sp β

+ kδY k

^p_Hp β

+ kδZk

^p_Hp β

≤ C

_p

n E

h

e

^βT

|δY

_T

|

²

p/2

i + E

h Z

T 0

e

^β2s

|δ

₂

f

_s

|ds

p

io

. (12) Proof. Throughout let t ∈ [0, T ], i ∈ {1, 2} and set D

₁

:= β − γ −

^αL˜_γ

and D

₂

:= 1 −

^αL˜_γ

. Recall (10) from Proposition 2.5:

e

^βt

|δY

_t

|

²

+ Z

T

t

(β − γ )e

^βs

|δY

_s

|

²

ds + Z

T

t

e

^βs

|δZ

_s

|

²

ds ≤ e

^βT

|δY

_T

|

²

+ 2 Z

T

t

e

^βs

δY

s

, δ

2

f

s

ds +

Z

T 0

˜ αL

γ e

^βs

|δY

_s

|

²

+ |δZ

_s

|

²

ds −

Z

T t

2e

^βs

hδY

_s

, δZ

_s

idW

_s

. (13) We say that a constant C > 0 depends on the data if C = C(β, γ, γ

2

, α, L, T ˜ ). We carry out the proof in several steps.

Step 1: We claim that E

Z

T 0

e

^βs

|δZ

_s

|

²

ds

p/2

≤ D

⁻¹₃

n

2

^p/2−1

E h

e

^βT

|δY

_T

|

²

p/2

i

+ 2

^3p/2−2

d

_p/2

γ

2

kδY k

^p_Sp β

+ 2

^3p/2−2

E h

Z

T

0

e

^βs

δY

_s

, δ

₂

f

_s

ds

p/2

io

, (14)

with D

3

:= 1 −

^αL˜_γ

p/2

−

^23p/2−2_γ

2

d

_p/2

, where d

_p/2

> 0 is a given constant appearing in the

Burkholder-Davis-Gundy inequality which only depends on p > 2. Estimate (14) can be deduced

as follows: putting t = 0 in (13) and noticing that by (7) the constants D

1

and D

2

are positive we get

1 − αL ˜ γ

Z

_T

0

e

^βs

|δZ

_s

|

²

ds ≤ β − γ − αL ˜ γ

Z

_T

0

e

^βs

|δY

_s

|

²

ds + 1 − αL ˜ γ

Z

_T

0

e

^βs

|δZ

_s

|

²

ds

≤ e

^βT

|δY

_T

|

²

+ 2 Z

T

0

e

^βs

δY

s

, δ

2

f

s

ds − 2 Z

T

0

e

^βs

hδY

_s

, δZ

s

idW

_s

. Now raising both sides to the power p/2 > 1, making use of the fact that for a, b, c ∈ R

a + 2b − 2c

p/2

≤ 2

^p/2−1

|a|

^p/2

+ |2b − 2c|

^p/2

≤ 2

^p/2−1

|a|

^p/2

+ 2

^p/2−1

|2b|

^p/2

+ |2c|

^p/2

= 2

^p/2−1

|a|

^p/2

+ 2

^3p/2−2

|b|

^p/2

+ 2

^3p/2−2

|c|

^p/2

and taking expectations, we get

1 − αL ˜ γ

p/2

E h Z

T

0

e

^βs

|δZ

_s

|

²

ds

p/2

i

≤ 2

^p/2−1

E h

e

^βT

|δY

_T

|

²

p/2

i + 2

^3p/2−2

E

h

Z

T 0

e

^βs

δY

_s

, δ

₂

f

_s

ds

p/2

i

+ 2

^3p/2−2

E h

Z

T 0

e

^βs

hδY

_s

, δZ

_s

dW

_s

i

p/2

i

. (15) An application of the BDG inequality yields that for a given constant d

_p/2

> 0, we have

E h

Z

T 0

e

^βs

hδY

_s

, δZ

_s

dW

_s

i

p/2

i

≤ d

_p/2

E h Z

T

0

e

^2βs

|δY

_s

|

²

|δZ

_s

|

²

ds

p/4

i

≤ d

_p/2

E h

sup

0≤t≤T

e

^βt

|δY

_t

|

²

p/4

Z

T 0

e

^βs

|δZ

_s

|

²

ds

p/4

i

≤ d

_p/2

n

γ

₂

kδY k

^p_Sp β

+ 1 γ

2

kδZk

^p_Hp β

o

, (16)

where the last line follows from Young’s inequality with γ

₂

> 0. Now we choose γ

₂

> 0 such that D

₃

> 0. Note that with this γ

₂

, if one replaces L by L

⁰

with L

⁰

< L then the quantity

1 −

^{2 ˜αL}_γ ⁰

p/2

−

^23p/2−2_γ

2

d

_p/2

> D

₃

is still positive. Plugging (16) into (15), we get D

3

kδZk

^p_Hp

β

≤ 2

^p/2−1

E h

e

^βT

|δY

_T

|

²

p/2

i

+ 2

^3p/2−2

E h

Z

T 0

e

^βs

δY

s

, δ

2

f

s

ds

p/2

i

+ 2

^3p/2−2

d

_p/2

γ

₂

kδY k

^p_Sp β

, which proves the claim.

Step 2: We claim that D

4

kδY k

^p_Sp

β

≤ p

p − 2

p/2

(

2

^p−2

+ 2

^3p/2−3

αL ˜ γ

p/2

D

₃⁻¹

E

h

e

^βT

|δY

_T

|

²

p/2

i + 2

^3p/2−2

+ 2

^5p/2−4

αL ˜

γ

p/2

D

₃⁻¹

E

h Z

T 0

e

^βs

δY

s

, δ

2

f

s

ds

p/2

i

)

, (17)

holds for

D

₄

:= 1 − 2

^5/4p−4

γ

₂

d

_p/2

p p − 2

p/2

αL ˜ γ

p/2

D

⁻¹₃

− αL ˜

γ T

p/2

p p − 2

p/2

2

^p−2

.

Before showing this estimate we stress that we can choose L and T such that D

4

> 0. More precisely, the constants β, γ, γ

₂

are already fixed and depend on L. But if one replaces L by L

⁰

with L

⁰

< L then it is clear that D

₁⁰

, D

₂⁰

, D

⁰₃

> 0 (where D

_i⁰

denotes D

_i

with L replaced by L

⁰

) for the same β, γ, γ

2

since D

_i⁰

> D

i

, i = 1, 2, 3. Thus we can choose L and T small enough making D

₁

, . . . , D

₄

> 0.

Now we prove (17). For this we go back to (13), take conditional expectation with respect to F

_t

and the sup in t ∈ [0, T ], raise to the power p/2, apply Doob’s inequality and find

E h

sup

0≤t≤T

e

^βt

|δY

_t

|

²

p/2

i

≤ E h

sup

0≤t≤T

E

h

e

^βT

|δY

_T

|

²

+ 2 Z

T

0

e

^βs

δY

s

, δ

2

f

s

ds +

Z

T 0

˜ αL

γ e

^βs

|δY

_s

|

²

+ |δZ

_s

|

²

ds

F

_t

i

p/2

i

≤ p

p − 2

p/2

E

e

^βT

|δY

_T

|

²

+ 2 Z

T

0

e

^βs

δY

s

, δ

2

f

s

ds +

Z

T 0

˜ αL

γ e

^βs

|δY

_s

|

²

+ |δZ

_s

|

²

ds

p/2

≤ p

p − 2

p/2

n

2

^p−2

E h

e

^βT

|δY

_T

|

²

p/2

i

+ 2

^3p/2−2

E h Z

T

0

e

^βs

δY

s

, δ

2

f

s

ds

p/2

i

+ 2

^p−2

E h Z

T

0

˜ αL

γ e

^βs

|δY

_s

|

²

ds

p/2

i

+ 2

^p−2

E h Z

T

0

˜ αL

γ e

^βs

|δZ

_s

|

²

ds

p/2

io

. (18)

Note that we made use of the fact that for a, b, c, d ∈ R and p > 2, we have

a + 2b + c + d

p/2

≤ 2

^p/2−1

|a + 2b|

^p/2

+ |c + d|

^p/2

≤ 2

^p−2

|a|

^p/2

+ 2

^3p/2−2

|b|

^p/2

+ 2

^p−2

|c|

^p/2

+ 2

^p−2

|d|

^p/2

. Plugging (14) into (18), we get

kδY k

^p_Sp β

≤ p

p − 2

p/2

2

^p−2

E h

e

^βT

|δY

_T

|

²

p/2

i

+ 2

^3p/2−2

E h Z

T

0

e

^βs

δY

_s

, δ

₂

f

_s

ds

p/2

i

+ 2

^p−2

αL ˜ γ

p/2

kδY k

^p_Hp β

+ αL ˜ γ

p/2

D

₃⁻¹

× 2

^p−2

n

2

^p/2−1

E h

e

^βT

|δY

_T

|

²

p/2

i + 2

^3p/2−2

E

h Z

_T

0

e

^βs

δY

_s

, δ

₂

f

_s

ds

p/2

i

+ 2

^3p/2−2

d

_p/2

γ

₂

kδY k

^p_Sp β

o

≤ p

p − 2

p/2

2

^p−2

+ 2

^3p/2−3

αL ˜ γ

p/2

D

⁻¹₃

E

h

e

^βT

|δY

_T

|

²

p/2

i + 2

^3/2p−2

+ 2

^5p/2−4

αL ˜

γ

p/2

D

⁻¹₃

E

h Z

T 0

e

^βs

δY

_s

, δ

₂

f

_s

ds

p/2

i

+

2

^p−2

αL ˜ γ T

p/2

+ 2

^5p/2−4

αL ˜ γ

p/2

D

⁻¹₃

d

_p/2

γ

₂

kδY k

^p_Sp β

,

from which (17) readily follows.

Step 3: At this stage, estimating E R

T

0

e

^βs

δY

s

, δ

2

f

s

ds

p/2

i

will yield (12) which follows as a consequence from equation(17). Applying Young’s inequality with some γ

3

> 0 we get

E Z

T 0

e

^βs

δY

s

, δ

2

f

s

ds

p/2

i

≤ E h Z

T

0

e

^βs

|δY

_s

| |δ

₂

f

s

|ds

p/2

i

≤ E h

sup

0≤t≤T

e

^β2t

|δY

_t

|

p/2

Z

T 0

e

^β2s

δ

₂

f

_s

|ds

p/2

i

≤ γ

3

kδY k

^p_Sp β

+ 1 γ

3

E

h Z

T 0

e

^β2s

δ

2

f

s

|ds

p

i

. (19)

Together with (17), inequality (19) leads to D

₅

kδY k

^p_Sp

β

≤ p

p − 2

p/2

(

2

^p−2

+ 2

^3p/2−3

αL ˜ γ

p/2

D

⁻¹₃

E

e

^βT

|δY

_T

|

²

p/2

+ 2

^3/2p−2

+ 2

^5p/2−4

αL ˜

γ

p/2

D

⁻¹₃

γ

₃⁻¹

E

h Z

T 0

e

^β2s

|δ

₂

f

_s

|ds

p

i )

, (20)

where D

5

:= D

4

−

_p−2^p

p/2

γ

3

2

^3/2p−2

+ 2

^{5p/2−4 αL˜}_γ

p/2

D

₃⁻¹

and γ

3

> 0 is chosen such that D

5

> 0. Finally putting together the previous steps, the estimate (12) follows. Note that (20) implies

kδY k

^p_Sp β

≤ C

_p

n E

h

e

^βT

|δY

_T

|

²

p/2

i + E

h Z

T 0

e

^β2s

|δ

₂

f

_s

|ds

p

io

, (21)

where C

_p

> 0 is a constant depending on the data. This leads to kδY k

^p_Hp

β

≤ T

^p/2

kδY k

^p_Sp β

≤ C

p

n E

h

e

^βT

|δY

_T

|

²

p/2

i + E

h Z

T 0

e

^β2s

|δ

₂

f

s

|ds

p

io . Finally, observing that (14) from step 1 yields

kδZk

^p_Hp β

≤ C

_p

n E

h

e

^βT

|δY

_T

|

²

p/2

i + E

h Z

T 0

e

^β2s

|δ

₂

f

_s

|ds

p

i

+ kδY k

^p_Sp β

o , and applying (21), it follows that (12) is valid. This finishes the proof.

As a by-product of the two previous theorems we obtain under their respective assumptions a result on the moment estimates for the solution of BSDE (4).

Corollary 2.7 (Moment estimates). Under the assumptions of Proposition 8 and Proposition 2.6, we have for p ≥ 2

kY k

^p_Sp β

+ kY k

^p_Hp β

+ kZ k

^p_Hp β

≤ C

p

n E

h

e

^βT

|δY

_T

|

²

p/2

i + E

h Z

T 0

e

^βs

|f (s, 0, 0)|

²

ds

p

io

.

The moment and a priori estimates in Delong and Imkeller (2010a) are tailor-made for a Picard

iteration procedure in H

²

× H

²

. The right-hand side of their estimates depends on the solution

process but such an estimate suffices in their context. The a priori estimates from Proposition

2.5 and Proposition 2.6 are somewhat “sharper” in the sense that they are the usual a priori

estimates one expects to obtain when dealing with BSDE, i.e. they exhibit a right-hand side which solely depends on the data δY

_T

and δ

₂

f

_t

.

With estimate (12) at hand, we now proceed to show the existence and uniqueness of solutions to (4) in S

_β^p

× H

^p_β

for p > 2. For p = 2, Theorem 2.1 from Delong and Imkeller (2010a) (recalled in our Theorem 2.1) yields a sufficient condition which guarantees the standard Picard iteration to converge and proves the existence and uniqueness of solutions to (4) . We will show in the following result that for p > 2, the convergence of the same Picard iteration is retained. What is needed to achieve this goal is to put up some extra effort for showing that the Picard iterates (Y

ⁿ

, Z

ⁿ

) satisfy the corresponding S

_β^p

, H

_β^p

-integrability properties.

Theorem 2.8. Let p > 2 and assume that (H1)-(H4) hold. Assume that β, γ, L, T, γ

₂

, γ

₃

are chosen like in Proposition 2.6 such that condition (7) holds and let C

p

denote the constant appearing in the a priori estimate (12). If

2

^p/2−1

C

_p

L α ˜

p/2

max{1, T }

^p

< 1, (22)

where α ˜ is given by (6), then the BSDE (4) admits a unique solution in S

_β^p

× H

_β^p

.

Proof. Throughout let t ∈ [0, T ] and p > 2. The proof is based on the standard Picard iteration:

we initialize by Y

⁰

= 0 and Z

⁰

= 0 and define recursively Y

_tⁿ⁺¹

= ξ +

Z

T t

f s, Γ

ⁿ

(s) ds −

Z

T t

Z

_sⁿ⁺¹

dW

_s

, 0 ≤ t ≤ T, (23) with Γ

ⁿ

(s) = R

0

−T

Y

_s+vⁿ

α

Y

(dv), R

0

−T

Z

_s+vⁿ

α

Z

(dv)

for s ∈ [0, T ] and n ∈ N . In the following, C > 0 will denote some generic constant which may vary from line to line but always independent of n ∈ N . We proceed by induction. For n ≥ 1, assume that (Y

ⁿ

, Z

ⁿ

) ∈ S

_β^p

× H

^p_β

is already shown, and we prove that (23) has a unique solution (Y

ⁿ⁺¹

, Z

ⁿ⁺¹

) ∈ S

_β^p

×H

^p_β

. Note that because of

E h Z

T

0

|f (s, Γ

ⁿ

(s))|ds

p

i

≤ E h Z

T

0

|f (s, 0, 0)|ds + Z

T

0

|f(s, Γ

ⁿ

(s)) − f (s, 0, 0)|ds

p

i

≤ 2

^p−1

E h Z

T

0

|f (s, 0, 0)|ds

p

+ T

Z

T 0

|f (s, Γ

ⁿ

(s)) − f (s, 0, 0)|

²

ds

p/2

i

≤ 2

^p−1

E h Z

T

0

|f(s, 0, 0)|ds

p

+ L

^p/2

T

^p/2

n Z

T 0

Z

0

−T

|Y

_s+vⁿ

|

²

α

Y

(dv)ds + Z

T

0

Z

0

−T

|Z

_s+vⁿ

|

²

α

Z

(dv)ds o

p/2

i

≤ 2

^p−1

E h Z

_T

0

|f (s, 0, 0)|ds

p

+ (αKT )

^p/2

n Z

_T

0

|Y

_sⁿ

|

²

ds + Z

_T

0

|Z

_sⁿ

|

²

ds o

p/2

i

≤ 2

^p−1

E h Z

T

0

|f (s, 0, 0)|ds

p

i

+ 2

^p/2−1

(2αKT )

^p/2

T

^p/2

kY

ⁿ

k

^p_Sp

0

+ kZ

ⁿ

k

^p_Hp 0

< ∞, (24)

the martingale representation yields a uniquely determined process Z

ⁿ⁺¹

∈ H

²₀

such that E

h ξ +

Z

T 0

f(s, Γ

ⁿ

(s))ds F

_t

i

= E ξ +

Z

T 0

f(s, Γ

ⁿ

(s))ds +

Z

T 0

Z

_sⁿ⁺¹

dW

_s

.