Random walk approximation of BSDEs with Hölder continuous terminal condition

(1)

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Preprint submitted on 14 Sep 2018

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Random walk approximation of BSDEs with Hölder continuous terminal condition

Christel Geiss, Céline Labart, Antti Luoto

To cite this version:

Christel Geiss, Céline Labart, Antti Luoto. Random walk approximation of BSDEs with Hölder

continuous terminal condition. 2018. �hal-01818668v2�

(2)

Random walk approximation of BSDEs with Hölder continuous terminal condition

Christel Geiss

¹

, Céline Labart

²

, Antti Luoto

³

Abstract

In this paper we consider the random walk approximation of the solution of a Markovian BSDE whose terminal condition is a locally Hölder continuous function of the Brownian motion.

We state the rate of the L

2

-convergence of the approximated solution to the true one. The proof relies in part on growth and smoothness properties of the solution u of the associated PDE. Here we improve existing results by showing some properties of the second derivative of u in space.

Keywords : Backward stochastic differential equations, numerical scheme, random walk approximation, speed of convergence

MSC codes : 65C30 60H35 60G50 65G99

1 Introduction

Let (Ω, F , P ) be a complete probability space carrying the standard Brownian motion B = (B

_t

)

_t≥0

and assume (F

_t

)

t≥0

is the augmented natural filtration. We consider the following backward stochastic differential equation (BSDE for short)

Y

s

= g(B

T

) + Z

T

s

f (r, B

r

, Y

r

, Z

r

)dr − Z

T

s

Z

r

dB

r

, 0 ≤ s ≤ T, (1) where f is Lipschitz continuous and g is a locally α-Hölder continuous and polynomially bounded function (see (3)). In this paper we are interested in the L

₂

-convergence of the numerical approximation of (1) by using a random walk. First results dealing with the numerical approximation of BSDEs date back to the late 1990s. Bally (see [2]) was the first to consider this problem by introducing random discretization, namely the jump times of a Poisson process. In his PhD thesis, Chevance (see [17]) proposed the following discretization

y

k

= E (y

k+1

+ hf (y

k+1

)|F

_kⁿ

), k = n − 1, · · · , 0, n ∈ N

^∗

and proved the convergence of (Y

_tⁿ

)

_t

:= (y

_[t/h]

)

_t

to Y . At the same time, Coquet, Mackevičius and Mémin [18] proved the convergence of Y

ⁿ

by using convergence of filtrations, still in the case of

1

Department of Mathematics and Statistics, P.O.Box 35 (MaD), FI-40014 University of Jyvaskyla, Finland christel.geiss@jyu.fi

2

Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France celine.labart@univ-smb.fr

3

Department of Mathematics and Statistics, P.O.Box 35 (MaD), FI-40014 University of Jyvaskyla, Finland

antti.k.luoto@student.jyu.fi

(3)

a generator independent from z. The general case (f depends on z, terminal condition ξ ∈ L

₂

) has been studied by Briand, Delyon and Mémin (see [5]). In that paper the authors define an approximated solution (Y

ⁿ

, Z

ⁿ

) based on random walk and prove weak convergence to (Y, Z) using convergence of filtrations. We also refer to [27], [29], [30], [31] for other numerical methods for BSDEs which use a random walk approach. The rate of convergence of this method was left as an open problem.

Introducing instead of random walk an approach based on the dynamic programming equation, Bouchard and Touzi in [8] and Zhang in [35] managed to establish a rate of convergence. However, to be fully implementable, this algorithm requires to have a good approximation of its associated conditional expectation. For this, various methods have been developed (see [24], [19], [15]). For- ward methods have also been introduced to approximate (1) : a branching diffusion method (see [26]), a multilevel Picard approximation (see [34]) and Wiener chaos expansion (see [7]). Many ex- tensions of (1) have also been considered : high order schemes (see [11], [10]), schemes for reflected BSDEs (see [3], [14]), for fully-coupled BSDEs (see [21], [9]), for quadratic BSDEs (see [13]), for BSDEs with jumps (see [23]) and for McKean-Vlasov BSDEs (see [1], [16], [12]).

From a numerical point of view, the random walk is of course not competitive with recent methods listed above. We emphasize that the aim of this paper is to give the convergence rate of the initial method based on random walk, which, to the best of our knowledge, has not been done so far.

As in [5], let us introduce the following approximation of B, based on a random walk:

B

_tⁿ

=

√ h

[t/h]

X

i=1

ε

i

, 0 ≤ t ≤ T,

where h =

^T_n

(n ∈ N

^∗

) and (ε

_i

)

_i=1,2,...

is a sequence of i.i.d. Rademacher random variables. Consider the following approximated solution (Y

ⁿ

, Z

ⁿ

) of (Y, Z)

Y

_tⁿ

k

= g(B

_Tⁿ

) + h

n−1

X

m=k

f (t

_m+1

, B

_tⁿ_m

, Y

_tⁿ_m

, Z

_tⁿ_m

) − √ h

n−1

X

m=k

Z

_tⁿ_m

ε

_m+1

, 0 ≤ k ≤ n − 1. (2) The main result of our paper gives the rate of convergence in L

₂

-norm of Y

_vⁿ

− Y

_v

and Z

_vⁿ

− Z

_v

for each v ∈ [0, T ) (see Theorem 3.1). Basically, we get that the L

₂

-norm of the error on Y is of order h

^α⁴

and the L

₂

-norm of the error on Z is of order

^h

α

√ 4

T−v

. The proof of this result is based on several ingredients. In particular, we need some estimates on the bound of the first and second derivatives of the solution of the PDE associated to the BSDE (1). We establish these bounds in the case of a forward backward SDE (FBSDE for short) whose terminal condition satisfies the Hölder continuity condition (3). This result extends Zhang [36, Theorem 3.2].

The rest of the paper is organized as follows. Section 2 introduces notations, assumptions and the representation for Z and Z

ⁿ

based on the Malliavin weights. Section 3 states the rate of convergence of the error on Y and Z in L

2

-norm, which is the main result of the paper. Section 4 presents numerical simulations and Section 5 recalls some properties of Malliavin weights, of the regularity of solutions to FBSDEs with a locally Hölder continuous terminal condition function and states some properties of the solutions to the PDEs associated to these FBSDEs.

2 Preliminaries

This section is dedicated to notations, assumptions and the representation of Z and Z

ⁿ

using the

Malliavin weights.

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Notation:

• G

_k

:= σ(ε

i

: 1 ≤ i ≤ k) and G

₀

= {∅, Ω}. The associated discrete-time random walk (B

ⁿ_t_k

)

ⁿ_k=0

is (G

_k

)

ⁿ_k=0

-adapted.

• k · k

_p

:= k · k

_Lp(P)

for p ≥ 1 and for p = 2 simply k · k. constant.

Assumption 2.1.

• g is locally Hölder continuous with order α ∈ (0, 1] and polynomially bounded (p

0

≥ 0, C

g

> 0) in the following sense

∀(x, y) ∈ R

²

, |g(x) − g(y)| ≤ C

g

(1 + |x|

^p⁰

+ |y|

^p⁰

)|x − y|

^α

. (3)

• The function [0, T ] × R

³

: (t, x, y, z) 7→ f (t, x, y, z) satisfies

|f (t, x, y, z) − f (t

⁰

, x

⁰

, y

⁰

, z

⁰

)| ≤ L

_f

( √

t − t

⁰

+ |x − x

⁰

| + |y − y

⁰

| + |z − z

⁰

|). (4) Notice that (3) implies

|g(x)| ≤ K(1 + |x|

^p⁰⁺¹

) =: Ψ(x). (5) In the rest of the paper, the study of the error (Y

ⁿ

− Y, Z

ⁿ

− Z) will either rely on (2) or on its integral version:

Y

_sⁿ

= g(B

_Tⁿ

) + Z

(s,T]

f (r, B

_rⁿ−

, Y

_rⁿ−

, Z

_rⁿ−

)d[B

ⁿ

, B

ⁿ

]

_r

− Z

(s,T]

Z

_rⁿ−

dB

ⁿ_r

, 0 ≤ s ≤ T, (6) where the backward equation (6) arises from (2) by setting Y

_rⁿ

:= Y

_tⁿ_m

and Z

_rⁿ

:= Z

_tⁿ_m

for r ∈ [t

m

, t

m+1

). For n large enough, (6) has a unique solution (Y

ⁿ

, Z

ⁿ

), and (Y

_tⁿ_m

, Z

_tⁿ_m

)

ⁿ⁻¹_m=0

is adapted to the filtration (G

_m

)

ⁿ⁻¹_m=0

. Let us now introduce the Malliavin representations for Z and Z

ⁿ

. They are the cornerstone of our study of the error on Z .

2.1 Representations for Z and Z

ⁿ

We will use the representation (see Ma and Zhang [28, Theorem 4.2]) Z

_t

= E

t

g(B

_T

)N

_T^t

+

Z

T t

f (s, B

_s

, Y

_s

, Z

_s

)N

_s^t

ds

!

, 0 ≤ t ≤ T, (7)

where E

t

[·] = E [·|F

_t

], and for all s ∈ (t, T ] we have N

_s^t

:= B

_s

− B

_t

s − t .

Lemma 2.2. Suppose that Assumption 2.1 holds. Then the process Z

ⁿ

given by (6) has the representation

Z

_tⁿ_k

= E

k

g(B

_Tⁿ

) B

_tⁿ_n

− B

ⁿ_t_k

t

_n

− t

_k

+ E

k



 h

n−1

X

m=k+1

f (t

m+1

, B

_tⁿ_m

, Y

_tⁿ_m

, Z

_tⁿ_m

) B

ⁿ_t_m

− B

_tⁿ_k

t

_m

− t

_k



 (8)

for k = 0, 1, . . . , n − 1, where E

k

[ · ] := E [ · |G

_k

].

(5)

Proof. We multiply equation (2) by ε

_k+1

and take the conditional expectation with respect to G

_k

. Since (Y

_tⁿ

k

, Z

_tⁿ

k

) is G

_k

-measurable, it holds for 0 ≤ k ≤ n − 1 that E

k

Y

_tⁿ_k

ε

_k+1

= E

k

(g(B

_Tⁿ

)ε

k+1

) + hE

k n−1

X

m=k

f(t

m+1

, B

ⁿ_t_m

, Y

_tⁿ_m

, Z

_tⁿ_m

)ε

k+1

!

− √ hE

k

n−1

X

m=k

Z

_tⁿ_m

ε

m+1

ε

k+1

!

=

√ h E

k

g(B

_Tⁿ

) B

_tⁿ_n

− B

_tⁿ_k

t

_n

− t

_k

+ h

^3/2

n−1

X

m=k+1

E

k

f (t

m+1

, B

_tⁿ_m

, Y

_tⁿ_m

, Z

_tⁿ_m

) B

_tⁿ_m

− B

ⁿ_t_k

t

_m

− t

_k

− √

hZ

_tⁿ_k

, (9) where the l.h.s. is equal to zero. Indeed, for m ≥ k + 1, we have

E

k

(Z

_tⁿ_m

ε

_m+1

ε

_k+1

) = E

k

(Z

_tⁿ_m

ε

_k+1

E

m

ε

_m+1

) = 0, and for m = k it holds E

k

(Z

_tⁿ_k

ε

²_k+1

) = Z

_tⁿ_k

. Moreover, the fact that B

_Tⁿ

= √

h ^P

ⁿ⁻¹_m=0

ε

_m+1

, where (ε

_m

)

_m=1,2...

are i.i.d., yields

E

k

(g(B

_Tⁿ

)ε

_k+1

) = E

k

g(B

_Tⁿ

)

n−1

X

m=k

ε

_k+1

n − k

!

= E

k

g(B

_Tⁿ

)

n−1

X

m=k

ε

_m+1

n − k

!

= √ h E

k

g(B

_Tⁿ

) B

_tⁿ_n

− B

ⁿ_t_k

t

n

− t

k

.

Similarly, for m ≥ k + 1, we get (using [5, Proposition 5.1], where it is stated that both Y

_tⁿ_m

and Z

_tⁿ_m

can be represented as functions of t

_m

and B

ⁿ_t_m

)

E

k

f(t

_m+1

, B

ⁿ_t_m

, Y

_tⁿ_m

, Z

_tⁿ_m

)ε

_k+1

=

√ h E

k

f (t

_m+1

, B

_tⁿ_m

, Y

_tⁿ_m

, Z

_tⁿ_m

) B

_tⁿ_m

− B

_tⁿ_k

t

_m

− t

_k

.

It remains to divide (9) by √

h and rearrange.

3 Main result

This section is devoted to the main result of the paper: the rate of the L

₂

-convergence of (Y

ⁿ

, Z

ⁿ

) to (Y, Z). The proof will rely on the fact that the random walk B

ⁿ

can be constructed from the Brownian motion B by Skorohod embedding. Let τ

0

:= 0 and define

τ

k

:= inf {t > τ

_k−1

: |B

_t

− B

τk−1

| =

√

h}, k ≥ 1.

Then (B

_τ_k

− B

_τ_k−1

)

^∞_k=1

is a sequence of i.i.d. random variables with P (B

_τ_k

− B

_τ_k−1

= ± √

h) =

¹₂

, which means that √

hε

_k

=

^d

B

τ_k

− B

τk−1

. We will use this random walk for our approximation, i.e.

we will require

B

_tⁿ

=

[t/h]

X

k=1

(B

_τ_k

− B

_τ_k−1

), 0 ≤ t ≤ T. (10)

Properties satisfied by τ

_k

and B

τ_k

are stated in Lemma A.1. We will denote by E

τ_k

the conditional

expectation w.r.t. F

_τ_k

.

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Theorem 3.1. Let Assumption 2.1 hold. If B

ⁿ

satisfies (10) then we have (for sufficiently large n) that

E |Y

_v

− Y

_vⁿ

|

²

≤ C

₀

h

^α²

for v ∈ [0, T ), E |Z

_v

− Z

_vⁿ

|

²

≤ C

0

h

^α²

T − t

k

+ C

1

h

^α²

(T − v)

¹⁻^α²

1

_v6=t_k

for v ∈ [t

k

, t

k+1

), k = 0, ..., n − 1, where we have the dependencies C

0

= C(T, p

0

, L

_f

, C

g

, C

_5.3^y

, C

_5.3^z

, K

_f

, c

5.4

, α), C

1

= C(T, p

0

, C

_5.3^z

, α) and K

_f

:= sup

_0≤t≤T

|f(t, 0, 0, 0)|.

Remark 3.2. Theorem 3.1 implies that sup

v∈[0,T)

E |Y

_v

− Y

_vⁿ

|

²

≤ C

₀

h

^α²

and E Z

T

0

|Z

_v

− Z

_vⁿ

|

²

dv ≤ C(C

₀

, C

₁

, β) h

^β

for β ∈ (0,

^α₂

).

Proof of Theorem 3.1. Let u : [0, T ) × R → R be the solution of the PDE associated to (1). Since by Theorem 5.4

Y

s

= u(s, B

s

), Z

s

= u

x

(s, B

s

), a.s.

we introduce

F(s, x) := f (s, x, u(s, x), u

_x

(s, x)),

so that F(s, B

s

) = f (s, B

s

, Y

s

, Z

s

). We first give some properties satisfied by F .

Lemma 3.3. If Assumption 2.1 holds then F is a Lipschitz continuous and polynomially bounded function in x :

|F (t, x

₁

) − F (t, x

₂

)| ≤ C(T, L

_f

, c

^2,3_5.4

)(1 + |x

₁

|

^p⁰⁺¹

+ |x

₂

|

^p⁰⁺¹

) |x

₁

− x

2

| (T − t)

¹⁻^α²

,

|F (t, x)| ≤ C(T, L

f

, c

^1,2_5.4

, K

f

) Ψ(x) (T − t)

^1−α²

, where Ψ(x) is given in (5).

Proof of Lemma 3.3. Thanks to the mean value theorem and Theorem 5.4-(ii-c) and (iii-b) we have for x

₁

, x

₂

∈ R that there exist ξ

₁

, ξ

₂

∈ [min{x

₁

, x

₂

}, max{x

₁

, x

₂

}] such that

|F (t, x

₁

) − F (t, x

₂

)| = |f (t, x

₁

, u(t, x

₁

), u

_x

(t, x

₁

)) − f (t, x

₂

, u(t, x

₂

), u

_x

(t, x

₂

))|

≤ L

_f

(|x

₁

− x

2

| + |u(t, x

₁

) − u(t, x

2

)| + |u

_x

(t, x

1

) − u

x

(t, x

2

)|)

≤ L

_f

1 + c

²_5.4

Ψ(ξ

₁

) (T − t)

^1−α²

+ c

³_5.4

Ψ(ξ

₂

) (T − t)

¹⁻^α²

!

|x

₁

− x

₂

|

≤ C(T, L

_f

, c

^2,3_5.4

)(1 + |x

₁

|

^p⁰⁺¹

+ |x

₂

|

^p⁰⁺¹

) |x

₁

− x

₂

|

(T − t)

¹⁻^α²

.

The second inequality can be shown similarly.

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For the estimate of E |Y

_t_k

− Y

_tⁿ_k

|

²

we will use (1) and (2): Since Y

_tⁿ_k

is F

_τ_k

-measurable we have kY

_t_k

− Y

_tⁿ_k

k ≤ k E

t_k

g(B

T

) − E

τ_k

g(B

_Tⁿ

)k

+

E

tk

Z

T tk

f (s, B

_s

, Y

_s

, Z

_s

)ds − h E

τk

n−1

X

m=k

f (t

_m+1

, B

_tⁿ_m

, Y

_tⁿ_m

, Z

_tⁿ_m

)

. (11) We frequently express conditional expectations with the help of an independent copy of B denoted by ˜ B, for example E

t

g(B

T

) = ˜ E g(B

t

+ ˜ B

T−t

).

By (3) and Lemma A.1,

k E

t_k

g(B

_T

) − E

τ_k

g(B

ⁿ_T

)k

²

= E | E ˜ g(B

t_k

+ ˜ B

_T−t_k

) − E ˜ g(B

τ_k

+ ˜ B

τ_˜n−k

)|

²

≤ ( E E ˜ (Ψ

₁

)

⁴

)

¹²

( E E ˜ |B

_t_k

− B

_τ_k

+ ˜ B

_T−t_k

− B ˜

_˜_τ_n−k

|

^4α

)

¹²

≤ C(C

_g

, T, p

₀

)(( E |B

_t_k

− B

_τ_k

|

^4α

)

¹²

+ ( E |B

_T_−t_k

− B

_τ_n−k

|

^4α

)

¹²

)

≤ C(C

g

, T, p

0

)h

^α²

, (12)

where Ψ

₁

:= C

_g

(1 + |B

_t_k

+ ˜ B

_T−t_k

|

^p⁰

+ |B

_τ_k

+ ˜ B

_˜_τ_n−k

|

^p⁰

). To estimate the other term in (11) we consider the decomposition

E

tk

f (s, B

s

, Y

s

, Z

s

) − E

τk

f (t

m+1

, B

_tⁿ_m

, Y

_tⁿ_m

, Z

_tⁿ_m

)

= ( E

tk

f (s, B

_s

, Y

_s

, Z

_s

) − E

tk

f (t

_m

, B

_t_m

, Y

_t_m

, Z

_t_m

)) + ( E

tk

F (t

_m

, B

_t_m

) − E

τk

F (t

_m

, B

_τ_m

))

+( E

τk

F (t

_m

, B

_τ_m

) − E

τk

F (t

_m

, B

_t_m

)) + ( E

τk

f (t

_m

, B

_t_m

, Y

_t_m

, Z

_t_m

) − E

τk

f (t

_m+1

, B

_tⁿ_m

, Y

_tⁿ_m

, Z

_tⁿ_m

))

=: D

₁

(s, m) + D

₂

(m) + ... + D

₄

(m) so that

E

t_k

Z

T tk

f (s, B

s

, Y

s

, Z

s

)ds − h E

τ_k n−1

X

m=k

f (t

m+1

, B

_tⁿ_m

, Y

_tⁿ_m

, Z

_tⁿ_m

)

≤

n−1

X

m=k

Z

tm+1

tm

D

₁

(s, m)ds

+ h

4

X

i=2

kD

_i

(m)k

! .

For D

1

we have by Theorem 5.3 that kD

₁

(s, m)k ≤ L

_f

( √

s − t

_m

+ kB

_s

− B

_t_m

k + kY

_s

− Y

_t_m

k + kZ

_s

− Z

_t_m

k)

≤ C(T, L

_f

, C

_5.3^y

, C

_5.3^z

, p

₀

) (T − s)

^α−2²

h

¹²

, (13) where the last inequality follows from kB

_s

− B

_t_m

k = √

s − t

_m

≤ h

¹²

for s ∈ [t

_m

, t

_m+1

] and kY

_s

− Y

_t_m

k + kZ

_s

− Z

_t_m

k ≤ ( E Ψ(B

_t_m

)

²

)

¹²

C

_5.3^y

Z

_s

tm

(T − r)

^α−1

dr

¹

2

+ C

_5.3^z

Z

_s

tm

(T − r)

^α−2

dr

¹

2

!

≤ C(T, C

_5.3^y

, C

_5.3^z

, p

₀

) √

s − t

m

((T − s)

^α−1²

+ (T − s)

^α−2²

).

We bound D

2

using Lemma 3.3 and Lemma A.1. Similar to (12) we conclude (setting Ψ

₂

:=

1 + |B

_t_k

+ ˜ B

_t_m−k

|

^p⁰⁺¹

+ |B

_τ_k

+ ˜ B

_τ_˜_m−k

|

^p⁰⁺¹

) that

kD

₂

(m)k = E | E

tk

F (t

_m

, B

_t_m

) − E

τk

F (t

_m

, B

_τ_m

)|

²

1 2

(8)

≤ C(T, L

_f

, c

^2,3_5.4

)( E E ˜ Ψ

⁴₂

)

¹⁴

1 (T − t

_m

)

¹⁻^α²

(t

_k

h + t

m−k

h)

¹⁴

≤ C(T, p

0

, L

f

, c

^2,3_5.4

) 1

(T − t

m

)

¹⁻^α²

h

¹⁴

. For D

₃

we apply again Lemma 3.3 and Lemma A.1,

kD

₃

(m)k ≤ kF (t

m

, B

tm

) − F (t

m

, B

τm

)k ≤ C(T, L

f

, c

^2,3_5.4

) 1

(T − t

m

)

¹⁻^α²

kΨ

₃

|B

_t_m

− B

τm

|k

≤ C(T, p

₀

, L

_f

, c

^2,3_5.4

) 1

(T − t

_m

)

¹⁻^α²

h

¹⁴

, where Ψ

₃

:= 1 + |B

_t_m

|

^p⁰⁺¹

+ |B

_τ_m

|

^p⁰⁺¹

. For the last term D

₄

we get

kD

₄

(m)k ≤ L

_f

(h

¹²

+ kB

_t_m

− B

_tⁿ_m

k + kY

_t_m

− Y

_tⁿ_m

k + kZ

_t_m

− Z

_tⁿ_m

k).

Finally, using the estimates for the terms D

₁

(s, m), D

₂

(m), ..., D

₄

(m) we arrive at kY

_t_k

− Y

_tⁿ_k

k ≤ C(C

_g

, T, p

₀

)h

^α⁴

+ C(T, L

_f

, C

_5.3^y

, C

_5.3^z

, p

₀

) h

¹²

Z

T tk

(T − s)

^α−2²

ds +C(T, p

0

, L

_f

, c

^2,3_5.4

)h

¹⁴

n−1

X

m=k

h

(T − t

m

)

¹⁻^α²

+ hL

_f

n−1

X

m=k

(kY

_t_m

− Y

_tⁿ_m

k + kZ

_t_m

− Z

_tⁿ_m

k)

≤ C(C

_g

, T, p

₀

, L

_f

, c

^2,3_5.4

, C

_5.3^y

, C

_5.3^z

)h

^α⁴

+ hL

_f

n−1

X

m=k

(kY

_t_m

− Y

_tⁿ_m

k + kZ

_t_m

− Z

_tⁿ_m

k). (14) For kZ

_t_k

− Z

_tⁿ

k

k we exploit the representations (7) and (8) and estimate kZ

_t_k

− Z

_tⁿ_k

k ≤ 1

T − t

_k

k E

t_k

g(B

T

)(B

T

− B

t_k

) − E

τ_k

g(B

τn

)(B

τn

− B

τ_k

)k + E

tk

Z

T tk+1

f (s, B

_s

, Y

_s

, Z

_s

) B

_s

− B

_t_k

s − t

_k

ds

!

− E

τk



 h

n−1

X

m=k+1

f (t

m+1

, B

_tⁿ_m

, Y

_tⁿ_m

, Z

_tⁿ_m

) B

ⁿ_t_m

− B

_tⁿ_k

t

m

− t

k





+ E

tk

Z

tk+1

tk

f (s, B

s

, Y

s

, Z

s

) B

s

− B

tk

s − t

_k

ds . Then, similar to (12), we have for the terminal condition by Lemma A.1 that

k E

t_k

[g(B

T

)(B

T

− B

t_k

)] − E

τ_k

[g(B

τn

)(B

τn

− B

τ_k

)]k

= k E ˜ [g(B

tk

+ ˜ B

T−tk

) − g(B

tk

)]( ˜ B

T−tk

− B ˜

τ˜n−k

) + ˜ E [g(B

tk

+ ˜ B

T−tk

) − g(B

τk

+ ˜ B

τ˜n−k

)] ˜ B

τ˜n−k

k

≤ C(C

g

, T, p

0

)h

¹⁴

(T − t

_k

)

^α²⁺¹⁴

+ C(C

g

, T, p

0

)h

^α⁴

(T − t

_k

)

¹²

≤ C(C

g

, T, p

0

)h

^α⁴

(T − t

_k

)

¹²

.

Here we have used that ˜ E [g(B

tk

)( ˜ B

T−tk

− B ˜

τ˜n−k

)] = 0. The term ˜ E [g(B

tk

+ ˜ B

T−tk

)−g(B

_t_k

)]( ˜ B

T−tk

− B ˜

_τ_˜_n−k

) provides us with the factor (T − t

_k

)

^α²

((T − t

_k

)h)

¹⁴

. For the next term of the estimate of kZ

_t_k

− Z

_tⁿ

k

k we use for s ∈ [t

_m

, t

_m+1

), where m ≥ k + 1, the decomposition E

t_k

f (s, B

s

, Y

s

, Z

s

)(B

s

− B

t_k

)

s − t

_k

− E

τk

f (t

m+1

, B

_tⁿ_m

, Y

_tⁿ_m

, Z

_tⁿ_m

)(B

_tⁿ_m

− B

_tⁿ_k

)

t

_m

− t

_k

(9)

= E

t_k

f (s, B

s

, Y

s

, Z

s

)(B

s

− B

t_k

)

s − t

_k

− E

t_k

f (t

m

, B

tm

, Y

tm

, Z

tm

)(B

tm

− B

t_k

) t

_m

− t

_k

+ E

tk

F(t

_m

, B

_t_m

)(B

_t_m

− B

_t_k

)

t

_m

− t

_k

− E

τk

F (t

_m

, B

_τ_m

)(B

_τ_m

− B

_τ_k

) t

_m

− t

_k

+ E

τk

[F (t

_m

, B

_τ_m

) − F (t

_m

, B

_t_m

)] B

_τ_m

− B

_τ_k

t

m

− t

k

+ E

τk

[f (t

m

, B

tm

, Y

tm

, Z

tm

) − f (t

m+1

, B

_tⁿ_m

, Y

_tⁿ_m

, Z

_tⁿ_m

)] B

_tⁿ_m

− B

_tⁿ

k

t

m

− t

_k

=: T

1

(s, m) + T

2

(m) + ... + T

4

(m).

Then by the conditional Hölder inequality and by (13) as well as by Lemma 3.3 we have kT

₁

(s, m)k ≤ kD

₁

(s, m)k kB

_s

− B

t_k

k

s − t

_k

+ kf (t

m

, B

tm

, Y

tm

, Z

tm

)k

B

s

− B

t_k

s − t

_k

− B

tm

− B

t_k

t

_m

− t

_k

≤ C(T, L

_f

, C

_5.3^y

, C

_5.3^z

, p

₀

) (T − s)

^α−2²

h

¹²

√ s − t

_k

+C(T, L

_f

, c

^1,2_5.4

, K

_f

) ( E Ψ(B

_t_m

)

²

)

¹²

(T − t

m

)

^1−α²

×

kB

_s

− B

tm

k

s − t

_k

+ kB

_t_m

− B

t_k

k

1 s − t

_k

− 1 t

_m

− t

_k

≤ C(T, L

_f

, K

_f

, C

_5.3^y

, C

_5.3^z

, c

^1,2_5.4

, p

₀

)(T − s)

^α−2²

h

¹⁴

(s − t

_k

)

³⁴

. Indeed,

kB

_s

− B

tm

k

s − t

_k

+ kB

_t_m

− B

tk

k

1 s − t

_k

− 1 t

m

− t

_k

≤

√ s − t

m

s − t

_k

+

√ t

m

− t

k

(s − t

m

)

(s − t

_k

)(t

m

− t

_k

) ≤ C h

¹⁴

(s − t

_k

)

³⁴

, where the last inequality follows from s− t

_m

≤ t

_m+1

−t

_m

= h and h ≤ t

_m

− t

_k

≤ s −t

_k

. We estimate T

2

with the help of Lemma 3.3 and Lemma A.1 as follows :

kT

₂

(m)k ≤ k D b

₂

(m)k kB

_t_m

− B

t_k

k

t

_m

− t

_k

+ kF (t

m

, B

τm

)k kB

_t_m−k

− B

τm−k

k t

_m

− t

_k

≤ C(T, p

₀

, L

_f

, K

_f

, c

_5.4

) 1 (T − t

_m

)

¹⁻^α²

h

¹⁴

(t

_m

− t

_k

)

³⁴

.

Here D b

₂

(m) := (˜ E |F (t

_m

, B

_t_k

+ ˜ B

_t_m−k

) − F (t

_m

, B

_τ_k

+ ˜ B

_τ_˜_m−k

)|

²

)

¹²

which can be estimated as D

₂

(m).

For T

3

the conditional Hölder inequality and Lemma A.1 yield kT

₃

(m)k ≤ k D b

3

(m)k

B

τm

− B

τ_k

t

_m

− t

_k

≤ C(T, p

0

, L

_f

, c

^2,3_5.4

) 1 (T − t

_m

)

¹⁻^α²

h

¹⁴

(t

_m

− t

_k

)

¹²

, where D b

3

(m) := F (t

m

, B

τm

) − F (t

m

, B

tm

) is estimated as D

3

(m). Finally,

kT

₄

(m)k ≤ L

_f

(h

¹²

+ kB

_t_m

− B

ⁿ_t_m

k + kY

_t_m

− Y

_tⁿ_m

k + kZ

_t_m

− Z

_tⁿ_m

k) 1

√ t

_m

− t

_k

.

(10)

For the estimate of E

tk

R

tk+1

tk

f (s, B

_s

, Y

_s

, Z

_s

)

^B^s_s−t^−B^tk

k

ds one notices that by the conditional Hölder inequality,

k E

tk

f (s, B

_s

, Y

_s

, Z

_s

)

^B^s_s−t^−B^tk

k

k = k E

tk

[(f (s, B

_s

, Y

_s

, Z

_s

) − f (s, B

_t_k

, Y

_t_k

, Z

_t_k

))

^B^s_s−t^−B^tk

k

]k

≤ kf(s, B

_s

, Y

s

, Z

s

) − f(s, B

tk

, Y

tk

, Z

tk

)k 1

√ s − t

_k

≤ C(T, L

_f

, C

_5.3^y

, C

_5.3^z

, p

0

) (T − s)

^α−2²

h

¹²

√ s − t

_k

, where the last inequality follows in the same way as in (13). Consequently, we have

kZ

_t_k

− Z

_tⁿ_k

k ≤ C(C

g

, T, p

0

)

(T − t

k

)

¹²

h

^α⁴

+ C(T, L

f

, K

f

, C

_5.3^y

, C

_5.3^z

, c

^1,2_5.4

, p

0

) Z

T

tk

ds

(T − s)

¹⁻^α²

(s − t

k

)

³⁴

h

¹⁴

+C(T, p

₀

, L

_f

, K

_f

, c

_5.4

) h

n−1

X

m=k+1

1 (T − t

_m

)

¹⁻^α²

h

¹⁴

(t

_m

− t

_k

)

³⁴

+L

f

h

n−1

X

m=k+1

(kB

_t_m

− B

_tⁿ_m

k + kY

_t_m

− Y

_tⁿ_m

k + kZ

_t_m

− Z

_tⁿ_m

k) 1

√ t

m−k

.

Lemma A.2 enables to bound the second and third term of the r.h.s. by C

^h

1 4

(T−t_k)³⁴⁻^α²

B (

^α₂

,

¹₄

), which is bounded by C

^h

α 4

(T−t_k)¹²⁻^α⁴

. Thus we get kZ

_t_k

− Z

_tⁿ_k

k ≤ C

0

h

^α⁴

(T − t

_k

)

¹²

+ L

f

h

n−1

X

m=k+1

(kY

_t_m

− Y

_tⁿ_m

k + kZ

_t_m

− Z

_tⁿ_m

k) 1

√ t

m−k

.

Then we use (14) and the above estimate to get kY

_t_k

− Y

_tⁿ_k

k + kZ

_t_k

− Z

_tⁿ_k

k ≤ C

0

h

^α⁴

(T − t

_k

)

¹²

+ C(L

_f

) h

n−1

X

m=k+1

(kY

_t_m

− Y

_tⁿ_m

k + kZ

_t_m

− Z

_tⁿ_m

k) 1

√ t

m−k

.

If this inequality is iterated, one gets a shape where the Gronwall lemma applies. Indeed, setting a

m

:= (kY

_t_m

− Y

_tⁿ_m

k + kZ

_t_m

− Z

_tⁿ_m

k) one has to consider the double sum

n−1

X

m=k+1





n−1

X

l=m+1

a

_l

h

√ t

l−m





√ h t

m−k

= h

n−1

X

l=k+1





l−1

X

m=k+1

√ h t

m−k

√ t

l−m



 a

_l

≤ Ch

n−1

X

l=k+1

a

_l

.

Consequently,

kY

_t_k

− Y

_tⁿ_k

k + kZ

_t_k

− Z

_tⁿ_k

k ≤ C

0

h

^α⁴

(T − t

_k

)

¹²

which gives the bound on the error on Z . Moreover, (14) yields

kY

_t_k

− Y

_tⁿ

k

k ≤ C

₀

h

^α⁴

. If v ∈ [t

_k

, t

_k+1

), we have by Theorem 5.3 that

kY

_v

− Y

_vⁿ

k ≤ kY

_v

− Y

_t_k

k + kY

_t_k

− Y

_tⁿ

k

k ≤ C(C

_5.3^y

, T, p

₀

) Z

_v

tk

(T − r)

^α−1

dr

¹₂

+ kY

_t_k

− Y

_tⁿ

k

k,