Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 98, pp. 1–16.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

NEWTON-KANTOROVITCH METHOD FOR DECOUPLED

FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL

EQUATIONS

DAI TAGUCHI, TAKAHIRO TSUCHIYA

Abstract. We formulate a Newton-Kantorovitch method for solving decou-

pled forward-backward stochastic differential equations involving smooth and
degenerate coefficients with uniformly bounded derivatives. We show that it

converges globally and its rate of convergence is exponential.

1. Introduction

In this article, we study a method for approximating non-Markovian and decou-
pled forward-backward stochastic differential equations (FBSDEs) of the following
form for arbitrary T > 0,

X(t) = X(0) +

∫ t

0

b(s,X(s)) ds+

∫ t

0

σ(s,X(s)) dW (s),

Y (t) = ϕ(X(T )) +

∫ T

t

f(s,X(s), Y (s), Z(s)) ds−
∫ T

t

Z(s) dW (s),

(1.1)

where the solution triplets (X,Y, Z) ≡
(
X(t), Y (t), Z(t)

)
t∈[0,T ]

take values in Rd ×
Rm×Rm×k, W is a k-dimensional Wiener process, and b, f , σ, and ϕ are measurable
functions that could in general be random and defined on a probability space.

We propose a scheme for approximating such FBSDEs with random coefficients
based on applying the Newton-Kantorovitch theory. Unlike a four-step scheme that
relies on the Markov structure (see e.g., Ma, Protter and Yong [13], and Delarue
[6]), this method allows us to find a non-Markovian approximation. In addition,
as our approach is a type of contraction mapping [3, 20], the approximation works
for arbitrary large durations with smooth random coefficients, and without mono-
tonicity conditions; for details, see Hu and Peng [9], Peng and Wu [21], and Yong
[25]. We also refer readers to [27, 15] for more details about the uniqueness and
the existence of FBSDE solutions.

Most numerical algorithms for FBSDEs are based on time-space discretization
schemes [7] for quasi-linear parabolic partial differential equations for coupled FB-
SDEs with monotonicity condition [5]. In contrast, to the best of our knowledge,

2010 Mathematics Subject Classification. 49M15, 65C30, 41A25, 60H35, 41A25, 60H10.
Key words and phrases. Stochastic differential equation; Newton-type methods;

rate of convergence.
©2021. This work is licensed under a CC BY 4.0 license.

Submitted January 19, 2021. Published December 20, 2021.

1



2 D. TAGUCHI, T. TSUCHIYA EJDE-2021/98

very little has been published on state space discretization based approaches. Vi-
dossich proved that the Chaplygin and Newton methods are equivalent for ordi-
nary differential equations [22], and Kawabata and Yamada studied a state space
discretization based on Newton-Kantorovitch approach for stochastic differential
equations (SDEs) [12]. Ouknine [17] showed that the coefficients can be relaxed
by a linear growth condition. Amano obtained a probabilistic second-order error
estimate [2]. Wrzosek [23] extended to stochastic functional differential equations.

The aim of this paper is to formulate Newton-Kantorovitch approximation of the
decoupled multi-dimensional FBSDEs with random coefficients and to show that it
converges globally (1.2) as follows.

Theorem 1.1. Suppose that b, σ, f , and ϕ are all C1, their derivatives are uni-
formly bounded (s, ω)-a.e., and b, σ, and f are square-integrable with respect to their
time variables. Then, there exists a positive constant C > 0 such that

‖(X −Xn+1, Y − Yn+1, Z − Zn+1)‖ ≤ C2−n n ∈ N ∪ {0}. (1.2)

As pointed out in [4, 8], solutions (X,Y, Z) of decoupled FBSDEs can be viewed
as fixed points of a map. Here, we consider an alternative mapping

Fϕ : Ω→M(S2d × S2m ×H2, S2d × L2
T ),

where M(A,B) stands for a set of maps from A to B. It is inspired by a one-
dimensional analog of Kawabata and Yamada [12] and Niwa [26], as follows. For
given ϕ and u = (x, y, z) ∈ S2d × S2m ×H2, we define

Fϕ(u)(t) =

(
x(t)− x(0)−

∫ t
0
b(s, x(s)) ds−

∫ t
0
σ(s, x(s)) dW (s)

y(t)− ϕ(x(T ))−
∫ T
t
f(s, u(s)) ds+

∫ T
t
z(s) dW (s)

)>
. (1.3)

Then, for any (X0, Y0, Z0) ∈ S2d × S2m ×H2 with X0(0) = X(0) and assuming that
the driver f is smooth, the Newton-Kantorovitch approximation process is given
by

Fϕ(Xn, Yn, Zn) + F ′ϕ(Xn, Yn, Zn)(Xn+1 −Xn, Yn+1 − Yn, Zn+1 − Zn) = 0,

Xn+1(0) = X(0),
(1.4)

for n ∈ N ∪ {0}, where F ′ϕ stands for a Gâteaux derivative,

F ′ϕ : Ω→ L
(
S2d × S2m ×H2,S2d × L2

T

)
where L(A,B) stands for a set of linear maps from A to B. This sequence is
well-posed iff a unique system of linear backward stochastic differential equations
(BSDEs) formally by Theorem 3.6. Note that, if the given duration T is arbitrarily,
then, even for linear FBSDEs with constant coefficients, the necessary and sufficient
conditions become more complicated when the diffusion coefficients also depend on
z, as was pointed out by Ma, Wu, Zhang and Zhang [14]. In this paper, we focus
on decoupled FBSDEs and obtain the convergence result, Theorem 4.5.

The rest of this paper is structured as follows. Section 2 introduces the no-
tations and assumptions used. Section 3 is devoted to formulating the Newton-
Kantorovitch approximation process. Finally, Section 4 proves the main theorem
for the decoupled FBSDEs.



EJDE-2021/98 NEWTON-KANTOROVITCH METHOD FOR FBSDES 3

2. Preliminaries

For x, y ∈ Rd, |x| denotes the Euclidian norm and 〈x, y〉 denotes the inner
product. Matrixes size of m × k will be represented as an element y ∈ Rm×k,

whose Euclidean norms are also given by |y| =
√
yy> and for which 〈y, z〉 =

trace(yz>) where A> is the transpose matrix of A. In this paper, we only consider
the derivatives with respect to the space variable.

We also use the following notation, based on that in [19]. Let (Ω,F ,P) be a
complete probability space and (Ft)t≥0 be a Brownian filtration. P = P((Ft)t≥0)
is the σ-algebra of the progressively measurable subset A ⊂ Ω×[0,∞) such that, for
all t ≥ 0, A∩([0, t]×Ω) ∈ B[0, t]⊗Ft. Let T > 0 be a fixed, and final, deterministic
time. For m, k ∈ N, we define

L2 = {ξ : Ω→ Rm is FT -measurable and ‖ξ‖L2 = {E[|ξ|2]}1/2 <∞},
L2
T = {Y : Ω× [0, T ]→ Rm continuous : Y (t) ∈ mFT ,∀t ∈ [0, T ], ‖Y ‖L2

T
<∞},

S2m = {Y : Ω× [0, T ]→ Rm continuous, adapted : ‖Y ‖S2m <∞},

H2 = {Z : Ω× [0, T ]→ Rm×k adapted : ‖Z‖H2 <∞},

where the norms ‖ · ‖L2
T

, ‖ · ‖S2m , and ‖ · ‖H2 are defined by

‖Y ‖L2
T

= ‖Y ‖S2m = {E[ sup
0≤s≤T

|Y (s)|2]}1/2, ‖Z‖H2 = {E[

∫ T

0

|Z(s)|2 ds]}1/2.

For the sake of simplicity, we also write the operator norm of the operator A as
‖A‖. The Banach spaces S2m ×H2 and S2d are defined by

‖(Y, Z)‖2 = ‖Y ‖2S2m + ‖Z‖2H2 , ‖X‖2 = ‖X‖2S2d .

For α ∈ R, we introduce the weighted norm

‖(Y,Z)‖2α = E[ sup
0≤s≤T

eαs|Y (s)|2] + E[

∫ T

0

eαs|Z(s)|2 ds].

For p, q ∈ N, Ck(Rp,Rq) is the set of functions of class Ck from Rp to Rq, and
Ckb (Rp,Rq) is the subset of these functions whose partial derivatives of order at
most of the k values are bounded. When the domain and range dimensions are
clear based on context and when there is no risk of confusion, we often eliminate
the spaces to simplify the notation.

For a smooth g such that g(·, ·, ·) ∈ C1(Rp × Rq × Rr,Rq), it is convenient to
obtain a concrete representation of g′ : Rp×Rq ×Rr → Rq. The Fréchet derivative
at u ∈ Rp × Rq × Rq is the matrix representation of g′(u) [16, page 60] and it can
be obtained using the Jacobian matrix:

g(u+ ∆u)− g(u) = g′(u)∆u+ Rg(u)∆u, ∆u ∈ Rp × Rq × Rr (2.1)

where the Lagrange remainder is

Rg(u)∆u =
(∫ 1

0

{g′(u+ θ∆u)− g′(u)}dθ
)

∆u.

Notice that, for all u = (x, y, z)>,∆u = (∆x,∆y,∆z)> ∈ Rp × Rq × Rr,

g′(u)∆u = g′x(u)∆x+ g′y(u)∆y + g′z(u)∆z, (2.2)



4 D. TAGUCHI, T. TSUCHIYA EJDE-2021/98

where we obtain g′x(u) : Rp → Rp × Rp such that

g′x(u)∆x = (
∂

∂xj
g(i)(x, y, z)∆x(i))i≤p, j≤p,

where ∆x(i) stands for the ith component for i ≤ p. We define g′y and g′z, similarly.
Finally, for all s ∈ [0, T ], we define

‖g′‖∞ = sup
(s,x,y,z)∈[0,T ]×Rp×Rq×Rr

|g′(s, x, y, z)|,

for g(s, ·, ·, ·) ∈ C1
b (Rp × Rq × Rr,Rq).

3. Newton-Kantorovitch scheme for FBSDEs

In this section, we formulate the Newton-Kantorovitch scheme of the following
system of FBSDEs:

X(t) = X(0) +

∫ t

0

b(s,X(s)) ds+

∫ t

0

σ(s,X(s)) dW (s),

Y (t) = ϕ(X(T )) +

∫ T

t

f(s,X(s), Y (s), Z(s)) ds−
∫ T

t

Z(s) dW (s),

(3.1)

where the solutions (X,Y, Z) = (X(t), Y (t), Z(t))t∈[0,T ] take values in Rd × Rm ×
Rm×k, W is the k-dimensional Wiener process, and the (progressively) measurable
functions b, f , σ and ϕ are defined on the probability space (Ω,F , (Ft)t≥0,P).

In this paper, we also assume the following.

• X(0) : Ω→ Rd is an F0-measurable and square-integrable random vector.
• b : [0, T ]× Ω× Rd → Rd is P ⊗ B(Rd)-measurable.
• σ : [0, T ]× Ω× Rd → Rd×k is P ⊗ B(Rd)-measurable.
• There exists a constant C > 0 such that, for any x ∈ Rd,

|b(s, x)| ≤ |b(s, 0)|+ C|x|, (s, ω)-a.e.,

|σ(s, x)| ≤ |σ(s, 0)|+ C|x|, (s, ω)-a.e.

• ϕ : Ω× Rd → Rm is FT ⊗ B(Rd)-measurable.
• f : [0, T ]×Ω×Rd×Rm×Rm×k → Rm is P⊗B(Rm)⊗B(Rm×k)-measurable.
• There exists a constant C > 0 such that for any (x, y, z) ∈ Rd×Rm×Rm×k,

|f(s, x, y, z)| ≤ |f(s, 0, 0, 0)|+ C(|x|+ |y|+ |z|), (s, ω)-a.e.

|ϕ(x)| ≤ |ϕ(0)|+ C|x|, ω-a.e.

In addition, in the BSDE,

Y (t) = ξ +

∫ T

t

f(s, Y (s), Z(s)) ds−
∫ T

t

Z(s) dW (s),

we replace the above assumption on ϕ with the condition

• ξ : Ω → Rm is an FT -measurable and square integrable random vector;
ξ ∈ L2.

We also introduce the following assumptions.

Assumption 3.1. b(·, 0), σ(·, 0), f(·, 0, 0, 0) ∈ H2, i.e.,

E
[ ∫ T

0

{|b(s, 0)|2 + |σ(s, 0)|+ |f(s, 0, 0, 0)|2} ds
]
<∞



EJDE-2021/98 NEWTON-KANTOROVITCH METHOD FOR FBSDES 5

and ϕ(0) ∈ L2.

Assumption 3.2. b(s, ·) ∈ C1
b (Rd,Rd), σ(s, ·) ∈ C1

b (Rd,Rd×k),
f(s, ·, ·, ·) ∈ C1

b (Rd × Rm × Rm × k,Rm), (s, ω)-a.e. and ϕ ∈ C1
b (Rd,Rm), ω-a.e.

For a given ϕ, we consider the operator Fϕ (defined by (1.3)),

Fϕ : Ω→M(S2d × S2m ×H2,S2d × L2
T )

where M(A,B) stands for a collection of maps from A to B, namely,

Fϕ(u)(t) =

(
x(t)− x(0)−

∫ t
0
b(s, x(s)) ds−

∫ t
0
σ(s, x(s)) dW (s)

y(t)− ϕ(x(T ))−
∫ T
t
f(s, u(s)) ds+

∫ T
t
z(s) dW (s)

)>
,

for u = (x, y, z) ∈ S2d × S2m ×H2. As an immediate consequence of the result given
in [12, Lemma 3.1], we obtain the following corresponding result for FBSDEs.

Lemma 3.3. If Assumption 3.1 holds, then the operator Fϕ defined by (1.3) maps
the space S2d×S2m×H2 into S2d×L2

T and t 7→ Fϕ(u)(t) is a continuous modification
for u ∈ S2d × S2m ×H2.

Proof. For any u = (x, y, z) ∈ S2d × S2m × H2, it follows, from ‖x‖S2d < ∞ and

‖z‖2H2 <∞, that the Itô integrals t 7→
∫ t
0
σ(s, x(s)) dW (s) and t 7→

∫ T
t
z(s) dW (s),

respectively, are continuous modifications. By the Jensen inequality, we obtain

E
[ ∫ t

0

|b(s, x(s))|2 ds
]
≤ 2E

[ ∫ t

0

|b(s, 0)|2 ds
]

+ 2C2T‖x‖2S2d <∞

and

E
[ ∫ T

t

|f(s, x(s), y(s), z(s))|2 ds
]

≤ 4E
[ ∫ T

t

|f(s, 0, 0, 0)|2 ds
]

+ 4C2
(
T{‖x‖2S2d + ‖y‖2S2m}+ ‖z‖2H2

)
<∞.

By Doob’s inequality and Itô’s isometry property, there exists a c > 0 such that

E[ sup
0≤t≤T

|
∫ t

0

σ(s, x(s)) dW (s)|2]

≤ cE
[
|
∫ T

0

σ(s, x(s)) dW (s)|2
]

≤ 2cE
[ ∫ T

0

|σ(s, 0)|2 ds
]

+ 2cC2T‖x‖2Sd <∞

and

E
[

sup
0≤t≤T

∣∣ ∫ T

t

z(s) dW (s)
∣∣2] ≤ cE[∣∣ ∫ T

0

z(s) dW (s)
∣∣2]

= cE
[ ∫ T

0

|z(s)|2 ds
]
<∞.

Using the Jensen inequality, we further obtain that

1

9
E
[

sup
0≤t≤T

|Fϕ(u)(t)|2
]



6 D. TAGUCHI, T. TSUCHIYA EJDE-2021/98

≤ ‖x‖2S2d + E[|x(0)|2] + E
[ ∫ t

0

|b(s, x(s))|2 ds
]

+ E
[

sup
0≤t≤T

|
∫ t

0

σ(s, x(s)) dW (s)|2
]

+ ‖y‖2S2m + E|[ϕ(0)|2 + C|x(T )|2]

+ E
[∣∣ ∫ T

t

f(s, x(s), y(s), z(s)) ds
∣∣2]+ E

[
sup

0≤t≤T

∣∣ ∫ T

t

z(s) dW (s)
∣∣2],

which is bounded by the above estimates. This completes the proof. �

Denote by L(A,B) a collection of linear maps from A to B. The following
lemma shows that the operator Fϕ : Ω→ L(S2d× S2m×H2,S2d×L2

T ) has a Gâteaux
derivative.

Lemma 3.4. If Assumptions 3.1 and 3.2 hold, then for all u = (x, y, z) ∈ S2d×S2m×
H2, the Gâteaux derivative F ′ϕ(u) : S2d×S2m×H2 → S2d×L2

T of Fϕ at u ∈ S2d×S2m×H2

in the direction u = (x, y, z) ∈ S2m ×H2 exists and is given for any t ∈ [0, T ], by

F ′ϕ(u)u(t)

=

(
x(t)− x(0)−

∫ t
0
b′x(s, x(s))x(s) ds−

∫ t
0
σ′x(s, x(s))x(s) dW (s)

y(t)− ϕ′(x(T ))x(T )−
∫ T
t
f ′(s, u(s))u(s) ds+

∫ T
t
z(s) dW (s)

)>
.

(3.2)

Proof. We denote the right-hand side of (3.2) by A(u)u(t) for u = (x, y, z) ∈
S2d × S2m ×H2. We note that because

E
[ ∫ T

t

{
|b′x(s, x(s))x(s)|2 + |σ′x(s, x(s))x(s)|2 + |f ′(s, u(s))u(s)|2

}
ds
]

≤ (1 ∨ T ){‖b′‖2∞ + ‖σ′‖2∞ + ‖f ′‖2∞}‖u‖2,

we obtain that A(u)u ∈ L2
T by the same argument of Lemma 3.3. It follows from

(2.1) that, for all s ∈ [0, T ],

b(s, x(s) + δx(s))− b(s, x(s)) = δb′x(s, x(s))x(s) + Rb(x(s))(δx)(s),

σ(s, x(s) + δx(s))− σ(s, x(s)) = δσ′x(s, x(s))x(s) + Rσ(x(s))(δx)(s),

f(s, u(s) + δu(s))− f(s, u(s)) = δf ′(s, u(s))u(s) + Rf (u(s))(δu)(s),

ϕ(x(T ) + δx(T ))− ϕ(x(T )) = δϕ′(x(T ))x(T ) + Rϕ(x(T ))(δx)(T ).

Hence, for all δ > 0 and t ∈ [0, T ], we have

Fϕ(u+ δu)(t)− Fϕ(u)(t)

δ

= A(u)u(t) +
1

δ

(
−
∫ t
0

Rb(x(s))(δx)(s) ds−
∫ t
0

Rσ(x(s))(δx)(s) dW (s)

−Rϕ(x(T ))(δx)(T )−
∫ T
t

Rf (u(s))(δu)(s) ds.

)>
.

Since b′x(s, ·), σ′x(s, ·), f ′(s, ·) and ϕ′ are bounded and continuous (s, ω)-a.e., by
using the dominated convergence theorem, we obtain

lim
δ→0

∥∥Fϕ(u+ δu)− Fϕ(u)

δ
−A(u)u

∥∥
S2d×L

2
T

= 0,

thus completing the proof. �

The following lemma is a key for defining the Newton-Kantorovitch approxima-
tion process.



EJDE-2021/98 NEWTON-KANTOROVITCH METHOD FOR FBSDES 7

Lemma 3.5. Suppose that Assumptions 3.1 and 3.2 hold and let u = (x, y, z) ∈
S2d × S2m ×H2. Then, there exists a unique ũ = (x̃, ỹ, z̃) ∈ S2d × S2m ×H2 such that
Fϕ(u) + F ′ϕ(u)(ũ− u) = 0 with initial condition x̃(0) = x(0).

Proof. Let u = (x, y, z) be in S2d × S2m × H2. We show that there exists a unique
ũ = (x̃, ỹ, z̃) ∈ S2d× S2m×H2 such that Fϕ(u) +F ′ϕ(u)(ũ− u) = 0 with x̃(0) = x(0),
i.e.,

x̃(t) = x(0) +

∫ t

0

{b(s, x(s)) + b′x(s, x(s))(x̃(s)− x(s))}ds

+

∫ t

0

{σ(s, x(s)) + σ′x(s, x(s))(x̃(s)− x(s))}dW (s),

ỹ(t) = ϕ(x(T )) + ϕ′(x(T ))(x̃(T )− x(T ))

+

∫ T

t

{f(s, u(s)) + f ′(s, u(s))(ũ(s)− u(s))} ds−
∫ T

t

z̃(s) dW (s),

(3.3)

Because the above equation is a linear decoupled FBSDE with uniformly bounded
coefficients, there exists a unique ũ ∈ S2d × S2m ×H2 as required. �

From Lemma 3.5, we can conclude that for any initial condition (X0, Y0, Z0) ∈
S2d × S2m × H2, we can define the Newton-Kantorovitch approximation process
(Xn+1, Yn+1, Zn+1) ∈ S2d × S2m ×H2 as solving the equation

Fϕ(Xn, Yn, Zn) + F ′ϕ(Xn, Yn, Zn)(Xn+1 −Xn, Yn+1 − Yn, Zn+1 − Zn) = 0,

Xn+1(0) = X0(0),
(3.4)

for n ∈ N ∪ {0}, which is equivalent to

(Xn+1, Yn+1, Zn+1) = (Xn, Yn, Zn)− F ′ϕ(Xn, Yn, Zn)
−1
Fϕ(Xn, Yn, Zn).

The following theorem shows that (3.4) has a unique solution that satisfies a linear
decoupled FBSDE with uniformly bounded derivatives of coefficients.

Theorem 3.6. If Assumptions 3.1 and 3.2 hold, then (Xn+1, Yn+1, Zn+1) satisfies
the following linear decoupled FBSDE for 0 ≤ t ≤ T :

Xn+1(t) = Xn+1(0) +

∫ t

0

bn(s,Xn+1(s)) ds+

∫ t

0

σn(s,Xn+1(s)) dW (s),

Yn+1(t) = ϕn(Xn+1(T )) +

∫ T

t

fn(s,Xn+1(s), Yn+1(s), Zn+1(s)) ds

−
∫ T

t

Zn+1(s) dW (s),

(3.5)

where we define for 0 ≤ s ≤ T and (x, y, z) ∈ Rd × Rm × Rm×k,

bn(s, x) = b(s,Xn(s)) + b′x(s,Xn(s))(x−Xn(s)),

σn(s, x) = σ(s,Xn(s)) + σ′x(s,Xn(s))(x−Xn(s)),

fn(s, x, y, z) = f(s,Xn(s), Yn(s), Zn(s)),

+ f ′(Xn(s), Yn(s), Zn(s))(x−Xn(s), y − Yn(s), z − Zn(s)),

ϕn(x) = ϕ(Xn(T )) + ϕ′x(Xn(T ))(x−Xn(T )).

In particular, if X0(0) = X(0), then Xn+1(0) = X(0) for all n ∈ N ∪ {0}.



8 D. TAGUCHI, T. TSUCHIYA EJDE-2021/98

Proof. The existence and uniqueness of (Xn+1(t), Yn+1(t), Zn+1(t))t∈[0,T ] follows
from Lemma 3.5, where, for all n ∈ N∪{0}. The equation (3.3) can be re-expressed
as the desired FBSDE for 0 ≤ t ≤ T . Finally, by the initial condition Xn+1(0) =
Xn(0), if X0(0) = X(0), then Xn+1(0) = X(0), for all n ∈ N ∪ {0}. �

4. Convergence of the Newton-Kantorovitch approximation process
for FBSDEs

In this section, we investigate the convergence of the Newton-Kantorovitch ap-
proximation processes defined by (3.4).

4.1. Convergence of the Newton-Kantorovitch approximation process for
SDEs. In this subsection, we consider the Newton-Kantorovitch scheme for the
forward process X and extend earlier studies of Newton-Kantorovitch methods for
SDEs,

X(t) = X(0) +

∫ t

0

b(s,X(s)) ds+

∫ t

0

σ(s,X(s)) dW (s). (4.1)

Theorem 4.1. Let X be a solution of (4.1). If Assumptions 3.1 and 3.2 hold,
then, for any X0 ∈ S2d where X0(0) = X(0), we obtain that, for all n ∈ N∪{0} and
ε ∈ (0, 1),

‖X −Xn+1‖2 ≤
Cn+1

0

(n+ 1)!
‖X −X0‖2 ≤ εn+1eC0T/ε‖X −X0‖2, (4.2)

where the constant C0 is given by

C0 = 8cb,σT exp(4cb,σT ), cb,σ = ‖b′‖∞ + 18‖σ′‖∞ + ‖σ′‖2∞.

Remark 4.2. An estimate was initially given in [12] for one-dimensional SDEs
with uniformly bounded coefficients and an alternative estimation was proposed by
[1].

Proof of Theorem 4.1. The proof is based on [1]. By a fundamental result in [10],
X exists and is unique. For 0 ≤ s ≤ T and n ∈ N, define

Xn(s) = X(s)−Xn(s),

we obtain for all s ∈ [0, T ] and n ∈ N, P-a.s. By the mean value theorem, for
s ∈ [0, T ] and n ∈ N, we obtain

b(s,X(s))− bn(s,Xn+1(s))

= {b(s,X(s))− b(s,Xn(s))− b′x(s,Xn(s))Xn(s)}+ b′x(s,Xn(s))Xn+1(s)

= Rb(Xn(s))Xn(s) + b′x(s,Xn(s))Xn+1(s)

and

σ(s,X(s))− σn(s,Xn+1(s))

= {σ(s,X(s))− σ(s,Xn(s))− σ′x(s,Xn(s))Xn(s)}+ σ′x(s,Xn(s))Xn+1(s)

= Rσ(Xn(s))Xn(s) + σ′x(s,Xn(s))Xn+1(s).

Recall (2.1) and we have, for s ∈ [0, T ], n ∈ N and h ∈ S2d,

Rb(Xn(s))h(s) =
{∫ 1

0

b′x(s, (Xn(s)) + θh(s))dθ − b′x(s,Xn(s))
}
h(s),



EJDE-2021/98 NEWTON-KANTOROVITCH METHOD FOR FBSDES 9

Rσ(Xn(s))h(s) =
{∫ 1

0

σ′x(s, (Xn(s)) + θh(s))dθ − σ′x(s,Xn(s))
}
h(s).

This allows us to obtain

Xn+1(t) =

∫ t

0

{
b′x(s,Xn(s))Xn+1(s) + Rb(Xn(s))Xn(s)

}
ds

+

∫ t

0

{σ′x(s,Xn(s))Xn+1(s) + Rσ(Xn(s))Xn(s)}dW (s),

which, by applying Itô’s formula yields

|Xn+1(t)|2 = 2

∫ t

0

〈Xn+1(s), b′x(s,Xn(s))Xn+1(s) + Rb(Xn(s))Xn(s)〉ds

+ 2

∫ t

0

〈Xn+1(s), σ′x(s,Xn(s))Xn+1(s) + Rσ(Xn(s))Xn(s) dW (s)〉

+

∫ t

0

|σ′x(s,Xn(s))Xn+1(s) + Rσ(Xn(s))Xn(s)|2 ds.

Note that we obtain

2〈Xn+1(s), b′x(s,Xn(s))Xn+1(s) + Rb(Xn(s))Xn(s)〉
≤ 2‖b′‖∞|Xn+1(t)|2 + 4‖b′‖∞|Xn(t)|2

and

|σ′x(s,Xn(s))Xn+1(s) + Rσ(Xn(s))Xn(s)|2

≤ 2‖σ′‖2∞|Xn+1(t)|2 + 4‖σ′‖2∞|Xn(t)|2.
The Burkholder-Davis-Gundy inequality implies that there exists a c0 such that

E
[∣∣2 ∫ t

0

〈Xn+1(t), σ′x(s,Xn(s))Xn+1(s) dW (s)〉
∣∣]

≤ E
[
2
(1

4
sup

0≤s≤t
|Xn+1(s)|2

)1/2(
4c20‖σ′‖∞

∫ t

0

|Xn+1(s)|2 ds
)1/2]

≤ 1

4
E
[

sup
0≤s≤t

|Xn+1(s)|2
]

+ 4c20‖σ′‖∞E
[ ∫ t

0

|Xn+1(s)|2 ds
]
,

where we obtain the last inequality by applying the inequality ab ≤ (a2/2) + (b2/2)
for all a, b ∈ R. Similarly, we have

E
[∣∣2∫ t

0

〈Xn+1(t),Rσ(Xn(s))Xn(s) dW (s)〉
∣∣]

≤ E
[
2
(1

4
sup

0≤s≤t
|Xn+1(s)|2

)1/2(
8c20‖σ′‖∞

∫ t

0

|Xn(s)|2 ds
)1/2]

≤ 1

4
E
[

sup
0≤s≤t

|Xn+1(s)|2
]

+ 8c20‖σ′‖∞E
[ ∫ t

0

|Xn(s)|2 ds
]
,

where we note that an explicit upper bounded of c0 can be obtained by 3; see [11,
Theorem 3.28]. By setting cb,σ = ‖b′‖∞ + 2c20‖σ′‖∞ + ‖σ′‖2∞, we obtain, for any
t′ ∈ [0, T ],

E
[

sup
0≤t≤t′

|Xn+1(t)|2
]



10 D. TAGUCHI, T. TSUCHIYA EJDE-2021/98

≤ 4cb,σ

∫ t′

0

E[ sup
0≤t≤s

|Xn+1(t)|2] ds+ 8cb,σ

∫ t′

0

E[ sup
0≤t≤s

|Xn(t)|2] ds.

Gronwall’s inequality further implies that

E
[

sup
0≤t≤t′

|Xn+1(t)|2
]
≤ C0

∫ t′

0

E
[

sup
0≤t≤s

|Xn(t)|2
]

ds, (4.3)

where C0 = 8cb,σ exp(4cb,σT ). Iterating (4.3) yields

E
[

sup
0≤t≤T

|Xn+1(t)|2
]
≤ C2

0

∫ T

0

ds1

∫ s1

0

ds2E
[

sup
0≤t≤s2

|Xn−1(t)|2
]

≤ Cn+1
0 E

[
sup

0≤t≤T
|X0(t)|2

] ∫ T

0

ds1

∫ s1

0

ds2 · · ·
∫ sn

0

dsn+1

=
(TC0)n+1

(n+ 1)!
E
[

sup
0≤t≤T

|X0(t)|2
]
.

In addition, we obtain

ε−(n+1) (TC0)n+1

(n+ 1)!
≤
∞∑
k=0

(TC0/ε)
k

k!
= eC0T/ε,

thus completing the proof. �

We can also use the same argument to show that the approximation process is
a Cauchy sequence in S2d.

4.2. Toward decoupled forward-backward stochastic differential equations.
The inequality in Theorem 4.3 below indicates that approximating the terminal
condition is the key to estimating the error between the solution and the Newton-
Kantorovitch approximation process. In this section, we consider the Newton-
Kantorovitch approximation with respect to the terminal condition. First, we show
that it converges with respect to the weighted norm ‖ · ‖α.

Theorem 4.3. Let (X,Y, Z) be a solution of the FBSDE (1.1). If Assumptions
3.1 and 3.2 hold and (X0, Y0, Z0) ∈ S2d×S2m×H2 such that X0(0) = X(0), then, for
all n ∈ N ∪ {0}, ε ∈ (0, 1) and T > 0, we obtain

‖(Y − Yn+1, Z − Zn+1)‖2α ≤ εn+1{C1‖X −X0‖2S2d + ‖(Y − Y0, Z − Z0)‖2α}, (4.4)

where C1 ≡ C1(ε) = ε−1{1 + (1 + 4(2 +TC0)‖ϕ′‖2∞)(1 + 4c20)}eαT+C0/ε, c0 = 3 and
C0 is given by Theorem 4.1 and α = 2‖f ′‖∞+4‖f ′‖2∞+12‖f ′‖∞(1+4c20)(1∨T )ε−1.

Proof. By the fundamental result in [18], the solution (X,Y, Z) exists and is unique.
For s ∈ [0, T ] and n ∈ N, we define

hn(s) = (Xn(s), Y n(s), Zn(s)) = (X(s)−Xn(s), Y (s)− Yn(s), Z(s)− Zn(s)),

and apply the mean value theorem, yielding

f(s,X(s), Y (s), Z(s))− fn(s,Xn+1(s), Yn+1(s), Zn+1(s))

= f ′(s,Xn(s), Yn(s), Zn(s))hn+1(s) + {f(s,X(s), Y (s), Z(s))

− f(s,Xn(s), Yn(s), Zn(s))− f ′(s,Xn(s), Yn(s), Zn(s))hn(s)}
= f ′(s,Xn(s), Yn(s), Zn(s))hn+1(s) + Rf (Xn(s), Yn(s), Zn(s))hn(s).



EJDE-2021/98 NEWTON-KANTOROVITCH METHOD FOR FBSDES 11

Recall (2.1) and we have, for s ∈ [0, T ], n ∈ N and h ∈ S2d × S2m ×H2,

Rf (Xn(s), Yn(s), Zn(s))h(s)

=
{∫ 1

0

f ′(s, (Xn(s), Yn(s), Zn(s)) + θh(s))dθ − f ′(s,Xn(s), Yn(s), Zn(s))
}
h(s).

This allows us to show that (Y n+1, Zn+1) satisfies the following linear BSDE

Y n+1(t)− Y n+1(T ) +

∫ T

t

Zn+1(s) dW (s)

=

∫ T

t

{f ′(s,Xn(s), Yn(s), Zn(s))hn+1(s) + Rf (Xn(s), Yn(s), Zn(s))hn(s)} ds.

Applying Itô’s formula to eαt|Y n+1(t)|2 for all α ∈ R, we obtain

eαt|Y n+1(t)|2 − eαT |Y n+1(T )|2 +

∫ T

t

eαs|Zn+1(s)|2 ds

=

∫ T

t

eαs(−α)|Y n+1(s)|2 ds− 2

∫ T

t

eαs〈Y n+1(s), Zn+1(s) dW (s)〉

+ 2

∫ T

t

eαs〈Y n+1(s), f ′(s,Xn(s), Yn(s), Zn(s))hn+1(s)

+ Rf (Xn(s), Yn(s), Zn(s))hn(s)〉ds.

(4.5)

From the Cauchy-Schwarz inequality and the inequality 2ab ≤ δ−1|a|2 + δ|b|2 for
all δ > 0, we have

2|〈Y n+1(s), f ′x(s,Xn(s), Yn(s), Zn(s))Xn+1(s)〉|

≤ 2‖f ′‖2∞|Y n+1(s)|2 +
1

2
|Xn+1(s)|2,

2|〈Y n+1(s), f ′y(s,Xn(s), Yn(s), Zn(s))Y n+1(s)〉| ≤ 2‖f ′‖∞|Y n+1(s)|2,
2|〈Y n+1(s), f ′z(s,Xn(s), Yn(s), Zn(s))Zn+1(s)〉|

≤ 2‖f ′‖2∞|Y n+1(s)|2 +
1

2
|Zn+1(s)|2,

2|〈Y n+1(s),Rf (Xn(s), Yn(s), Zn(s))hn(s)〉|
≤ δ−1|Y n+1(s)|2 + δ|Rf (Xn(s), Yn(s), Zn(s))hn(s)|2,

and

|Rf (Xn(s), Yn(s), Zn(s))hn(s)| ≤ 2‖f ′‖∞|hn(s)|. (4.6)

The right-hand side of (4.5) is therefore less than or equal to∫ T

t

eαs · {(−α) + 2‖f ′‖∞ + 4‖f ′‖2∞ + δ−1}|Y n+1(s)|2 ds

+

∫ T

t

eαs{1

2
|Zn+1(s)|2 +

1

2
|Xn+1(s)|2 + δ|Rf (Xn(s), Yn(s), Zn(s))hn(s)|2}ds

− 2

∫ T

t

eαs〈Y n+1(s), Zn+1(s) dW (s)〉.



12 D. TAGUCHI, T. TSUCHIYA EJDE-2021/98

Setting α ≡ α(δ) = 2‖f ′‖∞ + 4‖f ′‖2∞ + δ−1, we obtain

eαt|Y n+1(t)|2 − eαT |Y n+1(T )|2 +
1

2

∫ T

t

eαs{|Zn+1(s)|2 − |Xn+1(s)|2} ds

≤ δ
∫ T

t

eαs|Rf (Xn(s), Yn(s), Zn(s))hn(s)|2 ds

− 2

∫ T

t

eαs〈Y n+1(s), Zn+1(s) dW (s)〉.

(4.7)

As (Y,Z), (Yn, Zn) ∈ S2m ×H2, the local martingale{∫ t

0

eαs〈Y n+1(s), Zn+1(s) dW (s)〉
}
t∈[0,T ]

vanishes at 0. Thus, in particular, by setting t = 0, we obtain

E
[ ∫ T

0

eαs{|Zn+1(s)|2 − |Xn+1(s)|2}ds
]

≤ 2δE
[ ∫ T

0

eαs|Rf (Xn(s), Yn(s), Zn(s))hn(s)|2 ds
]

+ 2E[eαT |Y n+1(T )|2].

(4.8)

Note that∫ T

t

eαs〈Y n+1(s), Zn+1(s) dW (s)〉 =

∫ T

0

1[t,T ](s)e
αs〈Y n+1(s), Zn+1(s) dW (s)〉.

An application of the Burkholder-Davis-Gundy inequality indicates that there is a
universal c0 such that

2E
[

sup
0≤t≤T

∣∣ ∫ T

t

eαs〈Y n+1(s), Zn+1(s) dW (s)〉
∣∣]

≤ 2c0E
[( ∫ T

0

1[t,T ](s)e
2αs|Y n+1(s)|2|Zn+1(s)|2 ds

)1/2]
≤ E

[
sup

0≤t≤T
eαt/2|Y n+1(t)|

(
4c20

∫ T

0

eαs|Zn+1(s)|2 ds
)1/2]

,

where we note that an explicit upper bounded of c0 can be obtained by 3; refer to
[11, Theorem 3.28]. Hence, by considering the supremum of (4.7) and using the
inequality ab ≤ (a2/2) + (b2/2) for all a, b ∈ R, we obtain

E
[

sup
0≤t≤T

eαt|Y n+1(t)|2 − eαT |Y n+1(T )|2

+
1

2

∫ T

0

eαs{|Zn+1(s)|2 − |Xn+1(s)|2} ds
]

≤ δ
∫ T

0

eαsE[|Rf (Xn(s), Yn(s), Zn(s))hn(s)|2] ds

+
1

2
E
[

sup
0≤t≤T

eαt|Y n+1(t)|2
]

+ 2c20E
[ ∫ T

0

eαs|Zn+1(s)|2 ds
]
,

which implies that

E
[

sup
0≤t≤T

eαt|Y n+1(t)|2 − 2eαT |Y n+1(T )|2



EJDE-2021/98 NEWTON-KANTOROVITCH METHOD FOR FBSDES 13

+

∫ T

0

eαs{|Zn+1(s)|2 − |Xn+1(s)|2} ds
]

≤ 2δ

∫ T

0

eαsE[|Rf (Xn(s), Yn(s), Zn(s))hn(s)|2] ds

+ 4c20E
[ ∫ T

0

eαs|Zn+1(s)|2 ds
]
.

Then by taking the inequality (4.8) into consideration, we observe that

E
[

sup
0≤t≤T

eαt|Y n+1(t)|2 +

∫ T

0

eαs{|Zn+1(s)|2 − |Xn+1(s)|2} ds
]

≤ 2δ(1 + 4c20)

∫ T

0

eαsE[|Rf (Xn(s), Yn(s), Zn(s))hn(s)|2] ds

+ 2(1 + 4c20)eαTE[|Y n+1(T )|2] + 4c20E
[ ∫ T

0

eαs|Xn+1(s)|2 ds
]
.

If we also consider the inequality (4.6), we obtain∫ T

0

eαsE[|Rf (Xn(s), Yn(s), Zn(s))hn(s)|2] ds

≤ 6‖f ′‖∞(1 ∨ T )
{
E
[

sup
0≤t≤T

eαt|Xn(t)|2 + sup
0≤t≤T

eαt|Y n(t)|2

+

∫ T

0

eαs|Zn(s)|2 ds
]}
.

Selecting δ such that 12‖f ′‖∞(1 + 4c20)(1 ∨ T )δ = ε, we obtain

α ≡ α(δ) = 2‖f ′‖∞ + 4‖f ′‖2∞ + 12‖f ′‖∞(1 + 4c20)(1 ∨ T )ε−1.

This leads to

‖(Y n+1, Zn+1)‖2α
≤ ε‖(Xn, Y n, Zn)‖2α + 2(1 + 4c20)eαT ‖Y n+1(T )‖2L2 + (1 + 4c20)‖Xn+1‖2α.

Hence, using (4.3),

E[|Y n+1(T )|2] ≤ ‖ϕ′‖2∞{4E[|Xn(T )|2] + 2E[|Xn+1(T )|2]}
≤ 2(2 + TC0)‖ϕ′‖2∞E[|Xn(T )|2].

Thus, we obtain

2(1 + 4c20)eαT ‖Y n+1(T )‖2L2 ≤ 4(2 + TC0)‖ϕ′‖2∞(1 + 4c20)eαT ‖Xn‖2S2d .

Applying inequality (4.2) for c1 = eαT + (1 + 4(2 +TC0)‖ϕ′‖2∞)(1 + 4c20)eαT , yields

‖(Y n+1, Zn+1)‖2α ≤ ε‖(Y n, Zn)‖2α + c1‖X −X0‖2
Cn0
n!
.

For any positive sequence {an}, {bn} and ε > 0 such that an+1 ≤ bn + εan for all
n ∈ N ∪ {0}, we obtain

an+1 ≤ bn + ε(εan−1 + bn−1)

= ε2(ε−2bn + ε−1bn−1 + an−1) ≤ · · ·



14 D. TAGUCHI, T. TSUCHIYA EJDE-2021/98

≤ εn+1
{ n∑
k=0

εk−(n+1)bn−k + a0

}
.

Replacing bn = c1‖X −X0‖2C
n
0

n! for all n ∈ N, we obtain

n∑
k=0

εk−(n+1)bn−k ≤ ε−1c1‖X −X0‖2
n∑
k=0

(C0/ε)
k

k!
≤ ε−1c1‖X −X0‖2eC0/ε,

for n ∈ N ∪ {0}. Thus, we obtain

‖(Y n+1, Zn+1)‖2α ≤ εn+1(C1‖X −X0‖2 + ‖(Y 0, Z0)‖2α),

where

C1 = ε−1c1e
C0/ε = ε−1{1 + (1 + 4(2 + TC0)‖ϕ′‖2∞)(1 + 4c20)}eαT+C0/ε.

This completes the proof. �

Remark 4.4. We can also prove the so called “semilocal theorem”, which states
that the approximation process is a Cauchy sequence in S2m × H2 by the same
argument. For the definition of “semilocal”, refer to [24].

We can now prove our main result based on the following theorem.

Theorem 4.5. Let (X,Y, Z) be a solution of the FBSDE (1.1). If Assumptions
3.1 and 3.2 hold and (X0, Y0, Z0) ∈ S2d × S2m × H2 with X0(0) = X(0), then, there
exists a C3 > 0 such that, for all n ∈ N ∪ {0}, we obtain

‖(X −Xn+1, Y − Yn+1, Z − Zn+1)‖2

≤ εn+1C3‖(X −X0, Y − Y0, Z − Z0)‖2,
(4.9)

where the constant C3 = C3(ε) is bounded by the coefficients and independent of n.

Proof. By inequalities (4.2) and (4.4), we obtain

‖X −Xn+1‖2 ≤ εn+1eC0T/ε‖X −X0‖2,

and

‖(Y − Yn+1, Z − Zn+1)‖2 ≤ ‖(Y − Yn+1, Z − Zn+1)‖2α
≤ εn+1

{
C1‖X −X0‖2S2d + ‖(Y − Y0, Z − Z0)‖2α

}
,

respectively, where

C0 = 8cb,σT exp(4cb,σT ), cb,σ = ‖b′‖∞ + 18‖σ′‖∞ + ‖σ′‖2∞,

C1 ≡ C1(ε) = ε−1{1 + (1 + 4(2 + TC0)‖ϕ′‖2∞)(1 + 4c20)}eαT+C0/ε,

c0 = 3, α = 2‖f ′‖∞ + 4‖f ′‖2∞ + 12‖f ′‖∞(1 + 4c20)(1 ∨ T )ε−1.

Defining

C3 = {(C1 + eC0T/ε) ∨ eαT },
we complete the proof. �

This finally allows us to prove our main theorem.

Proof of Theorem 1.1. By setting ε = 1/2, the desired result can be obtained from
Theorem 4.5. �



EJDE-2021/98 NEWTON-KANTOROVITCH METHOD FOR FBSDES 15

Acknowledgements. We would like to express our sincere appreciation to Pro-
fessor Toshio Yamada for insightful comments. This work was supported by JSPS
KAKENHI Grant Numbers 16J00894 and 17H06833.

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Dai Taguchi

Research Institute for Interdisciplinary Science, Okayama University 3-1-1 Tsushima-

naka, Kita-ku Okayama 700-8530, Japan
Email address: dai.taguchi.dai@gmail.com

Takahiro Tsuchiya

School of Computer Science and Engineering, The University of Aizu, Japan
Email address: suci@probab.com


	1. Introduction
	2. Preliminaries
	3. Newton-Kantorovitch scheme for FBSDEs
	4. Convergence of the Newton-Kantorovitch approximation process for FBSDEs
	4.1. Convergence of the Newton-Kantorovitch approximation process for SDEs
	4.2. Toward decoupled forward-backward stochastic differential equations
	Acknowledgements

	References