Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 113, pp. 1–26.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.113

APPROXIMATIONS OF EULER-PEANO SCHEME FOR REFLECTED

STOCHASTIC DIFFERENTIAL EQUATIONS WITH NON-LIPSCHITZ

COEFFICIENTS

MINGBO ZHANG

Abstract. This article concerns reflected stochastic differential equations with reflecting bound-

ary conditions that were introduced by Lions and Sznitman [5]. We establish a theorem on the

existence and uniqueness of the strong solution when the coefficients satisfy non-Lipschitz con-
ditions. We further show that the solutions depend continuously on the initial data. Also

we construct a measurable flow of the solution, and prove that the solution is a Markov pro-

cess. The analytical solution of stochastic differential equations is generally very difficult to
obtain, so numerical approximations are important in applications. Their convergence rates are

very important for improving efficiency and for designing algorithms. So we characterize the

convergence rate of Euler’s approximations under some restrictions on the coefficients.

1. Introduction

Let (Ω,F , P ) be a complete probability space and {Ft}t≥0 be the right continuous filtration with
the property that Ft contains all null set of (Ω,F , P ). Let B = (Bt) be an {Ft} Brownian motion
taking values in Rn. In this article we are concerned with the reflected stochastic differential
equations (RSDEs)

x(t) = x0 +

∫ t

0

σ(s, x(s))dB(s) +

∫ t

0

b(s, x(s))ds+ ϕ(t), x0 ∈ D̄,

ϕ(t) =

∫ t

0

n(s)d∥ϕ∥[0,s], ∥ϕ∥[0,t] =
∫ t

0

1(x(s)∈∂D)d∥ϕ∥[0,s],
(1.1)

where t ∈ [0,+∞) and, d × n-dimensional matrix σ(t, x) and d-vector b(t, x) are continuous in x
for any t, measurable in (t, x).

By a direct approach based on the Skorokhod problem (SP), Lions and Sznitman [5] proved
the existence and uniqueness of the solution of equation (1.1) with reflecting boundary conditions,
where the coefficients σ and b satisfy the Lipschitz conditions. Aida and Sasaki [1] showed the
convergence in Lp of Euler type approximation of equation (1.1). When the coefficients σ and
b are Lipschitz continuous, the solvability of Itô stochastic equations is obvious. Recently, Ren
and Wu [9] studied approximate continuity and the support of reflected stochastic differential
equations. When the coefficients σ and b do not satisfy the Lipschitz conditions, the solvability
of Itô stochastic equations has been studied extensively, see for example [4, 15]. In [15], the
authors studied the pathwise uniqueness of solution of vector stochastic differential equations with
conditions weaker than Lipschitz conditions on the coefficients. Using Euler’s method, Krylov [4]
investigated the Itô stochastic equations with the coefficients σ and b satisfying one side Lipschitz
continuous condition. Based on the Euler scheme, Wang [16] constructed a measurable flow of the
solution for stochastic differential equation with non-Lipschitz coefficients, and proved that the
solution is a Markov process.

2020 Mathematics Subject Classification. 60F25, 60H20, 60G17.
Key words and phrases. Reflected stochastic differential equations; Euler-Peano’s method;

non-Lipschitz continuous coefficients.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted April 10, 2025. Published December 2, 2025.

1



2 M. ZHANG EJDE-2025/113

First, we introduce some notation and conditions on the domain D, and the coefficients σ and
b. We set

∥ϖ∥∞,[s,t] = max
s≤u≤v≤t

|ϖ(u)−ϖ(v)|,

∥ϖ∥[s,t] = sup
∆

N∑
k=1

|ϖ(tk)−ϖ(tk−1)|,

where ∆ = {(t0, . . . , tN ) : s = t0 < . . . < tN = t} is a partition of the interval [s, t]. We assume
that D is a connected domain in Rd, and define the set Nx of inward unit normal vectors at the
boundary point x ∈ ∂D by

Nx = ∪r>0Nx,r,

Nx,r = {n ∈ Rd; |n| = 1, B(x− rn, r) ∩D = ∅}.

where B(z, r) = {y ∈ Rd; |y − z| < r}, z ∈ Rd, and r > 0.
The reflecting boundary conditions on D were firstly introduced by Lions [5] and Aida [1], as

follows:

(A1) (uniform exterior sphere condition). There exists a constant r0 > 0, such that for any
x ∈ ∂D,

Nx = Nx,r0 ̸= ∅,
(A2) There exist constants δ > 0 and β ≥ 1 satisfying: for any x ∈ ∂D, there exists a unit

vector lx, such that

(lx,n) ≥
1

β
,

for any n ∈ ∪y∈B(x,δ)∩∂DNy.

(A3) There exists a C2
b function f and a positive constant γ such that for any x ∈ ∂D, y ∈ D̄,

n ∈ Nx, it holds that

⟨y − x,n⟩+ 1

γ
⟨(Df)(x),n⟩|x− y|2 ≥ 0. (1.2)

Remark 1.1. We first observe that when D is a convex domain, condition (A1) holds for arbitrary
r0 > 0, while condition (A3) is automatically satisfied when f ≡ 0. The admissibility condition on
Dintroduced in [5] requires that the domain can be approximated by smooth-boundary domains
in a specific sense. While we do not utilize this property in our current work, we refer interested
readers to [5] for details.

Notably, Saisho and Tanaka [13] provide an explicit example of a domain satisfying conditions
(A1) and (A2). Furthermore, for piecewise smooth bounded domains in Rd that satisfy (A1) and
(A2), [5, Remark 3.1 ] establishes that condition (A3) holds locally.

Remark 1.2. D satisfies (A2) if it satisfies the following uniform interior cone condition: There
exist δ > 0 and α ∈ (0, 1) with the following property: for any x ∈ ∂D there exists a unit vector
lx such that

C(y, lx, α) ∩B(x, δ) ⊂ D̄, y ∈ B(x, δ) ∩ ∂D,

where C(y, lx, α) is the convex cone with vertex y, defined by

C(y, lx, α) = {z ∈ Rd : ⟨z − y, lx⟩ ≥ α|z − y|}.

Let

|f(x)| ∨ |Df(x)| ∨ ∥D2f(x)∥ ∨ ∥(Dif(x)Djf(x)i,j∥ ≤ C,

c0 ≤ exp(− 2

γ
f(x)) + exp(− 2

γ
(f(x) + f(y))) ≤ C0,

where C, c0, C0 are positive constants. Fixed R > 0 arbitrarily, KR : R+ → R+ is an increasing
concave continuous function satisfying ∫

0+

du

KR(u)
= ∞.



EJDE-2025/113 REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS 3

Next, we introduce the following assumptions on the coefficients σ and b.

(A4) Let R > 0 be fixed arbitrarily. D0 is a positive constant such that 4C
γ + 2 ≤ D0. For all

t ∈ [0,∞), |x| ∨ |y| ≤ R, the following locally weak monotonicity condition

2⟨x− y, b(t, x)− b(t, y)⟩+D0∥σ(t, x)− σ(t, y)∥2 ≤ g(t)KR(|x− y|2) (1.3)

is satisfied, where g is a non-negative function such that∫ +∞

0

g(s)ds < ∞

(A5) For each 0 ≤ s ≤ t < ∞ and 0 < R < +∞, there exists a positive function λ(·) such that∫ t

s

sup
|x|≤R

(|b(u, x)|+ ∥σ(u, x)∥2)du ≤
∫ t

s

λ(u)du ≤
∫ +∞

0

λ(u)du < +∞.

Motivated by the work of Krylov [4], we introduce the non-Lipschitz continuous coefficients’
conditions (A4) and (A5). Our aim of the present paper is to study the solvability of equation
(1.1) with coefficients σ and b satisfying (A4) and (A5). We also want to explore the properties
of the solution of equation (1.1). As we know, if the solution constitutes a stochastic flow of
homeomorphisms, many useful properties, such as Markov property, can be constructed by the
flow property. In our case, however, the existence of such a well-behaved modification cannot be
guaranteed, as the validity of the flow property remains uncertain. Therefore, we investigate the
Markov property of solutions to (1.1) under our assumptions.

In the convergence analysis of numerical methods for stochastic differential equations with
reflection term (see (??) below), the core challenge lies in the circular dependency between the
reflection term and the state variables: moment estimation of the state variables {xn} depends on
estimates of the reflection term, while the reflection term’s estimation reciprocally relies on {xn}’s
estimates. This bidirectional dependence renders traditional approaches (e.g., directly adopting
estimation frameworks from references [4] and [12]) inapplicable. To resolve this fundamental
difficulty, it is essential to fully leverage the geometric and analytical properties of the reflection
domain D by constructing non-negative functions. Through applying the Itô formula and stopping
time techniques, the influence of the reflection term can be transformed into a controllable negative
term, effectively decoupling the circular dependency between state variables and reflection terms.
Subsequently, Bihari’s inequality is employed instead of Gronwall’s inequality to handle non-
Lipschitz coefficient scenarios, thereby overcoming the key difficulty in the proof.

This article is organized as follows. We recall in Section 2 the existence of solution of Skorohod
problem with reflection boundary conditions and the basic results of Skorohod problem. In Section
3, we prove the existence and uniqueness of the solution of equation (1.1) under the assumptions
(A1)–(A5). Furthermore, we show that the solution of equation (1.1) is non-explosive and contin-
uous with respect to the initial data. On the other hand, when the domain D and coefficients σ, b
are bounded, we also obtain the solvability of equation (1.1) under the assumptions (A1), (A2),
(A4) and (A5). In Section 4, we present the stability of solution of equation (1.1). In Section 5, we
construct a measurable flow of equation (1.1), and also prove that the solution is a Markov process.
In Section 6, we assume that the coefficients b and σ satisfy some slightly stronger conditions than
(A4) and (A5), then we get a characterization of convergence rate of Euler’s approximations.

2. Skorohod problem

In this section, we recall some basic results of the Skorohod problem with reflection boundary
conditions. For details, we refer to Aida and Sasaki [1]. For any fixed 0 < T < ∞, let ϖ ∈
C([0, T ];Rd) with ϖ(0) ∈ D̄. The pair of paths (x, ϕ) ∈ C([0, T ];Rd)× C([0, T ];Rd) is said to be a
solution of Skorohod problem with reflecting boundary conditions associated with ϖ, if:

• x is a continuous function and x(t) ∈ D̄, for all t ∈ [0, T ], with x0 := x(0) = ϖ(0),
• x(t) = ϖ(t) + ϕ(t) for every t ≥ 0,



4 M. ZHANG EJDE-2025/113

• ϕ is an Rd valued continuous function with bounded variations on each finite interval,
ϕ(0) = 0 and

ϕ(t) =

∫ t

0

n(s)d∥ϕ∥[0,s], (2.1)

∥ϕ∥[0,t] =
∫ t

0

1(x(s)∈∂D)d∥ϕ∥[0,s], (2.2)

where n(t) ∈ Nx if x(t) ∈ ∂D. We denote this Skorohod problem as (ϖ,D,n). The following
lemma is important for our study.

Lemma 2.1 ([1, Theorem 2.2]). For any fixed 0 < T < ∞, let ϖ ∈ C([0, T ];Rd) and x0 ∈ D̄.
Then under assumptions (A1) and (A2), there exists a unique solution to the Skorohod problem
(ϖ,D,n) for any continuous path ϖ. Moreover, the mapping κ : ϖ 7→ x is continuous in the
uniform convergence topology.

3. Solvability of RSDEs

In this section, we study the existence and uniqueness of the solutions of reflected stochastic
differential equations driven by Brownian motion under our conditions. To consider the equation
(1.1) with initial random variable x0 : Ω → D̄ being F0 measurable. A pair of Ft adapted
continuous processes (x(t, x0), ϕ(t, x0)) is called a solution to (1.1) if the following holds: for

W (t, x0) = x0 +

∫ t

0

σ(s, x(s, x0))dB(s) +

∫ t

0

b(s, x(s, x0))ds. (3.1)

the pair (x(x0, ., ω), ϕ(x0, ., ω)) is a solution of the Skorohod problem (W (·, ω), D,n) for almost
all ω ∈ Ω.

We apply the method of Euler approximation to discuss the existence and uniqueness of the
solutions of RSDEs. Let n ∈ N. Let (xn(t, x0), ϕn(t, x0)) (0 ≤ t ≤ T ) be the solution to the
Skorohod problem inductively which is given by xn(0, x0) = x0 and

xn(t, x0) = x0 +

∫ t

0

b(s, xn(πn(s), x0))ds+

∫ t

0

σ(s, xn(πn(s), x0))dB(s) + ϕn(t, x0), (3.2)

where xn(0, x0) = x0 and πn(s) =
k
2n for s ∈ ( k

2n ,
(k+1)
2n ], k = 0, 1, . . . , 2n, . . . . By Theorem 2.1,

we have a unique solution of (3.2). In fact, once xn(t, x0) is obtained for 0 ≤ t ≤ k2−n, xn(t, x0)
for k2−n ≤ t ≤ (k + 1)2−n is uniquely determined as the solution of the Skorohod equation

xn(t, x0) = xn(k2
−n, x0)+

∫ t

k2−n

b(s, xn(k2
−n, x0))ds+

∫ t

k2−n

σ(s, xn(k2
−n, x0))dB(s)+ϕn(t, x0).

The following theorems are main results in this section.

Theorem 3.1. Assume (A1)–(A5). Let x0 : Ω → D̄ be F0 measurable and such that E[|x0|2] < ∞.
Then the equation (1.1) with initial random variable x0 has a solution, which is unique up to
indistinguishability and non-explosive.

Theorem 3.2. Assume (A1), (A2), (A4), (A5). Suppose that D is a bounded domain and σ, b
are bounded functions. Then the equation (1.1) with initial random variablex0 : Ω → D̄ has a
solution, which is unique up to indistinguishability.

Before proving the above theorems, we introduce the following helpful lemmas.

Lemma 3.3 (Bihari’s inequality [3]). Let ρ : R+ 7→ R+ be a continuous non-decreasing function.
If h(s), χ(s) are two strictly positive functions on R+ such that

χ(t) ≤ ϑ+

∫ t

0

ρ(χ(s))h(s)ds, t ≥ 0,

then

χ(t) ≤ Φ−1(Φ(ϑ) +

∫ t

0

h(s)ds),



EJDE-2025/113 REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS 5

where Φ(x) := −
∫ a

x
du
ρ(u) is well defined for some a > 0.

Remark 3.4. Let ρ : R+ 7→ R+ be a continuous non-decreasing function. If p(s), q(s) and χ(s)
are strictly positive functions on R+ such that

χ(t) ≤ ϑ+

∫ t

0

χ(s)p(s)ds+

∫ t

0

ρ(χ(s))q(s)ds

≤ ϑ+

∫ t

0

(ρ+ I)(χ(s))(p(s) + q(s))ds, t ≥ 0,

where I denotes the identity mapping. Then by Bihari’s inequality, we have

χ(t) ≤ Φ−1(Φ(ϑ) +

∫ t

0

p(s) + q(s)ds), t ≥ 0,

where Φ(x) := −
∫ a

x
du

ρ(u)+u . In the proof of Proposition 3.7 and Theorem 5.2 we will use this fact.

Lemma 3.5 ([8, Lemma 3.1.3]]). Let w(t) be a continuous nonnegative, R+-valued, Ft adapted
process, and ι be a stopping time. For ε ∈ (0,+∞), set

ιε = ι ∧ inf{t ≥ 0 : w(t) ≥ ε},
where as usual we set inf{∅} = ∞, then

P{ sup
0≤t≤ι

w(t) ≥ ε} ≤ 1

ε
E[w(ιε)].

Lemma 3.6. Removing both (A1) and (A2) in Theorem 3.1, we have

sup
n

E[|xn(t, x0)|2] < ∞, t ∈ [0,+∞).

Proof. Denote θt := |xn(t, x0) − x0|2, and τ∗n(R) = inf{t > 0 : |xn(t, x0)| ≥ R}, for R > 0. For
simplicity, we denote xn(s, x0) by xn(s). Applying Itô’s formula, one can write

z1(xn(t ∧ τ∗n(R)))θu∧τ∗
n(R) =

5∑
i=1

Λi,

where z1(x) = exp(− 2
γ f(x)) and

Λ1 = 2

∫ t∧τ∗
n(R)

0

z1(xn(s))⟨xn(s)− x0, σ(s, xn(πn(s)))dB(s)⟩

− 2

γ

∫ t∧τ∗
n(R)

0

z1(xn(s))θs⟨Df(xn(s)), σ(s, xn(πn(s)))dB(s)⟩,

Λ2 = 2

∫ t∧τ∗
n(R)

0

z1(xn(s))⟨xn(s)− x0, b(s, xn(πn(s)))⟩ds

+

∫ t∧τ∗
n(R)

0

tr(z1(xn(s))(σ(s, xn(πn(s))))
Tσ(s, xn(πn(s))))ds

− 2

γ

∫ t∧τ∗
n(R)

0

z1(xn(s))θs⟨Df(xn(s)), b(s, xn(πn(s)))⟩ds,

Λ3 = 2

∫ t∧τ∗
n(R)

0

z1(xn(s))⟨xn(s)− x0, dϕn(s))⟩

− 2

γ

∫ t∧τ∗
n(R)

0

z1(xn(s))θs⟨Df(xn(s)), dϕn(s))⟩,

Λ4 = − 1

γ

∫ t∧τ∗
n(R)

0

tr(z1(xn(s))θsσ
T (xn(πn(s)))D

2f(xn(s))σ(s, xn(πn(s))))ds

+
2

γ2

∫ t∧τ∗
n(R)

0

z1(xn(s))θs|(σ(s, xn(πn(s))))
TDf(xn(s))|2ds,



6 M. ZHANG EJDE-2025/113

Λ5 = − 4

γ

∫ t∧τ∗
n(R)

0

z1(xn(s))⟨σT (xn(πn(s)))Df(xn(s)), (σ(s, xn(πn(s))))
T (xn(s)− x0)⟩ds.

For the terms Λ2 and Λ4, we have the following estimate

Λ2 + Λ4 ≤
∫ t∧τ∗

n(R)

0

z1(xn(s))(∥σ(s, xn(πn(s)))∥2 + |b(s, xn(πn(s)))|)ds

+ (
2C

γ2
+

2C

γ
+ 1)

∫ t∧τ∗
n(R)

0

z1(xn(s))θs(∥σ(s, xn(πn(s)))∥2 + |b(s, xn(πn(s)))|)ds.

It is obvious that the term Λ3 ≤ 0 by condition (A3). For the term Λ5, we have

Λ5 ≤ 4C

γ

∫ t∧τ∗
n(R)

0

z1(xn(s))∥σT (xn(πn(s)))∥2|xn(s)− x(0)|ds

≤ 2C

γ

∫ t∧τ∗
n(R)

0

z1(xn(s))θs∥σT (xn(πn(s)))∥2ds+
2C

γ

∫ t∧τ∗
n(R)

0

z1(xn(s))∥σT (xn(πn(s)))∥2ds.

With these estimates, taking the expectation both side gives

E[θt∧τ∗
n(R)] ≤ α1E[

∫ t∧τ∗
n(R)

0

θs(|b(s, xn(πn(s)))|+ ∥σ(s, xn(πn(s)))∥2)ds]

+ α2E[

∫ t∧τ∗
n(R)

0

(∥σ(s, xn(πn(s)))∥2 + |b(s, xn(πn(s)))|)ds],

where α1 = C0

c0
( 2Cγ2 + 4C

γ + 1) and α2 = C0

c0
( 2Cγ + 1). Subsequently,

E[θt∧τ∗
n(R)] ≤ α1E[

∫ t

0

θs∧τ∗(R) sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds]

+ α2

∫ +∞

0

sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds.

By Gronwall’s inequality and assumption (A5), we obtain

E[θt∧τ∗
n(R)] ≤ α2 exp

[
α1

∫ t

0

sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds
] ∫ +∞

0

sup
|x|≤R

(∥σ(s, x)∥2 + |b(s, x)|)ds

≤ α2 exp
[
α1

∫ t

0

λ(s)ds
] ∫ +∞

0

λ(s)ds

:= C(t) < ∞,

where C(t) is independent of R and n.
Let

lim inf
R→∞

τ∗n(R) = η.

Suppose that P (η < ∞) > 0. Then for some T > 0, P (η < T ) > 0. Lemma 3.5, we have

0 < P (η < T )

≤ P (τ∗n(R) < T )

= P ( sup
t∈(0,T ]

|xn(t ∧ τ∗n(R), x0)| ≥ R)

≤ E[|xn(T ∧ τ∗n(R), x0)|2]
R2

≤ 2E[(|xn(T ∧ τ∗n(R), x0)− x0|)2 + 2E[|x0|2]]
R2

=
2E[θT∧τ∗

n(R)] + 2E[|x0|2]
R2

≤ 2C(T ) + 2E[|x0|2]
R2

→ 0, as R → ∞.



EJDE-2025/113 REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS 7

Then η = ∞. By Fatou’s Lemma

E[θt] ≤ C(t) < ∞. (3.3)

Then the claim follows. □

We denote pn(s) = xn(πn(s), x0)− xn(s, x0). Clearly, xn(s, x0) satisfies

xn(t, x0) = x0 +

∫ t

0

b(s, xn(s) + pn(s))ds+

∫ t

0

σ(s, xn(s) + pn(s))dBs + ϕn(t), (3.4)

where xn(s) := xn(s, x0).
Let x0, y0 : Ω → D̄ be F0 measurable. For ϵ0 > 0, R > 0, we define

τn,m(R, x0, y0) = inf{t : |xn(t, x0)| ∨ |xm(t, y0)| ≥ R},
τ̃m,n(x0, y0) = inf{t : |xn(t, x0)− xm(t, y0)|2 ≥ ϵ0},
Γm,n(R, x0, y0) = τ̃m,n(x0, y0) ∧ τn,m(R, x0, y0).

Proposition 3.7. Assume (A3)–(A5). Let x0, y0 : Ω → D̄ be F0 measurable such that E[|x0|2] <
∞, E[|y0|2] < ∞. Then there exists a positive function η(u) on [0,+∞) which is continuous at
origin and η(0) = 0, such that

E[|xn(t ∧ Γm,n(R, x0, y0), x0)− xm(t ∧ Γm,n(R, x0, y0), y0)|2]

≤ F−1
(
F
(
η(

1

2n
) + η(

1

2m
)
)
+ 3α

∫ t

0

(g(s) + λ(s))ds
)

+ F−1
(
F
(C0

c0
E[|x0 − y0|2] + 2η(

1

2m
)
)
+ 3α

∫ t

0

(g(s) + λ(s))ds
)
,

where α = 3C0

c0
( 8Cγ + 4C

γ2 + 1
3 ), and

F (u) = −
∫ a

u

dv

KR(v) + v
. (3.5)

Proof. Fix m,n and R temporarily and let t ∈ [0,+∞). For simplicity, we denote xn(t, x0),
xm(t, x0) and Γm,n(R, x0, y0) by xn(t), xm(t) and Γm,n, respectively. Set

yt = xn(t)− xm(t), ζt = |yt|2.
Let

z(x, y) = exp(− 2

γ
(f(x) + f(y))),

zt = z(xm(t), xn(t))ζt.

By Itô formula, we have

zt∧Γm,n
:=

6∑
i=1

Ii,

where

I1 = 2

∫ t∧Γm,n

0

z(xm(s), xn(s))⟨ys, (σ(s, xn(πn(s)))− σ(s, xm(πm(s)))dB(s)⟩

− 2

γ

∫ t∧Γm,n

0

zs[⟨Df(xn(s)), (σ(s, xn(πn(s)))dB(s)⟩

+ ⟨Df(xm(s)), σ(s, xm(πm(s)))dB(s)⟩],

I2 = 2

∫ t∧Γm,n

0

z(xm(s), xn(s))⟨ys, b(s, xn(πn(s)))− b(s, xm(πm(s)))⟩ds

+

∫ t∧Γm,n

0

tr{z(xm(s), xn(s))(σ(s, xn(πn(s)))− σ(s, xm(πm(s)))T

× (σ(s, xn(πn(s)))− σ(s, xm(πm(s)))}ds,



8 M. ZHANG EJDE-2025/113

I3 = − 2

γ

∫ t∧Γm,n

0

zs[⟨Df(xn(s)), b(s, xn(πn(s)))⟩

+ ⟨Df(xm(s)), b(s, xm(πm(s)))⟩]ds,

I4 = 2

∫ t∧Γm,n

0

z(xm(s), xn(s))⟨ys, dϕn(s)− dϕm(s)⟩

− 2

γ

∫ t∧Γm,n

0

zs[⟨Df(xn(s)), dϕn(s)⟩+ ⟨Df(xm(s)), dϕm(s)⟩],

I5 = − 1

γ

∫ t∧Γm,n

0

zs tr[σ
T (s, xn(πn(s)))D

2f(xn(s))σ(s, xn(πn(s)))

+ σT (s, xm(πm(s)))D2f(xm(s))σT (s, xm(πm(s)))]ds

+
2

γ2

∫ t∧Γm,n

0

zs[|σT (s, xn(πn(s)))Df(xn(s))|2

+ |σT (s, xm(πm(s)))(Df(xm(s)))|2

+ 2⟨σT (s, xn(πn(s)))Df(xn(s)), σ
T (s, xm(πm(s)))Df(xm(s))⟩]ds,

I6 = − 4

γ

∫ t∧Γm,n

0

tr
{
z(xm(s), xn(s))

×
∣∣〈σ(s, xn(πn(s)))− σ(s, xm(πm(s)))T (xn(s)− xm(s)), σ(s, xn(πn(s)))Df(xn(s))

〉
+
〈
(σ(s, xn(πn(s)))− σ(s, xm(πm(s)))T (xn(s)− xm(s)), σ(s, xm(πm(s)))Df(xm(s))

〉]}
ds.

For the terms I3 and I5, we have

I3 + I5 ≤ (
4C

γ
+

8C

γ2
)

∫ t∧Γm,n

0

zs sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds.

For the term I4, the condition (A3) implies

I4 = 2

∫ t∧Γm,n

0

z(xm(s), xn(s))[⟨ys, n(xn(s))⟩ −
1

γ
ζs⟨Df(xn(s)), n(xn(s))⟩]d∥ϕn∥[0,s]

+ 2

∫ t∧Γm,n

0

z(xm(s), xn(s))[−⟨ys, n(xm(s))⟩ − 1

γ
ζs⟨Df(xm(s)), n(xm(s))⟩]d∥ϕm∥[0,s] ≤ 0.

Hence, we can omit this term. We estimate the term I6 as follows,

I6 ≤ 4C

γ

∫ t∧Γm,n

0

z(xm(s), xn(s))∥σ(s, xn(πn(s)))− σ(s, xm(πm(s)))∥

× |xn(s)− xm(s)|(∥σ(s, xn(πn(s)))∥+ ∥σ(s, xm(πm(s)))∥)ds

≤ 2C

γ

∫ t∧Γm,n

0

z(xm(s), xn(s))∥σ(s, xn(πn(s)))− σ(s, xm(πm(s)))∥2

+
4C

γ

∫ t∧Γm,n

0

zs(∥σ(s, xn(πn(s)))∥2 + ∥σ(s, xm(πm(s)))∥2)ds.



EJDE-2025/113 REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS 9

These estimations lead to

zt∧Γm,n
≤ 2

∫ t∧Γm,n

0

z(xm(s), xn(s))⟨ys, b(s, xn(πn(s)))− b(s, xm(πm(s)))⟩ds

+ (
2C

γ
+ 1)

∫ t∧Γm,n

0

z(xm(s), xn(s))∥σ(s, xn(πn(s)))− σ(s, xm(πm(s)))∥2ds

+ (
8C

γ
+

8C

γ2
)

∫ t∧Γm,n

0

zs sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds+ I1

≤ 4C0

∫ t∧Γm,n

0

(|pn(s)|+ |pm(s)|) sup
|x|≤R

|b(s, x))|ds

+ C0

∫ t∧Γm,n

0

g(s)KR(|xn(πn(s))− xm(πm(s))|2)ds

+ C0(
8C

γ
+

8C

γ2
)

∫ t∧Γm,n

0

ζs sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds+ I1.

(3.6)

From

|xn(πn(s))− xm(πm(s))|2 ≤ 3(|xn(s)− xm(s)|2 + |xn(πn(s))− xn(s)|2 + |xm(πm(s))− xm(s, x)|2)

and

ζs ≤ |xn(s)− xm(s)|2 + |xn(πn(s))− xn(s)|2 + |xm(πm(s))− xm(s, x)|2,

we obtain

E[ζt∧Γm,n
]

≤ 4
C0

c0
E[

∫ t∧Γm,n

0

(|pn(s)|+ |pm(s)|) sup
|x|≤R

|b(s, x))|ds]

+ α

∫ t

0

(g(s) + sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2))
{
KR

(
3E[|xn(s ∧ Γm,n)− xm(s ∧ Γm,n)|2]

+ 3E[|pn(s ∧ Γm,n)|2 + |pm(s ∧ Γm,n)|2]
)
+ 3E[|xn(s ∧ Γm,n)− xm(s ∧ Γm,n)|2]

+ 3E[|pn(s ∧ Γm,n)|2 + |pm(s ∧ Γm,n)|2]
}
ds,

where α = 3C0

c0
( 8Cγ + 8C

γ2 + 1
3 ). Applying Itô formula as the proof in Lemma 3.6 and by simple

calculations, thus

E[|pn(t ∧ Γm,n)|2] ≤ ς1E[

∫ t∧Γm,n

πn(t)∧Γm,n

|pn(s)|2 sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds]

+ ς2E[

∫ t∧Γm,n

πn(t)∧Γm,n

sup
|x|≤R

(∥σ(s, x)∥2 + |b(s, x)|)ds]

≤ ς1E[

∫ t

πn(t)

|pn(s ∧ Γm,n)|2 sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds]

+ ς2

∫ t

πn(t)

sup
|x|≤R

(∥σ(s, x)∥2 + |b(s, x)|)ds

≤ ς1E[

∫ t

πn(t)

|pn(s ∧ Γm,n)|2 sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds]

+ ς2 sup
|u−v|≤ 1

2n

∫ v

u

sup
|x|≤R

(∥σ(s, x)∥2 + |b(s, x)|)ds,



10 M. ZHANG EJDE-2025/113

where ς1 = C0

c0
( 2Cγ2 + 3C

γ +1) and ς2 = C0

c0
( 2Cγ +1) . Then by Gronwall’s inequality and assumption

(A5) we rewrite

E[|pn(t ∧ Γm,n)|2]

≤ ς2 exp
(
ς1

∫ t

πn(t)

sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds
)

sup
|u−v|≤ 1

2n

∫ v

u

sup
|x|≤R

(∥σ(s, x)∥2 + |b(s, x)|)ds

≤ ς2 exp
(
ς1

∫ t

πn(t)

λ(s)ds
)

sup
|u−v|≤ 1

2n

∫ v

u

λ(s)ds

≤ sup
|u−v|≤ 1

2n

ς2 exp
(
ς1

∫ v

u

λ(s)ds
)∫ v

u

λ(s)ds.

(3.7)
We set

η0(δ) := sup
|u−v|≤δ

ς2 exp(ς1

∫ v

u

λ(s)ds)

∫ v

u

λ(s)ds,

h1(t) := 3(η0(
1

2n
) + η0(

1

2m
)) + 3E[|xn(s ∧ Γm,n)− xm(s ∧ Γm,n)|2].

Then it follows from the assumption (A5) and (3.7) that

h1(t) ≤ 3(η0(
1

2n
) + η0(

1

2m
))

+ 12
C0

c0
E

∫ t

0

(|pn(s ∧ Γm,n)|+ |pm(s ∧ Γm,n)|) sup
|x|≤R

|b(s, x)|ds]

+ 3α

∫ t

0

(g(s) + sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2))(KR + I)(h1(s))ds

≤ η(
1

2n
) + η(

1

2m
) + 3α

∫ t

0

(g(s) + sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2))(KR(h1(s)) + h1(s))ds,

where

η(u) := 3η0(u) +
(
12

C0

c0

√
η0(u)

)∫ ∞

0

λ(s)ds, u ≥ 0.

Combining this with Bihari’s inequality and assumption (A5), we have

h1(t) ≤ F−1
(
F
(
η(

1

2n
) + η(

1

2m
)
)
+ 3α

∫ t

0

(g(s) + λ(s))ds
)
. (3.8)

Similarly, one can write

E[|xm(t ∧ Γm,n, x0)− xm(t ∧ Γm,n, y0)|2]

≤ C0

c0
E[|x0 − y0|2] + 4

C0

c0
E

∫ t∧Γm,n

0

(|pm(x0, s)|+ |pm(y0, s)|) sup
|x|≤R

|b(s, x)|ds]

+ α

∫ t

0

(g(s) + sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2))

×
{
KR

(
3E[|xm(s ∧ Γm,n, x0)− xm(s ∧ Γm,n, y0)|2]

+ 3E[|pm(x0, s ∧ Γm,n)|2 + |pm(y0, s ∧ Γm,n)|2]
)

+ 3E[|xm(s ∧ Γm,n, x0)− xm(s ∧ Γm,n, y0)|2]

+ 3E[|pm(x0, s ∧ Γm,n)|2 + |pm(y0, s ∧ Γm,n)|2]
}
ds,

where

pm(x0, s) = xm(πn(s), x0)− xm(s, x0), pm(y0, s) = xm(πn(s), y0)− xm(s, y0).



EJDE-2025/113 REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS 11

We denote

h2(t) := 6η0(
1

2m
) + 3E[|xm(s ∧ Γm,n, x0)− xm(s ∧ Γm,n, y0)|2.

From this and (3.7), we have

h2(t) ≤
C0

c0
E[|x0 − y0|2] + 2η(

1

2m
)

+ 3α

∫ t

0

(g(s) + sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2))(KR + I)(h2(s))ds.

Then Bihari’s inequality leads to

h2(t) ≤ F−1
(
F
(C0

c0
E[|x0 − y0|2] + 2η(

1

2m
)
)
+ 3α

∫ t

0

(g(s) + λ(s))ds
)
. (3.9)

The claim follows from (3.8) and (3.9). □

Corollary 3.8. For E[|x0|2] < ∞, E[|y0|2] < ∞, T > 0 and ε > 0, then we have

lim
m,n→∞,E[|x0−y0|2]→0

P (sup
t≤T

|xn(t, x0)− xm(t, y0)| ≥ ε) = 0. (3.10)

In particular, we have

lim
m,n→∞

P (sup
t≤T

|xn(t, x0)− xm(t, x0)| ≥ ε) = 0. (3.11)

Proof. By applying Lemma 3.5, we obtain that for any ε > 0,

P (sup
s≤t

|xn(s, x0)− xm(s, y0)|2 ≥ ε)

≤ P ( sup
s≤t∧τn,m(R,x0,y0)

|xn(s, x0)− xm(s, y0)|2 ≥ ε) + P (τn,m(R, x0, y0) ≤ t)

≤ P ( sup
s≤t∧τn,m(R,x0,y0)

|xn(s, x0)− xm(s, y0)|2 ≥ ε) + P (|xn(t ∧ τn,m(R, x0, y0), x0)| ≥ R)

+ P (|xm(t ∧ τn,m(R, x0, y0), x0)| ≥ R)

≤ P ( sup
s≤t∧τn,m(R,x0,y0)

|xn(s, x0)− xm(s, y0)|2 ≥ ε) +
E[|xn(t ∧ τn,m(R, x0, y0), x0)|2]

R2

+
E[|xm(t ∧ τn,m(R, x0, y0), x0)|2]

R2

≤ 1

ε2
E[|xn(t ∧ Γm,n, x0)− xm(t ∧ Γm,n, y0)|2] +

E[|xn(t ∧ τn,m(R, x0, y0), x0)|2]
R2

+
E[|xm(t ∧ τn,m(R, x0, y0), x0)|2]

R2
.

Furthermore, by Proposition 3.7, one can easily write

P (sup
s≤t

|xn(s, x0)− xm(s, y0)|2 ≥ ε)

≤ 1

ε2
F−1

(
F
(
η(

1

2n
) + η(

1

2m
)
)
+ 3α

∫ t

0

(g(s) + λ(s))ds
)

+
1

ε2
F−1

(
F
(C0

c0
E[|x0 − y0|2] + 2η(

1

2n
)
)
+ 3α

∫ t

0

(g(s) + λ(s))ds
)

+
E[|xn(t ∧ τn,m(R, x0, y0), x0)|2]

R2
+

E[|xn(t ∧ τn,m(R, x0, y0), x0)|2]
R2

.

Therefore, letting R → ∞ and applying Lemma 3.6, we have

sup
E[|x0−y0|2]≤δ

P (sup
s≤t

|xn(s, x0)− xm(s, y0)|2 ≥ ε)

≤ 1

ε2
F−1

(
F
(
η(

1

2n
) + η(

1

2m
)
)
+ 3α

∫ t

0

(g(s) + λ(s))ds
)



12 M. ZHANG EJDE-2025/113

+
1

ε2
F−1

(
F
(C0

c0
δ + 2η(

1

2n
)
)
+ 3α

∫ t

0

(g(s) + λ(s))ds
)
,

which implies

lim
m,n→∞,E[|x0−y0|2]→0

P (sup
s≤t

|xn(s, x0)− xm(s, y0)|2 ≥ ε) = 0.

Considering that t is arbitrary, this completes the proof. □

Proof of Theorem 3.1. We denote the P-limit of {xn(t, x0)}n by x(t, x0) in C([0, T ];Rd). We now
can select a subsequence ( denoted by {xnk

(t, x0)}), such that

sup
t≤T

∣∣∣xnk
(t, x0)− x(t, x0)|

a.s−−→ 0, as k → ∞. (3.12)

Next we check that x(t, x0) is a solution of equation (1.1).. By equicontinuity of {xnk
(t, x0)} in t

and (3.12), we have

sup
t≤T

|xnk
(πnk

(t), x0)− x(t, x0)|
a.s−−→ 0, as k → ∞. (3.13)

It is obvious that

Wnk
(t, x0) :=

∫ t

0

b(s, xnk
(πnk

(s), x0))ds+

∫ t

0

σ(s, xnk
(πnk

(s), x0))dB(s)

→
∫ t

0

b(s, x(s, x0))ds+

∫ t

0

σ(s, x(s, x0))dB(s) =: W (t, x0), in L2(Ω),

as k → ∞, by the fact that b and σ are both continuous with respect to x. Then there exists a
subsequence {nk} such that

sup
0≤t≤T

∣∣∣ ∫ t

0

b(s, xnk
(πnk

(s), x0))ds+

∫ t

0

σ(s, xnk
(πnk

(s), x0))dB(s)

−
∫ t

0

b(s, x(s, x0))ds+

∫ t

0

σ(s, x(s, x0))dB(s)
∣∣∣ → 0, a.s. t ≤ T,

(3.14)

as k → ∞. Consequently, by Lemma 2.1 we obtain that there exists an Ft adapted process
ϕ(t, x0) ∈ C([0, T ];Rd) with the properties (2.1) and (2.2) such that, for a.s. ω ∈ Ω, (x(t, x0)(ω), ϕ(t, x0)(ω))
is a solution of the Skorohod problem

x(t, x0)(ω) = W (t, x0)(ω) + ϕ(t, x0)(ω).

Since

ϕnk
(t, x0) = Wnk

(t, x0)− xnk
(t, x0),

we apply (3.13) and (3.14) to obtain

sup
0≤t≤T

|ϕnk
(t, x0)− ϕ(t, x0)| → 0 a.s.for t ≤ T, as k → ∞,

and then

x(t, x0) =

∫ t

0

b(s, x(s, x0))ds+

∫ t

0

σ(s, x(s, x0))dB(s) + ϕ(t, x0), t ≤ T.

The uniqueness and continuous dependence with respect to x0 of the solution of equation (1.1)
will be presented in Theorem 3.10. This completes the proof. □

Remark 3.9. Now we see that (x(t, x0), ϕ(t, x0)) indeed is a solution of equation (1.1) with initial
data x0. By Lemma 2.1 and (3.14), it is easy to see that for any t ≤ T , and fixed x0,

[0, t]× Ω ∋ (s, ω) 7→ x(s, x0, ω),

[0, t]× Ω ∋ (s, ω) 7→ ϕ(s, x0, ω)

are B[0,t] ×Ft measurable.



EJDE-2025/113 REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS 13

Let x0, y0 : Ω → D̄ be F0 measurable. Let x1(t, x0) and x2(t, y0) be the solutions of equation
(1.1) with initial conditions x1(0, x0) = x0 and x2(0, y0) = y0, respectively, and ϵ0 > 0, E[|x0|2] ∨
E[|y0|2] < ∞. We define

τx(R, x0, y0) = inf{t : |x1(t, x0)| ∨ |x2(t, y0)| ≥ R},
τ̃x(x0, y0) = inf{t : |x1(t, x0)− x2(t, y0)|2 ≥ ϵ0},

Γx(R, x0, y0) = τ̃x(x0, y0) ∧ τx(R, x0, y0).

Now we prove that (x(t), ϕ(t)) is the unique solution. Indeed, we have the following result.

Theorem 3.10. Let x1(t, x0) and x2(t, y0) be the solutions of (1.1) with initial conditions x1(0, x0) =
x0 and x2(0, y0) = y0, respectively. Then for any ε > 0, we have

P (sup
s≤t

|x1(s, x0)− x2(s, y0)|2 ≥ ε) ≤ 1

ε
F−1

(
F (

C0

c0
E[|x0 − y0|2]) + α̃

∫ t

0

(g(s) + λ(s))ds
)
, (3.15)

where the function F is defined by (3.5) and α̃ = C0

c0
( 8Cγ + 8C

γ2 ).

Proof. Using Itô formula for the processes

Zt := exp(− 2

γ
(f(x(t, x0)) + f(x(t, y0)))|x1(t, x0)− x2(t, y0)|2.

and estimating as (3.6), there is a positive constant C such that

Zt∧Γx(R,x0,y0)

≤ C0|x0 − y0|2 + C0

∫ t∧Γx(R,x0,y0)

0

g(s)KR(|x1(s, x0)− x2(s, y0)|2)ds

+ C0(
8C

γ
+

8C

γ2
)

∫ t∧Γx(R,x0,y0)

0

|x1(s, x0)− x2(s, y0)|2 sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds+R1(t),

where R1(t) is a martingale with zero expectation. From the fact that exp(− 2
γ (f(x) + f(y)) ≥ c0

and the function KR being concave, we obtain

E[|x1(t ∧ Γx(R, x0, y0), x0)− x2(t ∧ Γx(R, x0, y0), y0)|2]

≤ C0

c0
E[|x0 − y0|2] +

C0

c0

∫ t∧Γx(R,x0,y0)

0

g(s)KR(E[|x1(s, x0)− x2(s, y0)|2])ds

+ α̃

∫ t∧Γx(R,x0,y0)

0

E[|x1(s, x0)− x2(s, y0)|2] sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds

≤ C0

c0
E[|x0 − y0|2]

+
C0

c0

∫ t

0

g(s)KR(E[|x1(s ∧ ∧Γx(R, x0, y0), x0)− x2(s ∧ ∧Γx(R, x0, y0), y0)|2])ds

+ α̃

∫ t

0

E[|x1(s ∧ ∧Γx(R, x0, y0), x0)− x2(s ∧ ∧Γx(R, x0, y0), y0)|2]

× sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)ds

By remark 3.4, we have

E[|x1(t ∧ Γx(R, x0, y0), x0)− x2(t ∧ Γx(R, x0, y0), y0)|2]

≤ F−1
(
F (

C0

c0
E[|x0 − y0|2]) + α̃

∫ t

0

(g(s) + λ(s))ds
)

By Lemma 3.5, for any ε > 0, we write

P (sup
s≤t

|x1(s, x0)− x2(s, y0)|2 ≥ ε)



14 M. ZHANG EJDE-2025/113

≤ P ( sup
s≤t∧τx(R,x0,y0)

|x1(x0, s)− x2(y0, s)|2 ≥ ε) + P (τx(R, x0, y0) ≤ t)

≤ E[|x1(t ∧ Γx(R, x0, y0), x0)− x2(t ∧ Γx(R, x0, y0), y0)|2]
ε

+
E[|x1

n(t ∧ τx(R, x0, y0), x0)|2]
R2

+
E[|x2

n(t ∧ τx(R, x0, y0), y0)|2]
R2

≤ 1

ε
F−1

(
F (

C0

c0
E[|x0 − y0|2]) + α̃

∫ t

0

(g(s) + λ(s))ds
)
+

E[|x1
n(t ∧ τx(R, x0, y0), x0)|2]

R2

+
E[|x2

n(t ∧ τx(R, x0, y0), y0)|2]
R2

.

Letting R → ∞, we have

P (sup
s≤t

|x1(s, x0)− x2(s, y0)|2 ≥ ε) ≤ 1

ε
F−1

(
F (

C0

c0
E[|x0 − y0|2]) + α̃

∫ t

0

(g(s) + λ(s))ds
)
.

This completes the proof. □

Theorem 3.10 tells us that the solution of equation (1.1) continuously depends on the initial
random variable. That is to say we have the following result.

Corollary 3.11. Let x0 : Ω → D̄ and xn(0) :→ D̄ be F0 measurable random variables and
assume that E[|x0|2] < ∞ and supn E[|xn(0)|2] < ∞ and that E[|xn(0) − x0|2] → 0, as n → ∞.
Assume that (x(t, x0), ϕ(t, x0)) and (xn(t, xn(0)), ϕn(t, xn(0))) are the solutions of equation (1.1)
with initial random variables x0 and xn(0), respectively. Then

P - lim
n→∞

sup
t≤T

|xn(t, xn(0))− x(t, x0)| = 0, (3.16)

where P -lim means a limit in probability.

To prove Theorem 3.2, we introduce the following condition and lemma.

(A6) There exist positive constants δ′ and γ such that for each x0 ∈ ∂D we can find a function
f ∈ C2(Rd) satisfying (1.2) for any x ∈ B(x0, δ

′) ∩ ∂D, y ∈ B(x0, δ
′) ∩ D̄ and n ∈ Nx.

Lemma 3.12 ([12, Lemma 5.3]). Assume (A1) and (A2). Then the domain D satisfies (A6) with
γ = 2r0β

−1.

Proof of Theorem 3.2. We use the techniques in Pardoux and Rascanu [7, Proposition 3.38 ].
By a standard compactness argument, there exists a colection of sets {Ui : Ui = B(xi, δ

′), i =
1, 2, . . . , k} covering ∂D and there exists an open set U0 ⊂ D such that D̄ ⊂ ∪k

i=0{Ui}. For
F0 measurable random variable x0 ∈ D̄ a.s., we consider the equation (1.1) with initial value
xi
0 := x0IUi(x0) (where IU is the indicator function of set U). We see that Euler approximation

(3.2) which is well-posed by [12, Theorem 4.1]. We denote

τ̂ in,m = inf{t : xn(t, x
i
0) /∈ D̄ ∩ Ui} ∧ inf{t : xm(t, xi

0) /∈ D̄ ∩ Ui}, i = 0, 1, . . . , k,

ῑim,n = inf{t : |xn(t, x
i
0)− xm(t, xi

0)|2 ≥ ϵ0}, i = 0, 1, . . . , k.

Let

τ̂ i = lim inf
m,n→∞

τ̂ in,m,

τ̂(ω) = min{τ̂ i(ω) ̸= 0 : i = 0, 1, . . . , k}, ω ∈ Ω,

Γ̂m,n = ῑm,n ∧ τ̂ .

We may assume that τ̂ < ∞ a.s.; otherwise, consider τ̂ ∧N . Applying Lemma 3.12 and the proof
of Proposition 3.7, we can write

E[|xn(t ∧ Γ̂m,n, x
i
0)− xm(t ∧ Γ̂m,n, x

i
0)|2] ≤ F−1

(
F
(
η(

1

2n
) + η(

1

2m
)
)
+ 3α

∫ t

0

(g(s) + λ(s))ds
)
,



EJDE-2025/113 REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS 15

where α, F (u) and η(u) are defined as in Proposition 3.7. According to Lemma 3.5 and the proof
of Corollary 3.8, we have for any ε > 0,

P ( sup
s≤t∧τ̂

|xn(s, x
i
0)− xm(s, xi

0)|2 ≥ ε) ≤ E[|xn(t ∧ Γ̂m,n, x
i
0)− xm(t ∧ Γ̂m,n, x

i
0)|2]

ε

≤ 1

ε
F−1

(
F
(
η(

1

2n
) + η(

1

2m
)
)
+ 3α

∫ t

0

(g(s) + λ(s))ds
)
.

Hence,

lim
m,n→∞

P ( sup
0≤s≤t∧τ̂

|xn(s, x
i
0)− xm(s, xi

0)| ≥ ε) = 0. (3.17)

The proof of Theorem 3.1 says that there exists a unique strong solution (x̂i(t), ϕ̂i(t)) (t ≤ τ̂) of
equation (1.1) with initial random variable xi

0.

We denote α(x) =
∑k

i=0 IUi
(x), x ∈ D̄, and

x̂(t) =

k∑
i=0

(α(x0))
−1x̂i(t)IUi

(x0), t ≤ τ̂ ,

ϕ̂(t) =

k∑
i=0

(α(x0))
−1ϕ̂i(t)IUi

(x0), t ≤ τ̂ .

Then (x̂(t), ϕ̂(t)) (t ≤ τ̂) is the unique strong solution of equation (1.1) with initial random variable
x0. Indeed, by the fact

x̂i(t) = x̂(t)IUi
(x0),

ϕ̂i(t) = ϕ̂(t)IUi
(x0),

for t ≤ τ̂ , we have

x̂(t) =

k∑
i=0

(α(x0))
−1x̂i(t)IUi

(x0)

=

k∑
i=0

(α(x0))
−1xi

0 +

∫ t

0

k∑
i=0

(α(x0))
−1σ(s, x̂(t)IUi(x0))IUi(x0)dB(s)

+

∫ t

0

k∑
i=0

(α(x0))
−1b(s, x̂(t)IUi

(x0))IUi
(x0)ds+

k∑
i=0

(α(x0))
−1ϕ̂i(t)IUi

(x0)

=

k∑
i=0

(α(x0))
−1xi

0 +

∫ t

0

k∑
i=0

(α(x0))
−1σ(s, x̂(t))IUi(x0)dB(s)

+

∫ t

0

k∑
i=0

(α(x0))
−1b(s, x̂(t))IUi

(x0)ds+ ϕ̂(t)

= x0 +

∫ t

0

σ(s, x̂(t))dB(s) +

∫ t

0

b(s, x̂(t))ds+ ϕ̂(t).

Assume that there exists a stopping time τ̂l such that x(t) (0 ≤ t ≤ τ̂l) is the unique solution of
equation (1.1) with initial random variable x0. We may assume τ̂l < ∞ a.s.; otherwise, consider

τ̂l ∧ N . Next, define B̂(u) = B(τ̂l + u) − B(τl) and F̂u = Fτ̂l+u = {Λ ∈ F : Λ ∩ {τ̂l + u ≤ t} ∈
Ft,∀t ≥ 0}, u ≥ 0. Then it is easy to see that B̂ is an (F̂u)u≥0 Brownain motion and τ̄l =: t− τ̂l∧ t

is an (F̂u)u≥0 stopping time for each fixed t ≥ 0. Thus we consider the SDE

Ŷ (t) = x(τ̂l) +

∫ t

0

σ(t, Ŷ (s))dB̂(t) +

∫ t

0

b(t, Ŷ (s))dt+ ϕ̂(t). (3.18)



16 M. ZHANG EJDE-2025/113

As proved as above, there exist a stopping time τ̂l+1 and a unique strong solution Ŷ (t) (t ≤ τ̂l+1)
of (3.18) with initial value x(τ̂l). Let

x(t) =

{
x̂(t), if t ≤ τ̂l,

Ŷ (t− τ̂l), if τ̂l ≤ t ≤ τ̂l+1.

Then x(t) (0 ≤ t ≤ τl+1) is the unique strong solution of (1.1) with initial value x0. This completes
the proof. □

4. Stability results

In this section, we consider the stability of the solution of equation (1.1). Suppose that σn(t, x)
and bn(t, x) are functions satisfying the assumptions (A4) and (A5), where the function λ does
not depend on n and for any 0 < R < +∞ and 0 < T < +∞,∫ T

0

sup
|x|≤R

(∥σn(t, x)− σ(t, x)∥2 + |bn(t, x)− b(t, x)|)dt → 0. (4.1)

Let 0 ≤ χ ∈ C∞
c (Rd) with

∫
Rd χ(d)dx = 1 and χn(x) = ndχ(nx). If b and σ satisfying the

assumptions (A4) and (A5), bn := χn ∗ b(t, ·) and σn := χn ∗ σ(t, ·) satisfy the condition (4.1).
For every n ∈ N, let (yn(t, yn(0)), ϕn(t, yn(0))) be the solution of the Skorohod problem,

yn(t, yn(0)) = yn(0)+

∫ t

0

σn(s, yn(s, yn(0)))dB(s)+

∫ t

0

bn(s, yn(s, yn(0)))ds+ϕn(t, yn(0)). (4.2)

Our main object in this section is to prove the following theorem.

Theorem 4.1. Assume (A1)–(A3) hold. Suppose that E[|yn(0) − x0|2] → 0 as n → ∞. Then
yn(t, x0) → x(t, x0) uniformly on [0, T ] in probability, where x(t, x0) satisfies the equation (1.1)
with initial random variable x0.

Proof. We denote yn(t, yn(0)) and x(t, x0) by yn(t) and x(t), respectively. We define

ς̂n = inf{t > 0 : |yn(t)| ∨ |x(t)| ≥ R}.

Let

v(t) = yn(t)− x(t), ζt = |v(t)|2, zt = z(yn(t), x(t))ζt,

λ̂n := inf{t > 0, ζt ≥ ϵ0}, Λn = λ̂n ∧ ς̂n,

where z(x, y) is defined in the proof of Proposition 3.7.
Using Itô’s formula for zt∧Λn

, and following the same as proof in Proposition 3.7, we have

zt∧Λn ≤ z0 + 2

∫ t∧Λn

0

z(yn(s), x(s))⟨vs, bn(s, yn(s))− b(s, x(s)))⟩ds

+ (
2C

γ
+ 1)

∫ t∧Γn(R)

0

z(yn(s), x(s))∥σn(s, yn(s))− σ(s, x(s))∥2ds

+ (
6C

γ
+

4C

γ2
)

∫ t∧Λn

0

zs[ sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)

+ sup
|x|≤R

(|bn(s, x)|+ ∥σn(s, x)∥2)]ds+ I ′1

= z0 + 2

∫ t∧Λn

0

z(yn(s), x(s))⟨vs, bn(s, yn(s))− bn(s, x(s)))⟩ds

+ (
4C

γ
+ 2)

∫ t∧Λn

0

z(yn(s), x(s))∥σn(s, yn(s))− σn(s, x(s))∥2ds

+ 2

∫ t∧Λn

0

z(yn(s), x(s))⟨vs, bn(s, x(s))− b(s, x(s)))⟩ds



EJDE-2025/113 REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS 17

+ (
4C

γ
+ 2)

∫ t∧Λn

0

z(yn(s), x(s))∥σn(s, x(s))− σ(s, x(s))∥2ds

+ (
6C

γ
+

4C

γ2
)

∫ t∧τn

0

zs[ sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)

+ sup
|x|≤R

(|bn(s, x)|+ ∥σn(s, x)∥2)]ds+ I ′1

≤ z0 +

∫ t∧Λn

0

z(yn(s), x(s))g(s)KR(ζs)ds

+

∫ t∧Λn

0

z(yn(s), x(s))|bn(s, yn(s))− b(s, x(s))|ds

+ (
4C

γ
+ 2)

∫ t∧Λn

0

z(yn(s), x(s))∥σn(s, yn(s))− σ(s, x(s))∥2ds

+ (
6C

γ
+

4C

γ2
+ 1)

∫ t∧Λn

0

zs[ sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)

+ sup
|x|≤R

(|bn(s, x)|+ ∥σn(s, x)∥2)]ds+ I ′1,

where I ′1 is a martingale with E[I ′1] = 0. Thus taking expectation on both side, hence

E[ζt∧Λn
] ≤ β2

∫ t

0

sup
|x|≤R

(∥σn(s, x)− σ(s, x)∥2 + |bn(s, x)− b(s, x)|)ds

+ β3E[

∫ t

0

(KR + I)(ζs∧Λn
)[ sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)

+ sup
|x|≤R

(|bn(s, x)|+ ∥σn(s, x)∥2) + g(s)]ds] +
C0

c0
E[|yn(0)− x0|2],

where β2 = C0

c0
( 8Cγ + 2) and β3 = C0

c0
( 6Cγ + 4C

γ2 + 1). By Bihari’s inequality, we obtain

E[ζt∧Λn
] ≤ F−1

(
F
(
β2

∫ t

0

sup
|x|≤R

(∥σn(s, x)− σ(s, x)∥2 + |bn(s, x)− b(s, x)|)ds

+
C0

c0
E[|yn(0)− x0|2]

)
+ β3

∫ t

0

[ sup
|x|≤R

(|b(s, x)|+ ∥σ(s, x)∥2)

+ sup
|x|≤R

(|bn(s, x)|+ ∥σn(s, x)∥2) + g(s)]ds
)

≤ F−1
(
F
(
β1

∫ t

0

sup
|x|≤R

(∥σn(s, x)− σ(s, x)∥2 + |bn(s, x)− b(s, x)|)ds

+
C0

c0
E[|yn(0)− x0|2]

)
+ β3

∫ t

0

[2λ(s) + g(s)]ds
)
,

where F is defined by (3.5).
As proof in Corollary 3.8, we claim that

P (sup
s≤T

ζs ≥ ε) → 0, as n → ∞,

and the theorem follows. □

5. Regular version

For 0 ≤ s < T < +∞, x0 ∈ D̄ being constant, under the conditions of Theorem 3.1, the
equation (1.1) admits a unique solution on [s, T ] with the initial value x(s) = x0. We will denote
it by (x(s, x0; t, ω), ϕ(s, x0; t, ω)). In this section, we will explore the Markov property of the
solution of equation (1.1).



18 M. ZHANG EJDE-2025/113

We will say that a function f : R+ ×Rd ×Ω 7→ Rd is progressively measurable if for any t > 0,
restricted on [0, t]× Rd × Ω it is B([0, t])× Bd ×Ft-measurable.

Now, the canonical space is used such that it holds B(t)(ω) = ω(t). To obtain the Markov
property of the solution x(s, x0; t, ω), we hope that for s ≤ s1 < s2 ≤ t, the identity

x(s1, x0; t, ω) = x(s2, x(s1, x0; s2, ω); t, θs2ω)

holds, where θsω(t) = ω(s+ t)− ω(s). Next we will show the following lemma.

Lemma 5.1. Let g1(s, x, ω) and g2(s, x, ω) be progressively measurable functions and almost surely
continuous with respect to x for any t satisfying

E[

∫ +∞

0

sup
|x|≤R

(|g2(s, x)|+ |g1(s, x)|2)ds] < ∞, R > 0. (5.1)

Then

(i) there exists a progressively measurable function G(x, t, ω) such that

G(x, t) =

∫ t

0

g1(s, x)dB(s) +

∫ t

0

g2(s, x)ds, ∀ x ∈ Rd;

(ii) if η is independent of (B(t))t≥0, then

G(η, t, ω) =

∫ t

0

g1(s, η, ω)dB(s) +

∫ t

0

g2(s, η, ω)ds.

Proof. For simplicity we assume that g2 = 0 because of the ordinary integral term produces no
difficulty. Set yk := 2−k[2ky]. For x0 = (x1, . . . , xd) ∈ Rd, set xk = (xk

1 , . . . , x
k
d). We denote

Gn(t, x) =

∫ t

0

g1(s, x
n)dB(s).

According to the Burkholder-Davis-Gundy inequality, we have

E[ sup
0≤t≤T

|Gn(x, t)−
∫ t

0

g1(s, x)dB(s)|2] ≤ E[

∫ T

0

|g1(s, xn)− g1(s, x)|2ds].

Combining this with

g1(s, x
n) → g1(s, x), a.e., as → ∞,

and (5.1), we have

lim sup
n→+∞

E[

∫ T

0

|g1(s, xn)− g1(s, x)|2ds] ≤ E[

∫ T

0

lim sup
n→+∞

|g1(s, xn)− g1(s, x)|2ds]

= E[

∫ T

0

lim
n→+∞

|g1(s, xn)− g1(s, x)|2ds] = 0,

which leads to

lim
n→+∞

E[

∫ T

0

|g1(s, xn)− g1(s, x)|2ds] = 0. (5.2)

Then, there exists a subsequence {Gnk
} (still denote {Gn}), such that

lim
n→∞

Gn(t, x) =

∫ t

0

g1(s, x)dB(s) a.s. in C([0, T ]), ∀x ∈ Rd.

Consequently we define

G(x, ·) =

{
limn→∞ Gn(., x), if the limit exists,

0, otherwise.

Then G satisfies (i).



EJDE-2025/113 REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS 19

To prove (ii), we first assume that s 7→ g1(s, x, ω) is almost surely continuous. Denote tk :=
(k/2n) ∧ t, k = 1, 2, . . . . Then we have (see [11, Proposition IV 2.13])

G(t, x) = P − lim
n→∞

∞∑
k=1

g1(tk, x)(B(tk+1)−B(tk)),∫ t

0

g1(s, η)dB(s) = P − lim
n→∞

∞∑
k=1

g1(tk, η)(B(tk+1)−B(tk)).

Then there exists a subsequence of {n} (we still denote by {n}) such that

∞∑
k=1

g1(tk, η)(B(tk+1)−B(tk)) →
∫ t

0

g1(s, η)dB(s), a.s., as n → ∞.

Because of the independence of η and (B(t))t≥0, for k = 1, 2, . . . , we have

P (G(t, η) =

∫ t

0

g1(s, η)dB(s))

= P (G(t, η) = lim
n→∞

∞∑
k=1

g1(tk, η)(B(tk+1)−B(tk)))

=

∫
Rd

P (G(t, η) = lim
n→∞

∞∑
k=1

g1(tk, η)(B(tk+1)−B(tk))|η = x)P (η ∈ dx)

=

∫
Rd

P (G(t, x) = lim
n→∞

∞∑
k=1

g1(tk, x)(B(tk+1)−B(tk))|η = x)P (η ∈ dx) = 1.

For the general case, we set g1(s, x, ω) = 0 for t < 0 and define

gn1 (t, x) = n

∫ 1/n

−1/n

ρ(ns)g1(t− s− 1

n
, x)ds,

where ρ(t) ∈ C∞ is a positive function with support in [−1, 1] and
∫ 1

−1
ρ(s)ds = 1. It is clear that

gn1 (t, x) is almost surely continuous with respect to t. Furthermore, by Minkowski’s inequality,∫ T

0

|gn1 (t, y)− gn1 (t, x)|2dt

=
[( ∫ T

0

|
∫ 1/n

− 1
n

nρ(ns)(g1(t− s− 1

n
, y)− g1(t− s− 1

n
, x))ds|2dt

)1/2]2
≤

[ ∫ 1/n

− 1
n

(∫ T

0

|nρ(ns)(g1(t− s− 1

n
, y)− g1(t− s− 1

n
, x))|2dt

)1/2

ds
]2

=
[ ∫ 1/n

− 1
n

nρ(ns)
(∫ T

0

|(g1(t− s− 1

n
, y)− g1(t− s− 1

n
, x))|2dt

)1/2

ds
]2

≤
(∫ 1/n

− 1
n

nρ(ns)ds
)2

∫ T

0

|(g1(t, y)− g1(t, x))|2dt.

Thus, we have

E[

∫ T

0

|gn1 (t, y)− gn1 (t, x)|2dt] ≤
(
n

∫ 1/n

− 1
n

ρ(nu)du
)2

E[

∫ T

0

|g1(t, y)− g1(t, x)|2dt]

= E[

∫ T

0

|g1(t, y)− g1(t, x)|2dt].
(5.3)

Then, for any n, gn1 (t, x, ω) is almost surely continuous with respect to t for any x and almost
surely continuous with respect to x for any t. By the proof above there exists a progressively



20 M. ZHANG EJDE-2025/113

measurable function Gn such that

Gn(t, x) =

∫ t

0

gn1 (s, x)dB(s),

Gn(t, η) =

∫ t

0

gn1 (s, η)dB(s).

We define Br = {x ∈ Rd : |xi| < r, i = 1, 2, . . . , d} for r > 0 and

Σn(r) = {(rk12−n, rk22
−n, . . . , rkd2

−n) : ki = −2n, . . . , 2n, i = 1, 2, . . . , d}.
From the definition of gn1 , we have for any x,

lim
n→∞

∫ T

0

|gn1 (s, x)− g1(s, x)|2ds = 0, a.s..

Then applying the Burkholder-Davis-Gundy inequality and the lemma of Fatou, we can write

lim sup
n→∞

E[ sup
0≤t≤T

|Gn(t, x)−
∫ t

0

g1(s, x)dB(s)|2]

≤ lim sup
n→∞

E[

∫ T

0

|gn1 (s, x)− g1(s, x)|2ds]

≤ E[lim sup
n→∞

∫ T

0

|gn1 (s, x)− g1(s, x)|2ds] = 0.

For any m, we take nm ≥ m such that

max
x∈Σnm (r)

E[ sup
0≤t≤T

|Gnm
(t, x)−

∫ t

0

g1(s, x)dB(s)|2] ≤ 2−2m. (5.4)

For any i, by (5.2), we can take mi ≥ i and xmi ∈ Σmi(r) with |xmi − x| ≤ r
2mi

, such that

E[

∫ T

0

|g1(t, xmi
)− g1(t, x)|2ds] ≤ 2−2i. (5.5)

Combining with (5.3), (5.4) and (5.5), we then have for any x ∈ Br,

E[ sup
0≤t≤T

|Gnmi
(t, x)−

∫ t

0

g1(s, x)dB(s)|2]

≤ 3E[ sup
0≤t≤T

|Gnmi
(t, x)− Gnmi

(t, xmi)|2] + 3E[ sup
0≤t≤T

|Gnmi
(t, xmi)−

∫ t

0

g1(xs,mi)dB(s)|2]

+ 3E[ sup
0≤t≤T

|
∫ t

0

g1(xs,mi
)− g1(s, x)dB(s)|2]

≤ 6E[

∫ T

0

|g1(s, xmi)− g1(s, x)|2ds] + 3× 2−2i

≤ 9× 2−2i.

By Chebyshev’s inequality, thus
∞∑
i=1

P ( sup
0≤t≤T

|Gnmi
(t, x)−

∫ t

0

g1(s, x)dB(s)| > 1

2i
) < ∞.

Then, from the Borel-Cantelli lemma, we have

lim
i→∞

Gnmi
(t, x) =

∫ t

0

g1(s, x)dB(s) in C([0, T ]), a.s., for all x ∈ Br. (5.6)

We can see that (5.6) still holds for |x| < ∞. We define

F (·, x) =

{
limi→∞ Gnmi

(., x), if the limit exists,

0, otherwise.



EJDE-2025/113 REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS 21

Then F is progressively measurable. We claim that

F (·, η) =
∫ .

0

g1(s, η)dB(s).

Indeed,

E[ sup
0≤t≤T

|F (t, η)−
∫ t

0

g1(s, η)dB(s)|2]

≤ lim
n→∞

E[ sup
0≤t≤T

|F (t, η)− Gn(t, η)|2] + lim
n→∞

E[ sup
0≤t≤T

|Gn(t, η)−
∫ t

0

gn1 (s, η)dB(s)|2]

+ lim
n→∞

E[ sup
0≤t≤T

|
∫ t

0

gn1 (s, η)dB(s)−
∫ t

0

g1(s, η)dB(s)|2].

Applying the Burkholder-Davis-Gundy inequality, the dominated convergence theorem and that

E[ sup
0≤t≤T

|Gn(t, η)−
∫ t

0

gn1 (s, η)dB(s)|2] = 0,

we have

E[ sup
0≤t≤T

|F (t, η)−
∫ t

0

g1(s, η)dB(s)|2]

≤ lim
n→∞

∫
Rd

P (η ∈ dx)E[ sup
0≤t≤T

|F (t, x)− Gn(t, x)|2] + lim
n→∞

E[

∫ T

0

|gn1 (s, η)− g1(s, η)|2ds]

≤ lim
n→∞

∫
Rd

P (η ∈ dx)E[ sup
0≤t≤T

|F (t, x)− Gn(t, x)|2]

+ lim
n→∞

∫
Rd

P (η ∈ dx)E[

∫ T

0

|gn1 (s, x)− g1(s, x)|2ds] = 0.

Thus the claim follows, and this completes the proof. □

We consider the equation

x(s, x0; t) = x0 +

∫ t

s

σ(s, x(s))dB(s) +

∫ t

s

b(s, x(s))ds+ ϕ(s, x0; t), x0 ∈ D̄ (5.7)

The following theorem is the main result in this section. We denote the Borel σ-algebra of the set
D̄ by BD̄.

Theorem 5.2. Assume (A1–(A5). Then, for any s ≥ 0, there exists a function (s, x0, ω) 7→
x(s, x0; t, ω) such that

(i) for any s ≤ t, D̄×[s, t]×Ω ∋ (x0, u, ω) 7→ x(s, x0;u, ω) is BD̄×B[s,t]×F[s,t]/Bd measurable,

also BD̄ × [s, t]× Ω ∋ (x0, u, ω) 7→ ϕ(s, x0;u, ω) is Bd × B[s,t] ×F[s,t]/Bd measurable.

(ii) for any x0 ∈ D̄, {(x(s, x0; t, ω), ϕ(s, x0;u, ω)), s ≤ t} is a unique solution of equation (5.7)
with initial value x0.

(iii)
x(s, x0; t, ω) = x(u, x(s, x0;u, ω); t, θuω), s ≤ u ≤ t.

Here, for t ≥ s ≥ 0, θsω(t) := ω(s+ t)− ω(s) , and for s ≤ t,

F[s,t] := σ{ω(u)− ω(v) : s ≤ v ≤ u ≤ t}.

Proof. For y ∈ R, set yk := 2−k[2ky]. For x0 = (x1, . . . , xd) ∈ Rd, set xk = (xk
1 , . . . , x

k
d).

We take the subsequence {nk}k of {n}, such that

F−1
(C0

c0
F (2−nk) + 3α

∫ T

0

(g(s) + λ(s))ds
)
<

1

22k
,

and take the subsequence {nki}i of {nk}k, such that

|xnki − x0| ≤
1

2nki
.



22 M. ZHANG EJDE-2025/113

Then by the proof of Theorem 3.1, we see that (x(s, xk;u, ω), ϕ(s, xk;u, ω)) is a unique solution
to equation (1.1) with initial value xk. By Theorem 3.10, we obtain

∞∑
k=1

P (sup
u≤T

|x(xk, s;u, ·)− x(x0, s;u, .)|2 ≥ 1

2k
) ≤

∞∑
k=1

2k
1

22k
≤

∞∑
k=1

1

2k
< ∞.

Then by the Borel-Cantelli lemma we obtain

lim
k→∞

x(s, xk;u, .) = x(s, x0;u, .), uniformly on [s, T ] a.s..

Consequently
lim
k→∞

ϕ(s, xk;u, .) = ϕ(s, x0;u, .), uniformly on [s, T ] a.s..

We define

x̃(s, x0;u, .) =

{
limk→∞ x(s, xk;u, .), if the limit exists

0, otherwise.

ϕ̃(s, x0;u, .) =

{
limk→∞ ϕ(s, xk;u, .), if the limit exists,

0, otherwise.

By Remark 3.9, it is clear that for each k, [s, t]×Ω 7→ x(s, xk; ·, ·) and [s, t]×Ω 7→ ϕ(s, xk; ·, ·) are
B[s,t] × F[s,t] measurable with respect to (u, ω). Since x(s, xk;u, ·) and ϕ(s, xk;u, ·) are piecewise

constant in x0 for every fixed (u, ω), then they are Bd ×B[s,t] ×F[s,t]/Bd measurable in (x0, u, ω).

Thus, the limits of x(s, xk;u, ·) and ϕ(s, xk;u, ·) have the same measurability. This shows (i).
(ii) is obvious because

x(s, x0;u, .) = x̃(s, x0;u, .), a.s., for every x0,

and
ϕ(s, x0;u, ·) = ϕ̃(s, x0;u, ·), a.s., for every x0.

Form now on we denote x̃(s, x0;u, ·) by x(s, x0;u, ·).
Now we start to prove (iii). By the Lemma 5.1, there exists a progressively measurable function

H such that

H(s2, y0; t, θs2(ω)) = y0 +

∫ t

s2

σ(u, x(s2, y0;u))dB(u) +

∫ t

s2

b(u, x(s2, y0;u))du, ∀y0.

Then, since {x(s2, y0;u), ϕ(s2, y0;u), u ≥ s2} satisfies the equation (5.7), we have

H(s2, y0; t, .) = x(s2, y0; t, .)− ϕ(s2, y0;u), ∀y0.
Both H(s2, y0; t, ·) and x(s2, y0; t, .) − ϕ(s2, y0; t, .) are independent of Bt∧s2 , because of both
H(s2, y0; t, .) and x(s2, y0; t, .)−ϕ(s2, y0; t, .) depending on ω though θs2(ω). Form (i), x(s2, y0; t, .)−
ϕ(s2, y0; t, .) is progressively measurable. Therefore we have for any µ ∈ Fs2 ,

P (H(s2, µ; t, .) = x(s2, µ; t, .)− ϕ(s2, µ; t, ·))

=

∫
P (µ ∈ dy)P (H(s2, y; t, .) = x(s2, y; t, .)− ϕ(s2, y; t, .)) = 1.

Taking µ = x(s1, x0; s2, .), thus

x(s2, x(s1, x0; s2, ω); t, θs2)− ϕ(s2, x(s1, x0; s2, ω); t, θs2)

= H(s2, x(s1, x0; s2, ω); t, θs2)

= x(s1, x0; s2) +

∫ t

s2

σ(u, x(s2;u))dBu +

∫ t

s2

b(u, x(s2;u))du.

But x(s1, x0; t) satisfies the same equation. Therefore, we have by the uniqueness

x(s1, x0; t, ω) = x(s2, x(s1, x0; s2, ω); t, θs2(ω)).

Then the proof is complete. □

Corollary 5.3. Under the conditions of Theorem 5.2, x(s, x0; t, ω) is a strong Markov process.



EJDE-2025/113 REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS 23

Proof. First, we know that x(s, x0; t, ω) is a Markov process. Indeed, for any x0, any s1 < s2 < t,
and any bounded Borel function f we have

E[f(x(s1, x0; t))|Fs2 ] = E[f(x(s2, x(s1, x0; s2); t))|Fs2 ]

= E[f(x(s2, y; t))]|y=x(s1,x0;s2)

= E[f(x(s2, x(s1, x0; s2); t))|x(s1, x0; s2)]

= E[f(x(s1, x0; t))|x(s1, x0; s2)].

Replacing s2 by a stopping time and using the fact that the Brownian motion starts afresh at a
stopping time, one can show that x(s, x0; t, ω) is a strong Markov process by the same arguments.

□

6. Rate of convergence of Euler’s approximations

In this section, we investigate the characterization of convergence rate of Euler’s approximations
of equation (1.1) under more restrictions on the coefficients b and σ. For simplicity, in this section
we let C1 stand for a positive constant depending only on some determinate parameters and its
value may be different in different appearance.

First, we introduce the following assumptions:

(A7) Suppose that σ and b are bounded and satisfy

|b(t, x)− b(t, y)| ≤ |x− y|K̃(|x− y|),

∥σ(t, x)− σ(t, y)∥2 ≤ |x− y|2K̃(|x− y|),

where K̃(x) = Ĉ(log( 1x ) ∨ C̃)
1
β̂ for some β̂ > 2 and the constants Ĉ, C̃ > 0.

The assumptions above arise in the study of Brownian motion on the group of diffeomorphisms of
the circle (cf. [2, 6]). For 0 < η̂ < 1

e , we define a concave function

ϱη̂(x) =

{
x log x−1, x ≤ η̂,

(log η̂−1 − 1)x+ η̂, x > η̂.

Remark 6.1 ([10, Example 2.2]). Taking the function ρ(x) = ϱη̂(x) in Lemma 3.3, we can describe
the Bihari’s inequality as follows: If h(s), χ(s) are two strictly positive functions on R+ such that

χ(t) ≤ ϑ+

∫ t

0

ϱη̂(χ(s))h(s)ds, t ≥ 0,

then

χ(t) ≤ ϑexp(−
∫ t
0
h(s)ds). (6.1)

Lemma 6.2. Assume that b and σ are bounded. Then for any p ≥ 1 and for any T ∈ [0,+∞),
there is a constant C1 > 0 such that for all n ∈ N and s, t ∈ [0, T ],

E[∥xn∥∞,[s,t]|2p] + E[∥x∥∞,[s,t]|2p] ≤ C1|t− s|p.

Proof. If σ is bounded, then the quadratic variation of M(t) :=
∫ t

0
σ(s, x(s))dB(s) is bounded and

Mn(t) :=
∫ t

0
σ(s, xn(πn(s)))dB(s) is a martingale whose quadratic variation is uniformly bounded

for n. Thus we see that

E[exp( max
0≤t≤T

|M(t)|)] < ∞,

sup
n

E[exp(â max
0≤t≤T

|Mn(t)|)] < ∞,

for all constant â > 0. The assertion follows from [1, Lemmas 2.3 and 2.7] and the Burkholder-
Davis-Gundy inequality. □



24 M. ZHANG EJDE-2025/113

Theorem 6.3. Assume (A1)–(A3) and (A7). Then for any T ∈ [0,+∞), there exist constants

C1 > 0 and δ̂ > 1 such that

E[ sup
0≤t≤T

|xn(t, x0)− x(t, x0)|4] ≤ C1δ̂
−n,

where n ≥ N0 is sufficiently large.

Proof. In the proof of Proposition 3.7, if we take Γm,n(R) = ∞, then we have

zt :=

6∑
i=1

Ii,

and I4 ≤ 0. Thus

sup
0≤s≤t

|xn(s)− xm(s)|4 ≤ C1

6∑
i=1,i̸=4

sup
0≤s≤t

I2i .

By the boundness of σ and b and Burkholder-Davis-Gundy inequality, we have the following
estimates

E[ sup
0≤s≤t

I21 + sup
0≤s≤t

I23 + sup
0≤s≤t

I25 ]

≤ C1E
[ ∫ t

0

|xn(s)− xm(s)|2∥σ(s, xn(πn(s)))− σ(s, xm(πm(s)))∥2 + |xn(s)− xm(s)|4ds
]

≤ C1E
[ ∫ t

0

∥σ(s, xn(πn(s)))− σ(s, xm(πm(s)))∥4 + |xn(s)− xm(s)|4ds
]
,

E[ sup
0≤s≤t

I22 ] ≤ C1E
[ ∫ t

0

|xn(s)− xm(s)|2|b(s, xn(πn(s)))− b(s, xm(πm(s)))|2

+ ∥σ(s, xn(πn(s)))− σ(s, xm(πm(s)))∥4ds
]
,

E[ sup
0≤s≤t

I26 ] ≤ C1E
[
(

∫ t

0

∥σ(s, xn(πn(s)))− σ(s, xm(πm(s)))∥|xn(s)− xm(s)|ds)2
]

≤ C1E
[ ∫ t

0

|xn(s)− xm(s)|4 + ∥σ(s, xn(πn(s)))− σ(s, xm(πm(s)))∥4ds
]
.

Combining this with (A7), we can write

E[ sup
0≤s≤t

|xn(s)− xm(s)|4] ≤ C1

6∑
i=1,i̸=4

E[ sup
0≤s≤t

I2i ]

≤ C1E
[ ∫ t

0

|xn(s)− xm(s)|4 + ∥σ(s, xn(πn(s)))− σ(s, xm(πm(s)))∥4

+ |xn(t)− xm(t)|2|b(s, xn(πn(s)))− b(s, xm(πm(s)))|2ds
]

≤ C1E
[ ∫ t

0

|xn(s)− xm(s)|4 + ∥σ(s, xn(s))− σ(s, xm(s))∥4

+ ∥σ(s, xn(s))− σ(s, xn(πn(s)))∥4 + ∥σ(s, xm(s))− σ(s, xm(πm(s)))∥4

+ |xn(s)− xm(s)|2|b(s, xn(s))− b(s, xm(s))|2

+ ∥b(s, xn(s))− b(s, xn(πn(s)))∥4 + ∥b(s, xm(s))− b(s, xm(πm(s)))∥4ds
]

≤ C1E
[ ∫ t

0

|xn(s)− xm(s)|4K̃2(|xn(s)− xm(s)|)

+ |xm(s)− xm(πm(s))|4K̃2(|xm(s)− xm(πm(s))|)

+ |xn(s)− xn(πn(s))|4K̃2(|xn(s)− xn(πn(s))|)

+ |xm(s)− xm(πm(s))|4K̃4(|xm(s)− xm(πm(s))|)



EJDE-2025/113 REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS 25

+ |xn(s)− xn(πn(s))|4K̃4(|xn(s)− xn(πn(s))|)

+ |xn(s)− xm(s)|4ds
]
.

Since

lim
x↓0

Cp(x
4K̃2(x) + x4)

x4 lnx−4
= 0, and lim

x↓0

Cpx
4K̃4(x)

(x2 lnx−2)2
= 0,

there exists a constant η̂ > 0 such that

Cpx
4 ≤ ϱη̂(x

4),

Cpx
4K̃4(x) ≤ ϱ2η̂(x

2),

Cpx
4K̃2(x) ≤ ϱη̂(x

4).

Furthermore, by Jensen’s inequality and Lemma 6.2, we have

E[ sup
0≤s≤t

|xn(s)− xm(s)|4] ≤ C1E
[ ∫ t

0

ϱη̂(|xn(s)− xm(s)|4) + ϱη̂(|xn(s)− xn(πn(s))|4)

+ ϱη̂(|xm(s)− xm(πm(s))|4) + ϱ2η̂(|xn(s)− xn(πn(s))|2)

+ ϱ2η̂(|xm(s)− xm(πm(s))|2)ds
]

≤ C1

∫ t

0

ϱη̂(E[|xn(s)− xn(πn(s))|4]) + ϱη̂(E[|xm(s)− xm(πm(s))|4])

+ E[ϱ2η̂(|xn(s)− xn(πn(s))|2)] + E[ϱ2η̂(|xm(s)− xm(πm(s))|2)]
+ ϱη̂(E[ sup

0≤u≤s
|xn(u)− xm(u)|4])ds

≤ C1(2
−2α̂nT + 2−2α̂mT +

∫ t

0

ϱη̂(E[ sup
0≤u≤s

|xn(u)− xm(u)|4])ds),

where 0 < α̂ < 1 and n,m are sufficiently large. The last inequality above is due to ϱη̂(x) ≤
C1(x

α̂ + x), for x > 0. It follows from Remark 6.1 that

E[ sup
0≤s≤t

|xn(s)− xm(s)|4] ≤ C1(2
−2αnT + 2−2αmT )exp(−C1T ). (6.2)

Hence, xn(t) converges in probability, as n → ∞, to a continuous stochastic process x(t), uniformly
in t ∈ [0, T ]. Clearly

sup
m

E[ sup
0≤s≤t

|xn(s)− xm(s)|4] ≤ C1(2
−2αnT + T )exp(−C1T ) < ∞.

Thus, letting m → ∞ in (6.2) completes the proof. □

Acknowledgments. This work was supported by the National Natural Science Foundation of
China (Grant No. 12361030).

References

[1] S. Aida, K. Sasaki; Wong-Zakai approximation of solutions to reflecting stochastic differential equations on

domains in Euclidean spaces. Stochastic Process. Appl., 123 (2013), 3800-3827.
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of the circle. J. Funct. Ana., 196 (2002), 395-426.
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Mingbo Zhang

School of Statistics, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 333000, China

Email address: zmb1982zqz@hotmail.com


	1. Introduction
	2. Skorohod problem
	3. Solvability of RSDEs
	4. Stability results
	5. Regular version
	6. Rate of convergence of Euler's approximations
	Acknowledgments

	References