Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 79, pp. 1–21.

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

STOCHASTIC NEWTONIAN EQUATIONS WITH MEAN

BOUNDARY CONDITIONS

YING-JIA GUO, XIAO-MENG JIANG

Abstract. This article concerns stochastic Newtonian equations driven by

the white noise with mean boundary conditions. We obtain sufficient con-
ditions of the existence and uniqueness of solutions, and then solutions are

adapted to the Brownian filtration. As applications, we show the existence

and uniqueness of several stochastic differential equations with mean bound-
ary conditions.

1. Introduction

We consider the stochastic Newtonian system

d2Xt

dt2
= f(t,Xt) + g(t,Xt)

dBt

dt
, (1.1)

with mean boundary conditions

X(a) = 0, EX(b) = 0, (1.2)

where the time parameter t runs on the interval [a, b] (b > a ≥ 0). The drift term
f : [a, b]×Rl → Rl and the diffusion term g : [a, b]×Rl → Rl×m are continuous. {Bt}
is a standard m-dimensional Brownian motion defined on the complete probability
space (Ω,F ,P) adapted to the filtration {Ft}t≥0.

In classical dynamics, the Newtonian equation

d2Xt

dt2
= f(t,Xt), (1.3)

is usually used to describe the effects of external forces. If the frictions are periodic
with respect to time, then such systems can describe the behavior of a large class of
oscillation processes such as the motion of spring pendulum or oscillating circuits.

Dirichlet boundary conditions are one of the basic boundary conditions appearing
in mathematical physics. For deterministic boundary value problem with Dirichlet
boundary condition (BVP)

y′′ + f(t, y, y′) = 0,

y(a) = A, y(b) = B,
(1.4)

2010 Mathematics Subject Classification. 60H10, 60J50, 60J65.
Key words and phrases. Stochastic Newtonian systems; mean boundary conditions;

existence and uniqueness.
c©2021. This work is licensed under a CC BY 4.0 license.

Submitted April 2, 2021. Published September 17, 2021.

1



2 Y.-J. GUO, X.-M. JIANG EJDE-2021/79

where A,B ∈ Rl, f : [a, b] × Rl × Rl → Rl, one can utilize Green’s function in
reformulating (1.4) as an integral equation. By studying the properties of Green’s
function and the nonlinear terms, some conditions concerning the existence and
uniqueness of the solutions can be obtained, see [4, 17, 18].

However, some mechanical processes in real world are always effected by unknown
randomness, for instance breeze or inhomogeneous medium. In the study of random
dynamical systems, these randomness is often modeled by white noise and the
mechanical processes can be described as stochastic Newtonian system (1.1).

Stochastic boundary value problems (SBVPs) appear naturally in a variety of
fields such as stochastic optimal control [27], valuation of boundary-linked assets
[11], smoothing [25], the study of reciprocal processes [1] and maximum a posteriori
estimation of trajectories of diffusions [26]. There has been a lot of literature on
stochastic boundary value problems, see for example [2, 3, 5, 8, 9, 10, 20, 21, 22,
23, 24]. Nualart and Pardoux [21] studied a second-order stochastic differential
equation

d2Xt

dt2
+ f

(
Xt,

dXt

dt

)
=
dWt

dt
, t ∈ [0, 1], (1.5)

subject to the Dirichlet type boundary condition X0 = a and X1 = b, where a and
b are fixed real numbers. Here {Wt} is a one-dimensional Brownian motion starting
at zero. They proved that pathwise existence and uniqueness of solutions assuming
some smoothness and monotonicity conditions on f , and they also studied the
Markov property of solutions using an extended version of the Girsanov theorem
due to Kusuoka [16].

However, as far as we know, for the study of stochastic differential equations
with boundary conditions, the solution process is usually not adapted to the Brow-
nian filtration because of the deterministic boundary value conditions. Naturally,
one asks that what type of boundary conditions would make the solution Xt of
stochastic Newtonian system (1.1) adapt to the Brownian filtration.

We consider that boundary conditions should be uncertain and random due to
noise. In this paper, we propose a kind of boundary conditions (1.2) with expecta-
tion at the end of the interval. Boundary conditions with expectation represent that
the start point is fixed and the end point is random. In fact, such boundary value
conditions are more natural than fixed ones. For instance, if a small ball in simple
harmonic motion is affected by random perturbations, then one cannot forecast the
location of the ball in stochastic systems. However, it is rational to give a “possible
location” such as expectation at a time t in a statistical twist. Moreover, we verify
that the process {Xt, a ≤ t ≤ b} is an adapted process to Brownian filtration and
the problem mentioned above is well-posed. These phenomena also lend support to
that it is appropriate to consider mean boundary conditions in stochastic systems.

For Dirichlet boundary value problem of ordinary differential equations, suitable
Lipschitz condition can guarantee the existence and uniqueness of solutions. How-
ever, for stochastic Newtonian system (1.1) with mean boundary conditions (1.2),
what are sufficient conditions for the existence and uniqueness of solutions? This is
exactly what we do in this present paper. In this paper, we give a new idea to study
the above problems. We divide stochastic differential equation (1.1) into the deter-
ministic part and the stochastic part. By constructing stochastic Picard sequences
[12] for the stochastic part, we prove the convergence of numerical integrations in



EJDE-2021/79 STOCHASTIC NEWTONIAN EQUATIONS 3

L2(P,Rl). We get some explicit conditions to obtain the existence and uniqueness
of solutions for stochastic Newtonian systems with mean boundary conditions.

The dynamics of stochastic differential equations such as the well-posedness of
periodic solutions has been the subject of much concerns recently. By finding
new analysis tools or technology, the difficulties of the randomness on the orbit of
stochastic system and the non-compactness of the space for stochastic processes are
surmounted effectively. For example, Liu and Wang [19] obtained the existence of
stochastic almost periodic solutions in distribution by Favard separation method.
Chen et al [7] gave a weak Halanay’s criterion and proved the existence of periodic
solutions in distribution for SDEs. Ji et al [13] studied existence of periodic solutions
in distribution for stochastic differential equations with irregular coefficients. Jiang
and Li [15] studied the existence of periodic solutions in distribution of dissipative
stochastic differential equations via the Wong-Zakai approximation method. They
also verified a stochastic Levinson type conjecture in [14]. Under some assumptions
on the coefficients, the existence of periodic solutions to semilinear SDEs has been
established, see [6].

This article is organized as follows. In Section 2, we review some concepts,
introduce some notation and state our main result, which shows the existence and
uniqueness of solutions for system (1.1). In Section 3, we give the proof of the main
result by constructing the stochastic Picard sequences. In particular, we obtain
some explicit sufficient conditions of the existence and uniqueness of solutions. In
Section 4, some examples are given to illustrate the theoretical results.

2. Preliminaries and main result

Throughout this paper, we assume that (Ω,F ,P) is a complete probability space
with a filtration {Ft}t≥0 satisfying the usual conditions (i.e. it is right continuous
and F0 contains all P -null sets). Let L2(P,Rl) stand for the space of all Rl-valued
random variables X such that

E|X|2 =

∫
Ω

|X|2 dP <∞.

For X ∈ L2(P,Rl), we let

‖X‖2 :=
(∫

Ω

|X|2 dP
)1/2

.

Then L2(P,Rl) is a Hilbert space equipped with the norm ‖ · ‖2.
For the stochastic Newtonian equation (1.1) with mean boundary conditions

(1.2), we have the following result. It shows that under some sufficient conditions,
there exists a unique solution.

Theorem 2.1. Assume that there exist positive constants L1, L2 and K such that

(i) (Lipschitz condition) for all x, y ∈ Rl and t ∈ [a, b],

|f(t, x)− f(t, y)| ≤ L1|x− y|, |g(t, x)− g(t, y)| ≤ L2|x− y|;
(ii) (Linear growth condition) for all (t, x) ∈ [a, b]× Rl,

|f(t, x)| ∨ |g(t, x)| ≤ K(1 + |x|);
(iii) Lipschitz constants L1 and L2 satisfy

(12 +
1

64
)L2

1(b− a)4 + 3L2
2(b− a)3 < 1.



4 Y.-J. GUO, X.-M. JIANG EJDE-2021/79

Then stochastic Newtonian system (1.1) with mean boundary conditions X(a) = 0
and EX(b) = 0 has a unique continuous solution Xt(ω) belonging to L2(P,Rl), and
Xt(ω) is adapted to the filtration {Ft}t≥0.

Considering the influence of white noise for stochastic Newtonian system (1.1),
the boundary conditions are uncertain and random. So we impose the boundary
conditions with expectation at the end of the interval t = b. Here X(a) = 0 is
deterministic.

The solution {Xt, a ≤ t ≤ b} of stochastic Newtonian system (1.1) with mean
boundary conditions (1.2) is adapted to the filtration {Ft}t≥0.

If the mean boundary conditions (1.2) are generalized as

X(a) = A, EX(b) = B,

where A and B are arbitrary constants, then we can translate the variable Xt, and
yields

Xt = Zt + u(t), u(t) = A+
B −A
b− a

(t− a).

Hence, Z(t) satisfies mean boundary conditions Z(a) = 0,EZ(b) = 0.

3. Proof of main results

Proof of Theorem 2.1. First of all, we rewrite stochastic Newtonian system (1.1) as

EẌ(t) + Ẍ(t)−EẌ(t) = f(t,EXt) + f(t,Xt)− f(t,EXt) + g(t,Xt)Ḃt. (3.1)

By the change of variables

x̄(t) = EX(t), y(t) = X(t)−EX(t),

system (3.1) can be written as

¨̄x(t) + ÿ(t) = f(t, x̄(t)) + f(t, x̄(t) + y(t))− f(t, x̄(t)) + g(t, x̄(t) + y(t))Ḃt. (3.2)

We divide equation (3.2) into two parts, the deterministic system and the sto-
chastic system. Here the deterministic system is

¨̄x(t) = f(t, x̄(t)), (3.3)

and the stochastic system is

ÿ(t) = f(t, x̄(t) + y(t))− f(t, x̄(t)) + g(t, x̄(t) + y(t))Ḃt. (3.4)

We divide the proof into 5 steps. First, using the Green’s function, we study the
existence of solutions of the deterministic system. Then we construct the stochastic
Picard iterative sequences and studying their convergence in L2(P,Rl), which shows
the existence of solutions to the stochastic system. Third, we prove the existence
of solutions of system (1.1). Fourth, the uniqueness of solutions of system (1.1) is
obtained by studying the uniqueness of the solutions of the deterministic system
and the stochastic system, respectively. Finally, we show that the solutions of
system (1.1) is adapted to the filtration.

Step 1: Existence of solutions of the deterministic system. We consider
the deterministic second-order boundary value problem

¨̄x = f(t, x̄(t)),

x̄(a) = 0, x̄(b) = 0,
(3.5)



EJDE-2021/79 STOCHASTIC NEWTONIAN EQUATIONS 5

where f : [a, b]× Rl → Rl is a continuous function. It is easy to check that

x̄(t) =

∫ b

a

G(t, s)f(s, x̄(s))ds, t ∈ [a, b], (3.6)

where G : [a, b]× [a, b]→ R is the Green’s function

G(t, s) =

{
− (b−t)(s−a)

b−a , a ≤ s ≤ t ≤ b,

− (b−s)(t−a)
b−a , a ≤ t ≤ s ≤ b.

(3.7)

Obviously, G(t, s) is non-positive and

a− b
4
≤ G(t, s) ≤ 0.

By the estimate for G(t, s), we have

∫ b

a

|G(t, s)|ds =

∫ t

a

(b− t)(s− a)

b− a
ds+

∫ b

t

(b− s)(t− a)

b− a
ds

=
b− t
b− a

∫ t

a

(s− a)ds+
t− a
b− a

∫ b

t

(b− s)ds

=
(b− t)(t− a)

2

≤ (b− a)2

8
.

(3.8)

We define a Picard iterative sequences {x̄n(t)} as follows

x̄0(t) = 0,

x̄n(t) =

∫ b

a

G(t, s)f(s, x̄n−1(s))ds, n = 1, 2, . . . ,
(3.9)

for t ∈ [a, b]. Here x̄n(t) ∈ C([a, b];Rl), and there exists a constant M > 0 such
that |f(t, x̄(t))| ≤M for all t ∈ [a, b]. Note that

|x̄1(t)− x̄0(t)| =
∣∣ ∫ b

a

G(t, s)f(s, x̄0(s))ds
∣∣ ≤ M(b− a)2

8
,

|x̄2(t)− x̄1(t)| =
∣∣∣ ∫ b

a

G(t, s)f(s, x̄1(s))ds−
∫ b

a

G(t, s)f(s, x̄0(s))ds
∣∣∣

≤ML1

[ (b− a)2

8

]2
.

Suppose that

|x̄n(t)− x̄n−1(t)| ≤MLn−1
1

[ (b− a)2

8

]n
.



6 Y.-J. GUO, X.-M. JIANG EJDE-2021/79

Then

|x̄n+1(t)− x̄n(t)| =
∣∣∣ ∫ b

a

G(t, s)f(s, x̄n(s))ds−
∫ b

a

G(t, s)f(s, x̄n−1(s))ds
∣∣∣

≤ L1

∣∣∣ ∫ b

a

G(t, s)[x̄n(s)− x̄n−1(s)]ds
∣∣∣

≤MLn
1

[ (b− a)2

8

]n∣∣∣ ∫ b

a

G(t, s)ds
∣∣∣

≤ M(b− a)2

8

[L1(b− a)2

8

]n
.

(3.10)

Setting the condition L1(b−a)2

8 < 1, the partial sums

x̄n(t) = x̄0(t) +

n−1∑
i=0

[x̄i+1(t)− x̄i(t)]

converge uniformly in t ∈ [a, b]. Let x̄n(t) → x̄(t) as n → ∞. Then x̄(t) ∈
C([a, b];Rl).

It remains to show that x̄(t) satisfies (3.6). Note that, for all ε > 0,∣∣∣ ∫ b

a

G(t, s)f(s, x̄n(s))ds−
∫ b

a

G(t, s)f(s, x̄(s))ds
∣∣∣

≤ L1

∣∣∣ ∫ b

a

G(t, s)|x̄n(s)− x̄(s)|ds
∣∣∣

≤ L1 max
a≤t≤b

|x̄n(s)− x̄(s)|
∣∣∣ ∫ b

a

G(t, s)ds
∣∣∣

<
L1(b− a)2

8
ε.

(3.11)

Letting n→∞ in (3.9), we obtain that

x̄(t) =

∫ b

a

G(t, s)f(s, x̄(s))ds, (3.12)

on a ≤ t ≤ b. Then x̄(t) is the solution of the deterministic system (3.5).

Step 2. Existence of solutions of the stochastic system. We consider the
stochastic differential equation with mean boundary conditions

ÿ(t) = f(t, x̄(t) + y(t))− f(t, x̄(t)) + g(t, x̄(t) + y(t))Ḃ(t),

y(a) = 0, Ey(b) = 0.
(3.13)

By a change of variables, equation (3.13) can be written as

ẏ(t) = Y (t), (3.14)

Ẏ (t) = f(t, x̄(t) + y(t))− f(t, x̄(t)) + g(t, x̄(t) + y(t))Ḃ(t). (3.15)

Integrating on both sides of (3.14) and (3.15) from a to t, t ∈ [a, b], we have

y(t) = y(a) +

∫ t

a

Y (s)ds =

∫ t

a

Y (s)ds, (3.16)



EJDE-2021/79 STOCHASTIC NEWTONIAN EQUATIONS 7

and

Y (t) = Y (a) +

∫ t

a

[f(s, x̄(s) + y(s))− f(s, x̄(s))]ds

+

∫ t

a

g(s, x̄(s) + y(s))dB(s).

(3.17)

Taking expectations on both sides of y(t), we have

Ey(t) = E

∫ t

a

Y (s)ds

= E

∫ t

a

{
Y (a) +

∫ s

a

[f(u, x̄(u) + y(u))− f(u, x̄(u))]du

+

∫ s

a

g(u, x̄(u) + y(u))dB(u)
}
ds.

(3.18)

Then

0 = Ey(b) = E

∫ b

a

Y (s)ds

=

∫ b

a

Y (a)dt+

∫ b

a

ds

∫ s

a

E[f(u, x̄(u) + y(u))− f(u, x̄(u))]du

+

∫ b

a

dsE

∫ s

a

g(u, x̄(u) + y(u))dB(u)

= Y (a)(b− a) +

∫ b

a

ds

∫ s

a

E[f(u, x̄(u) + y(u))− f(u, x̄(u))]du.

(3.19)

We obtain

Y (a) = − 1

b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄(u) + y(u))− f(u, x̄(u))]du. (3.20)

Hence, y(t) can be represented as

y(t) =

∫ t

a

Y (s)ds

=− t− a
b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄(u) + y(u))− f(u, x̄(u))]du

+

∫ t

a

ds

∫ s

a

[f(u, x̄(u) + y(u))− f(u, x̄(u))]du

+

∫ t

a

ds

∫ s

a

g(u, x̄(u) + y(u))dB(u).

(3.21)

We define the stochastic Picard iterative sequences as follows: y0(t) = 0 and

yn(t) =− t− a
b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄n−1(u) + yn−1(u))− f(u, x̄n−1(u))]du

+

∫ t

a

ds

∫ s

a

[f(u, x̄n−1(u) + yn−1(u))− f(u, x̄n−1(u))]du

+

∫ t

a

ds

∫ s

a

g(u, x̄n−1(u) + yn−1(u))dB(u),

(3.22)



8 Y.-J. GUO, X.-M. JIANG EJDE-2021/79

for t ∈ [a, b] and n = 1, 2, . . . . Note that

E|y1(t)− y0(t)|2

= E
∣∣∣− t− a

b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄0(u) + y0(u))− f(u, x̄0(u))]du

+

∫ t

a

ds

∫ s

a

[f(u, x̄0(u) + y0(u))− f(u, x̄0(u))]du

+

∫ t

a

ds

∫ s

a

g(u, x̄0(u) + y0(u))dB(u)
∣∣∣2

= E
∣∣∣ ∫ t

a

ds

∫ s

a

g(u, 0)dB(u)
∣∣∣2

≤ (t− a)

∫ t

a

E
∣∣∣ ∫ s

a

g(u, 0)dB(u)
∣∣∣2ds.

(3.23)

By Itô isometry formula and linear growth condition (ii), we obtain

E|y1(t)− y0(t)|2 ≤ (t− a)

∫ t

a

E

∫ s

a

|g(u, 0)|2 du ds

≤ (t− a)

∫ t

a

K2(s− a)ds

≤ K2

2
(b− a)3.

Next, for n ≥ 1 and t ∈ [a, b], we have

E|yn+1(t)− yn(t)|2

= E
∣∣∣− t− a

b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄n(u) + yn(u))− f(u, x̄n(u))]du

+

∫ t

a

ds

∫ s

a

[f(u, x̄n(u) + yn(u))− f(u, x̄n(u))]du

+

∫ t

a

ds

∫ s

a

g(u, x̄n(u) + yn(u))dB(u)

+
t− a
b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄n−1(u) + yn−1(u))− f(u, x̄n−1(u))]du

−
∫ t

a

ds

∫ s

a

[f(u, x̄n−1(u) + yn−1(u))− f(u, x̄n−1(u))]du

−
∫ t

a

ds

∫ s

a

g(u, x̄n−1(u) + yn−1(u))dB(u)
∣∣∣2

= E
∣∣∣− t− a

b− a

∫ b

a

ds

∫ s

a

{
E[f(u, x̄n(u) + yn(u))− f(u, x̄n−1(u) + yn−1(u))]

−E[f(u, x̄n(u))− f(u, x̄n−1(u))]
}
du

+

∫ t

a

ds

∫ s

a

{
[f(u, x̄n(u) + yn(u))− f(u, x̄n−1(u) + yn−1(u))]

− [f(u, x̄n(u))− f(u, x̄n−1(u))]
}
du



EJDE-2021/79 STOCHASTIC NEWTONIAN EQUATIONS 9

+

∫ t

a

ds

∫ s

a

[g(u, x̄n(u) + yn(u))− g(u, x̄n−1(u) + yn−1(u))]dB(u)
∣∣∣2

≤ 3
∣∣∣ t− a
b− a

∫ b

a

ds

∫ s

a

{
E[f(u, x̄n(u) + yn(u))− f(u, x̄n−1(u) + yn−1(u))]

−E[f(u, x̄n(u)− f(u, x̄n−1(u))]
}
du
∣∣∣2

+ 3E
∣∣∣ ∫ t

a

ds

∫ s

a

{
[f(u, x̄n(u) + yn(u))− f(u, x̄n−1(u) + yn−1(u))]

− [f(u, x̄n(u))− f(u, x̄n−1(u))]
}
du
∣∣∣2

+ 3E
∣∣∣ ∫ t

a

ds

∫ s

a

[g(u, x̄n(u) + yn(u))− g(u, x̄n−1(u) + yn−1(u))]dB(u)
∣∣∣2.

Applying Hölder’s inequality and Itô isometry formula, we have

E|yn+1(t)− yn(t)|2

≤ 3(
t− a
b− a

)2(b− a)

∫ b

a

∣∣∣ ∫ s

a

{
E[f(u, x̄n(u) + yn(u))− f(u, x̄n−1(u) + yn−1(u))]

−E[f(u, x̄n(u)− f(u, x̄n−1(u))]
}
du
∣∣∣2ds

+ 3(t− a)

∫ t

a

E
∣∣∣ ∫ s

a

{
[f(u, x̄n(u) + yn(u))− f(u, x̄n−1(u) + yn−1(u))]

− [f(u, x̄n(u))− f(u, x̄n−1(u))]
}
du
∣∣∣2ds

+ 3(t− a)

∫ t

a

E

∣∣∣∣∫ s

a

[g(u, x̄n(u) + yn(u))− g(u, x̄n−1(u) + yn−1(u))]dB(u)

∣∣∣∣2 ds
≤ 3(t− a)2

b− a

∫ b

a

(s− a)

∫ s

a

|E[f(u, x̄n(u) + yn(u))− f(u, x̄n−1(u) + yn−1(u))]

−E[f(u, x̄n(u)− f(u, x̄n−1(u))]|2 du ds

+ 3(t− a)

∫ t

a

(s− a)

∫ s

a

E|[f(u, x̄n(u) + yn(u))− f(u, x̄n−1(u) + yn−1(u))]

− [f(u, x̄n(u)− f(u, x̄n−1(u))] |2 du ds

+ 3(t− a)

∫ t

a

∫ s

a

E|g(u, x̄n(u) + yn(u))− g(u, x̄n−1(u) + yn−1(u))|2 du ds

≤ 6(t− a)2

b− a

∫ b

a

(s− a)

∫ s

a

[
E|f(u, x̄n(u) + yn(u))− f(u, x̄n−1(u) + yn−1(u))|2

+ E|f(u, x̄n(u)− f(u, x̄n−1(u))|2
]
du ds

+ 6(t− a)

∫ t

a

(s− a)

∫ s

a

[
E|f(u, x̄n(u) + yn(u))− f(u, x̄n−1(u) + yn−1(u))|2

+ E|f(u, x̄n(u)− f(u, x̄n−1(u))|2
]
du ds

+ 3L2
2(t− a)

∫ t

a

∫ s

a

E|x̄n(u) + yn(u)− x̄n−1(u)− yn−1(u)|2 du ds.



10 Y.-J. GUO, X.-M. JIANG EJDE-2021/79

By Lipchitz condition (i), we obtain

E|yn+1(t)− yn(t)|2

≤ 6L2
1(t− a)2

b− a

∫ b

a

(s− a)

∫ s

a

[
E|x̄n(u) + yn(u)− x̄n−1(u)− yn−1(u)|2

+ |x̄n(u)− x̄n−1(u)|2
]
du ds

+ 6L2
1(t− a)

∫ t

a

(s− a)

∫ s

a

[
E|x̄n(u) + yn(u)− x̄n−1(u)− yn−1(u)|2

+ |x̄n(u)− x̄n−1(u)|2
]
du ds

+ 6L2
2(t− a)

∫ t

a

∫ s

a

[
|x̄n(u)− x̄n−1(u)|2 + E|yn(u)− yn−1(u)|2] du ds

≤ 6L2
1(t− a)2

b− a

∫ b

a

(s− a)

∫ s

a

[3|x̄n(u)− x̄n−1(u)|2 + 2E|yn(u)− yn−1(u)|2
]
du ds

+ 6L2
1(t− a)

∫ t

a

(s− a)

∫ s

a

[
3|x̄n(u)− x̄n−1(u)|2 + 2E|yn(u)− yn−1(u)|2

]
du ds

+ 6L2
2(t− a)

∫ t

a

∫ s

a

[
|x̄n(u)− x̄n−1(u)|2 + E|yn(u)− yn−1(u)|2

]
du ds

≤ 18L2
1(t− a)

∫ b

a

(s− a)

∫ s

a

|x̄n(u)− x̄n−1(u)|2 du ds

+ 18L2
1(t− a)

∫ t

a

(s− a)

∫ s

a

|x̄n(u)− x̄n−1(u)|2 du ds

+ 6L2
2(t− a)

∫ t

a

∫ s

a

|x̄n(u)− x̄n−1(u)|2 du ds

+ 12L2
1(t− a)

∫ b

a

(s− a)

∫ s

a

E|yn(u)− yn−1(u)|2 du ds

+ 12L2
1(t− a)

∫ t

a

(s− a)

∫ s

a

E|yn(u)− yn−1(u)|2 du ds

+ 6L2
2(t− a)

∫ t

a

∫ s

a

E|yn(u)− yn−1(u)|2 du ds.

Note that

|x̄n(t)− x̄n−1(t)|2 ≤ M2(b− a)4

64
·
[L1(b− a)2

8

]2n−2
:=

M2(b− a)4

64
q2n−2, (3.24)

and q = L1(b−a)2

8 < 1.
Then we have the following estimates:

18L2
1(t− a)

∫ b

a

(s− a)

∫ s

a

|x̄n(u)− x̄n−1(u)|2 du ds

≤ 9L2
1M

2

32
(b− a)4(t− a)q2n−2

∫ b

a

(s− a)2ds

=
3L2

1M
2

32
(b− a)8q2n−2,



EJDE-2021/79 STOCHASTIC NEWTONIAN EQUATIONS 11

6L2
2(t− a)

∫ t

a

∫ s

a

|x̄n(u)− x̄n−1(u)|2 du ds ≤ 3L2
2M

2

64
(b− a)7q2n−2.

Then

18L2
1(t− a)

∫ b

a

(s− a)

∫ s

a

|x̄n(u)− x̄n−1(u)|2 du ds

+ 18L2
1(t− a)

∫ t

a

(s− a)

∫ s

a

|x̄n(u)− x̄n−1(u)|2 du ds

+ 6L2
2(t− a)

∫ t

a

∫ s

a

|x̄n(u)− x̄n−1(u)|2 du ds

≤
[3L2

1(b− a)8

16
+

3L2
2(b− a)7

64

]
M2q2n−2 := M1q

2n−2,

where

M1 =
[3L2

1(b− a)8

16
+

3L2
2(b− a)7

64

]
M2.

Hence, for n ≥ 1, we obtain that

E|yn+1(t)− yn(t)|2

≤M1q
2n−2 + 12L2

1(t− a)

∫ b

a

(b− a)

∫ b

a

E|yn(u)− yn−1(u)|2 du ds

+ 12L2
1(t− a)

∫ b

a

(b− a)

∫ b

a

E|yn(u)− yn−1(u)|2 du ds

+ 6L2
2(t− a)

∫ b

a

∫ b

a

E|yn(u)− yn−1(u)|2 du ds

= M1q
2n−2 + 24L2

1(b− a)2(t− a)

∫ b

a

E|yn(u)− yn−1(u)|2du

+ 6L2
2(t− a)(b− a)

∫ b

a

E|yn(u)− yn−1(u)|2du

= M1q
2n−2 +

[
24L2

1(b− a)2 + 6L2
2(b− a)

]
(t− a)

∫ b

a

E|yn(u)− yn−1(u)|2du

:= M1q
2n−2 +M2(t− a)

∫ b

a

E|yn(u)− yn−1(u)|2du,

where M2 = 24L2
1(b− a)2 + 6L2

2(b− a).
Moreover, we have

E|y2(t)− y1(t)|2 ≤M1q
0 +M2(t− a)

∫ b

a

K2

2
(b− a)3du

= M1q
0 +

M2K
2(b− a)4

2
(t− a),

E|y3(t)− y2(t)|2 ≤M1q
2 +M2(t− a)

∫ b

a

[
M1q

0 +
M2K

2(b− a)4

2
(u− a)

]
du

= M1q
2 +M2(t− a)

[
M1q

0(b− a) +
M2K

2(b− a)6

22

]
,



12 Y.-J. GUO, X.-M. JIANG EJDE-2021/79

E|y4(t)− y3(t)|2 ≤M1q
4 +M2(t− a)

∫ b

a

{
M1q

2 +M2

[
M1q

0(b− a)

+
M2K

2(b− a)6

22

]
(u− a)

}
du

= M1q
4 +M2(t− a)

[
M1q

2(b− a)

+
M2M1q

0(b− a)3

2
+
M2

2K
2(b− a)8

23

]
,

E|y5(t)− y4(t)|2 ≤M1q
6 +M2(t− a)

∫ b

a

{
M1q

4 +M2

[
M1q

2(b− a)

+
M2M1q

0(b− a)3

2
+
M2

2K
2(b− a)8

23

]
(u− a)

}
du

= M1q
6 +M2(t− a)

[
M1q

4(b− a) +
M2M1q

2(b− a)3

2

+
M2

2M1q
0(b− a)5

22
+
M3

2K
2(b− a)10

24

]
,

E|y6(t)− y5(t)|2 ≤M1q
8 +M2(t− a)

∫ b

a

{
M1q

6 +M2

[
M1q

4(b− a)

+
M2M1q

2(b− a)3

2
+
M2

2M1q
0(b− a)5

22

+
M3

2K
2(b− a)10

24

]
(u− a)

}
du

= M1q
8 +M2(t− a)

[
M1q

6(b− a) +
M2M1q

4(b− a)3

2

+
M2

2M1q
2(b− a)5

22
+
M3

2M1(b− a)7

23
+
M4

2K
2(b− a)12

25

]
.

By induction we obtain that for n > 2,

E|yn+1(t)− yn(t)|2 ≤M1q
2n−2 +M2

[
M1q

2n−4(b− a) +
Mn−1

2 K2(b− a)2n+2

2n

+

n−2∑
k=1

Mk
2M1q

2n−2k−4

2k
(b− a)2k+1

]
(t− a).

(3.25)
Notice that

n−2∑
k=1

Mk
2M1q

2n−2k−4

2k
(b− a)2k+1 = M1(b− a)

n−2∑
k=1

Mk
2 (b− a)2k

2k
· q2(n−2−k)

= M1(b− a)

n−2∑
k=1

[M2(b− a)2

2

]k
· q2(n−2−k)

≤M1(b− a)
[M2(b− a)2

2
+ q2

]n−2

.

Then

E|yn+1(t)− yn(t)|2 ≤M1q
2n−2 +M2

{
M1q

2n−4(b− a) +
Mn−1

2 K2(b− a)2n+2

2n



EJDE-2021/79 STOCHASTIC NEWTONIAN EQUATIONS 13

+M1(b− a)
[M2(b− a)2

2
+ q2

]n−2
}

(t− a).

If p > n > 2, we have

‖yp − yn‖L2

≤
p−1∑
k=n

‖yk+1 − yk‖L2

=

p−1∑
k=n

[
E

∫ b

a

|yk+1(t)− yk(t)|2dt
]1/2

≤
p−1∑
k=n

{
M1(b− a)q2k−2 +

M2(b− a)2

2
[M1q

2k−4(b− a)

+
Mk−1

2 K2(b− a)2k+2

2k
+M1(b− a)(

M2(b− a)2

2
+ q2)k−2

}1/2

≤
p−1∑
k=n

[√
M1(b− a)qk−1 +

√
M1M2(b− a)3

2
qk−2

+
K√

2
(b− a)2

(√M2

2
(b− a)

)k
+

√
2M1M2(b− a)3

M2(b− a)2 + 2q2

(√M2(b− a)2

2
+ q2

)k]
.

By the assumption M2(b−a)2

2 + q2 < 1, which means

(12 +
1

64
)L2

1(b− a)4 + 3L2
2(b− a)3 < 1, (3.26)

we have

‖yp − yn‖L2 → 0 (3.27)

as p, n→∞.
Therefore, {yn(t)}∞n=0 is a Cauchy sequence in L2(P,Rl) for all t ∈ [a, b]. Hence

{yn(t)}∞n=0 is convergent in L2(P,Rl). We define

yt := lim
n→∞

yn(t). (3.28)

Then yt is Ft−measurable for all t ∈ [a, b].
Next, for n ∈ N+ and t ∈ [a, b], we have

yn+1(t) = − t− a
b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄n(u) + yn(u))− f(u, x̄n(u))]du

+

∫ t

a

ds

∫ s

a

[f(u, x̄n(u) + yn(u))− f(u, x̄n(u))]du

+

∫ t

a

ds

∫ s

a

g(u, x̄n(u) + yn(u))dB(u).

Then by the Cauchy-Schwartz inequality, we have∣∣∣ t− a
b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄n(u) + yn(u))− f(u, x̄n(u))]du



14 Y.-J. GUO, X.-M. JIANG EJDE-2021/79

− t− a
b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄(u) + y(u))− f(u, x̄(u))]du
∣∣∣2

≤ (t− a)2

b− a

∫ b

a

∣∣∣ ∫ s

a

{E[f(u, x̄n(u) + yn(u))− f(u, x̄(u) + y(u))]

−E[f(u, x̄n(u))− f(u, x̄(u))]}du
∣∣∣2ds

≤ (t− a)2

b− a

∫ b

a

(s− a)

∫ s

a

|E[f(u, x̄n(u) + yn(u))− f(u, x̄(u) + y(u))]

−E[f(u, x̄n(u))− f(u, x̄(u))]|2 du ds

≤ 2(t− a)2

b− a

∫ b

a

(s− a)

∫ s

a

[E|f(u, x̄n(u) + yn(u))− f(u, x̄(u) + y(u))|2

+ E|f(u, x̄n(u))− f(u, x̄(u))|2] du ds

≤ 2L2
1(t− a)2

b− a

∫ b

a

(s− a)

∫ s

a

[E|x̄n(u) + yn(u)− x̄(u)− y(u)|2

+ E|x̄n(u)− x̄(u)|2] du ds

≤ 2L2
1(t− a)2

b− a

∫ b

a

(s− a)

∫ s

a

[3E|x̄n(u)− x̄(u)|2 + 2E|ȳn(u)− ȳ(u)|2] du ds

≤ 2L2
1(b− a)3

∫ b

a

[3E|x̄n(u)− x̄(u)|2 + 2E|ȳn(u)− ȳ(u)|2]du

→ 0 as n→∞. (3.29)

Using Hölder’s inequality, we obtain

E
∣∣∣ ∫ t

a

ds

∫ s

a

[f(u, x̄n(u) + yn(u))− f(u, x̄n(u))]du

−
∫ t

a

ds

∫ s

a

[f(u, x̄(u) + y(u))− f(u, x̄(u))]du
∣∣∣2

≤ (t− a)

∫ t

a

(s− a)E

∫ s

a

|[f(u, x̄n(u) + yn(u))− f(u, x̄(u) + y(u))]

− [f(u, x̄n(u))− f(u, x̄(u))]|2 du ds

≤ 2(t− a)

∫ t

a

(s− a)

∫ s

a

[E|f(u, x̄n(u) + yn(u))− f(u, x̄(u) + y(u))|2

+ E|f(u, x̄n(u))− f(u, x̄(u))|2] du ds

≤ 2L2
1(t− a)

∫ t

a

(s− a)

∫ s

a

[3E|x̄n(u)− x̄(u)|2 + 2E|ȳn(u)− ȳ(u)|2] du ds

≤ 2L2
1(b− a)3

∫ b

a

[3E|x̄n(u)− x̄(u)|2 + 2E|ȳn(u)− ȳ(u)|2]du

→ 0 in L2(P ). (3.30)

By Itô isometry property, we have

E
∣∣∣ ∫ t

a

ds

∫ s

a

g(u, x̄n(u) + yn(u))dB(u)−
∫ t

a

ds

∫ s

a

g(u, x̄(u) + y(u))dB(u)
∣∣∣2



EJDE-2021/79 STOCHASTIC NEWTONIAN EQUATIONS 15

≤ (t− a)

∫ t

a

E
∣∣∣ ∫ s

a

[g(u, x̄n(u) + yn(u))− g(u, x̄(u) + y(u))]dB(u)
∣∣∣2ds

= (t− a)

∫ t

a

E

∫ s

a

|g(u, x̄n(u) + yn(u))− g(u, x̄(u) + y(u))|2 du ds

≤ L2
2(t− a)

∫ t

a

E

∫ s

a

|x̄n(u) + yn(u)− x̄(u)− y(u)|2 du ds

≤ 2L2
2(t− a)

∫ t

a

∫ s

a

[E|x̄n(u)− x̄(u)|2 + E|yn(u)− y(u)|2] du ds

→ 0 in L2(P ). (3.31)

Combining (3.29), (3.30) and (3.31), we conclude that for all t ∈ [a, b],

y(t) = − t− a
b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄(u) + y(u))− f(u, x̄(u))]du

+

∫ t

a

ds

∫ s

a

[f(u, x̄(u) + y(u))− f(u, x̄(u))]du

+

∫ t

a

ds

∫ s

a

g(u, x̄(u) + y(u))dB(u) a.s.

(3.32)

Then y(t) is the solution of the stochastic system (3.13).

Step 3: Existence of solutions of stochastic Newtonian equations with
mean boundary conditions We define

X0(t) = 0,

Xn(t) = x̄n(t) + yn(t),
(3.33)

for n ≥ 1 and t ∈ [a, b]. If h > n > 2, we obtain

‖Xh −Xn‖L2 = ‖x̄h + yh − x̄n − yn‖L2

≤ ‖x̄h − x̄n‖L2 + ‖yh − yn‖L2 → 0 as h, n→∞.
(3.34)

Therefore, {Xn(t)}∞n=0 is a Cauchy sequence in L2(P,Rl) and

lim
n→∞

Xn(t) = lim
n→∞

(x̄n + yn) = x̄(t) + y(t) := Xt. (3.35)

We conclude that for all t ∈ [a, b],

Xt =

∫ b

a

G(t, s)f(s, x̄(s))ds− t− a
b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄(u) + y(u))− f(u, x̄(u))]du

+

∫ t

a

ds

∫ s

a

[f(u, x̄(u) + y(u))− f(u, x̄(u))]du

+

∫ t

a

ds

∫ s

a

g(u, x̄(u) + y(u))dB(u)

=

∫ b

a

G(t, s)f(s,EX(s))ds− t− a
b− a

∫ b

a

ds

∫ s

a

E[f(u,X(u))− f(u,EX(u))]du

+

∫ t

a

ds

∫ s

a

[f(u,X(u))− f(u,EX(u))]du+

∫ t

a

ds

∫ s

a

g(u,X(u))dB(u),

and Xt is the solution of stochastic Newtonian system (1.1) with mean boundary
conditions (1.2). The existence of solutions is complete.



16 Y.-J. GUO, X.-M. JIANG EJDE-2021/79

Step 4 Uniqueness. Let x̄(t) and x̂(t) be two solutions of the deterministic
second-order boundary value problem (3.5). Note that

|x̄(t)− x̂(t)| = |
∫ b

a

G(t, s)f(s, x̄(s))ds−
∫ b

a

G(t, s)f(s, x̂(s))ds|

≤
∫ b

a

|G(t, s)f(s, x̄(s))− f(s, x̂(s))|ds

≤ (b− a)L1

4

∫ b

a

|x̄(s)− x̂(s)|ds

≤ [
(b− a)L1

4
]2
∫ b

a

∫ b

a

|x̄(s)− x̂(s)|dsds1

≤ [
(b− a)L1

4
]k
∫ b

a

· · ·
∫ b

a

∫ b

a︸ ︷︷ ︸
k

|x̄(s)− x̂(s)|dsds1 . . . dsk−1.

(3.36)

Set A1 =
∫ b

a
|x̄(s)− x̂(s)|ds, then

|x̄(t)− x̂(t)| ≤ [
(b− a)L1

4
]k
∫ b

a

· · ·
∫ b

a

A1ds1 . . . dsk−1

= A1(b− a)k−1 · [ (b− a)L1

4
]k

=
A1

b− a
[
(b− a)2L1

4
]k.

(3.37)

Under the condition 12L2
1(b− a)4 < 1, we obtain

L1(b− a)2

4
< 1.

Then

|x̄(t)− x̂(t)| ≤ A1

b− a
[
(b− a)2L1

4
]k → 0 as k →∞, (3.38)

Hence, x̄(t) ≡ x̂(t) for all a ≤ t ≤ b.
Next, let y(t) and ŷ(t) be two solutions of the stochastic differential equation

(3.13). They can be represented as follows

y(t) =− t− a
b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄(u) + y(u))− f(u, x̄(u))]du

+

∫ t

a

ds

∫ s

a

[f(u, x̄(u) + y(u))− f(u, x̄(u))]du

+

∫ t

a

ds

∫ s

a

g(u, x̄(u) + y(u))dB(u),

and

ŷ(t) =− t− a
b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄(u) + ŷ(u))− f(u, x̄(u))]du

+

∫ t

a

ds

∫ s

a

[f(u, x̄(u) + ŷ(u))− f(u, x̄(u))]du



EJDE-2021/79 STOCHASTIC NEWTONIAN EQUATIONS 17

+

∫ t

a

ds

∫ s

a

g(u, x̄(u) + ŷ(u))dB(u),

respectively. By Lipschitz condition, Hölder’s inequality and Itô isometry property,
we have

E|y(t)− ŷ(t)|2

= E
∣∣∣− t− a

b− a

∫ b

a

ds

∫ s

a

E[f(u, x̄(u) + y(u))− f(u, x̄(u) + ŷ(u))]du

+

∫ t

a

ds

∫ s

a

[f(u, x̄(u) + y(u))− f(u, x̄(u) + ŷ(u))]du

+

∫ t

a

ds

∫ s

a

[g(u, x̄(u) + y(u))− g(u, x̄(u) + ŷ(u))]dB(u)
∣∣∣2

≤ 3(t− a)2

b− a

∫ b

a

(s− a)

∫ s

a

E|f(u, x̄(u) + y(u))− f(u, x̄(u) + ŷ(u))|2 du ds

+ 3(t− a)

∫ t

a

(s− a)

∫ s

a

E|f(u, x̄(u) + y(u))− f(u, x̄(u) + ŷ(u))|2 du ds

+ 3(t− a)

∫ t

a

E|
∫ s

a

[g(u, x̄(u) + y(u))− g(u, x̄(u) + ŷ(u))]dB(u)|2ds

≤ 3L2(t− a)2

b− a

∫ b

a

(s− a)

∫ s

a

E|y(u)− ŷ(u)|2 du ds

+ 3L2(t− a)

∫ t

a

(s− a)

∫ s

a

E|y(u)− ŷ(u)|2 du ds

+ 3L2(t− a)

∫ t

a

∫ s

a

E|y(u)− ŷ(u)|2 du ds

≤ [6L2
1(b− a)2 + 3L2

2(b− a)]

∫ b

a

∫ s

a

E|y(u)− ŷ(u)|2 du ds

≤ [6L2
1(b− a)2 + 3L2

2(b− a)]2
∫ b

a

∫ s1

a

[ ∫ b

a

∫ s

a

E|y(u)− ŷ(u)|2 du ds
]
du1ds1

≤ [6L2
1(b− a)2 + 3L2

2(b− a)]k

×
∫ b

a

∫ sk−1

a

· · ·
∫ b

a

∫ s1

a

[ ∫ b

a

∫ s

a︸ ︷︷ ︸
2k

E|y(u)− ŷ(u)|2 du ds
]
du1ds1 . . . duk−1dsk−1.

Setting A2 =
∫ b

a

∫ s

a
E|y(u)− ŷ(u)|2 du ds, we obtain

E|y(t)− ŷ(t)|2

≤ [6L2
1(b− a)2 + 3L2

2(b− a)]k
∫ b

a

∫ sk−1

a

· · ·
∫ b

a

∫ s1

a

A2du1ds1 . . . du−1dsk−1

≤ A2[6L2
1(b− a)2 + 3L2

2(b− a)]k · [ (b− a)2

2
]k−1

=
2A2

(b− a)2
[3L2

1(b− a)4 +
3

2
L2

2(b− a)3)]k → 0 as k →∞,



18 Y.-J. GUO, X.-M. JIANG EJDE-2021/79

then y(t) ≡ ŷ(t) for all a ≤ t ≤ b, a.s.. Hence

Xt = x̄(t) + y(t)

is a unique solution of the stochastic Newtonian system (1.1) with mean boundary
conditions (1.2).

Step 5: Adaptability of the solution Xt to the filtration. The solution of
the stochastic Newtonian system (1.1) with mean boundary conditions (1.2) can be
represented as

Xt =− t− a
b− a

∫ b

a

∫ v

a

Ef(s,Xs) ds dv +

∫ t

a

∫ v

a

f(s,Xs) ds dv

+

∫ t

a

∫ u

a

g(s,Xs)dBsdu,

(3.39)

for t ∈ [a, b]. Hence, the solution Xt is adapted to the filtration {Ft}t≥0. The proof
is complete. �

4. Examples

In this section, we give several examples to illustrate the theoretical results ob-
tained in this paper. First, we consider the simplest care of SDE with mean bound-
ary conditions.

Example 4.1. Consider one-dimensional stochastic differential equation on the
interval [0, 1] with mean boundary conditions

ÿ(t) = λẆ (t), t ∈ [0, 1],

y(0) = 0, Ey(1) = 0.
(4.1)

Here λ ∈ (0, 1/5) is a constant and W (t) is a one-dimensional Wiener process. Let

ẏ(t) = z(t), ż(t) = λẆ (t), t ∈ [0, 1].

Then z(t) = z(0) + λW (t) and

y(t) = y(0) +

∫ t

0

z(s)ds = tz(0) + λtW (t)− λ
∫ t

0

sdW (s). (4.2)

From the mean boundary condition Ey(1) = 0, we have Ey(1) = z(0) = 0. So

y(t) = λtW (t)− λ
∫ t

0

sdW (s). (4.3)

Therefore, by Theorem 2.1, there is a unique solution y(t) of the stochastic system
(4.1).

The stochastic system (4.1) can represent the oscillation of a wire blown by
the wind. Next, let us consider linear stochastic differential equation with mean
boundary conditions.

Example 4.2. We consider the linear SDE

ÿ(t) + λy(t) = λẆ (t), t ∈ [0, 1], (4.4)

with mean boundary conditions

y(0) = 0, Ey(1) = 0,



EJDE-2021/79 STOCHASTIC NEWTONIAN EQUATIONS 19

where λ ∈ (0, 1/5) is a constant and W (t) is a one-dimensional Wiener process. We
change the variable y(t), then

ẏ(t) = u(t), u̇(t) + λy(t) = λẆ (t), t ∈ [0, 1]. (4.5)

Integrating from 0 to t on both sides of the equation (4.5) yields

u(t) = u(0)− λ
∫ t

0

y(s)ds+ λW (t).

Hence

y(t) = y(0) +

∫ t

0

u(s)ds

= tu(0)− λ
∫ t

0

∫ v

0

y(s) ds dv + λtW (t)− λ
∫ t

0

sdW (s).

From the mean boundary condition Ey(1) = 0, we obtain

Ey(1) = u(0)− λ
∫ 1

0

∫ v

0

Ey(s) ds dv = 0.

Then

u(0) = λ

∫ 1

0

∫ v

0

Ey(s) ds dv.

So

y(t) = λt

∫ 1

0

∫ v

0

Ey(s) ds dv − λ
∫ t

0

∫ v

0

y(s) ds dv + λtW (t)− λ
∫ t

0

sdW (s).

When λ ∈ (0, 1/5), the conditions in Theorem 2.1 are satisfied, and we have exis-
tence and uniqueness of system (4.4) with mean boundary conditions.

Let us consider the dynamics of stochastic flutter for a harmonic oscillator.

Example 4.3. We consider one-dimensional SDE:

ÿ(t) + λy(t) = sinπt+ λẆ (t), t ∈ [0, 1], (4.6)

with mean boundary conditions

y(0) = 0, Ey(1) = 0.

Here λ ∈ (0, 1
5 ) is a constant and W (t) is a one-dimensional Wiener process. Let

ẏ(t) = v(t), v̇(t) + λy(t) = sinπt+ λẆ (t), t ∈ [0, 1].

Then

v(t) = v(0)− λ
∫ t

0

y(s)ds− 1

π
cosπt+

1

π
+ λW (t),

y(t) = y(0) +

∫ t

0

v(s)ds

= tv(0)− λ
∫ t

0

∫ r

0

y(s) ds dr − 1

π2
sinπt+

t

π
+ λ

∫ t

0

W (s)ds.

Since Ey(1) = 0, we have

Ey(1) = v(0)− λ
∫ 1

0

∫ r

0

Ey(s) ds dr +
1

π
= 0.



20 Y.-J. GUO, X.-M. JIANG EJDE-2021/79

Then

v(0) = λ

∫ 1

0

∫ r

0

Ey(s) ds dr − 1

π
.

Hence

y(t) = λt

∫ 1

0

∫ r

0

Ey(s) ds dr − λ
∫ t

0

∫ r

0

y(s) ds dr

− 1

π2
sinπt+ λtW (t)− λ

∫ t

0

sdW (s).

When λ ∈ (0, 1
5 ), the conditions in Theorem 2.1 are satisfied. Then there exists a

unique solution y(t) of system (4.6) with mean boundary conditions.

5. Conclusion

The main goal of this paper is to discuss the existence and uniqueness of solu-
tions for a kind of second-order stochastic differential equations with mean bound-
ary conditions. Based on the Picard iteration method, we introduce a new idea and
obtain the sufficient conditions of the existence and uniqueness of solutions for sto-
chastic Newtonian equations with stochastic boundary conditions. As applications,
we show several types of SDEs with mean boundary conditions to illustrate our
theoretical results. Comparing with the Dirichlet boundary conditions, stochastic
boundary conditions with expectation are more random. And under this kind of
stochastic boundary conditions, the solution of stochastic Newtonian systems is
adapted to the Brownian filtration.

Acknowledgments. This research was supported by the National Natural Sci-
ence Foundation of China (No. 12001017), by the Department of Education of
Jilin Province 13th Five-Year Plan to support scientific research projects (No.
JJKH20200027KJ), by the National Natural Science Foundation of China (No.
11901231), by the National Natural Science Foundation of China (No. 12071175).

The authors sincerely thank Professor Yong Li for his valuable discussions and
suggestions.

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Ying-Jia Guo

School of Mathematics and Statistics, Beihua University, Jilin, Jilin 132000, China
Email address: guoyingjia2021@163.com

Xiao-Meng Jiang
College of Mathematics, Jilin University, Changchun, Jilin 130012, China

Email address: jxmlucy@hotmail.com


	1. Introduction
	2. Preliminaries and main result
	3. Proof of main results
	4. Examples
	5. Conclusion
	Acknowledgments

	References