On Bipartite Fuzzy Stochastic Differential Equations
Marek T. Malinowski
Institute of Mathematics, Cracow University of Technology, ul. Warszawska 24, 31-155 Krak
´
ow, Poland
Keywords:
Fuzzy And Stochastic Uncertainties, Bipartite Fuzzy Stochastic Differential Equation, Existence and Unique-
ness of Solution, Fuzzy Stochastic Process, Fuzzy Random Variable, Fuzzy Differential Equation.
Abstract:
The paper contains a discussion on solutions to new type of fuzzy stochastic differential equations. The
equations under study possess drift and diffusion terms at both sides of equations. We claim that such the
equations have unique solutions in the case that equations’ coefficients satisfy a certain generalized Lipschitz
condition. We use approximation sequences to reach solutions.
1 INTRODUCTION
In modelling dynamical systems in presence of un-
certainty the stochastic differential equations are used
(Gihman and Skorohod, 1972; Mao, 2007). However,
in the real-life phenomena there is often a sources
of uncertainty that does not come from randomness
and stochastic noises. This uncertainty is well treated
by fuzzy set theory (Zadeh, 1965). The fuzzy sets
theory has been successfully applied to deterministic
fuzzy differential equations (Kaleva, 1987; Bede and
Gal, 2005; Nieto and Rodr
´
ıguez-L
´
opez, 2006). There
are also many attempts to use two kinds of uncertain-
ties in modelling real-world systems, for instance (Li
et al., 2003; M
¨
oller et al., 2003; Zme
ˇ
skal, 2010). An
apparatus to model dynamical systems with random-
ness and fuzziness in a form of random fuzzy dif-
ferential equations with fuzzy derivative were stud-
ied too (Feng, 2000; Malinowski, 2009; Malinowski,
2012b; Park and Jeong, 2013; Malinowski, 2015c).
However, these models are not enough when some
stochastic noises in terms of Brownian motions ap-
pear. In such the situations fuzzy stochastic differen-
tial equations are more appropriate to be applied (Ma-
linowski, 2012c; Malinowski, 2013a; Malinowski,
2013b; Malinowski, 2015d; Malinowski, 2015e; Ma-
linowski, 2015a; Malinowski, 2015b; Malinowski
and Agarwal, 2015).
The latter topic on fuzzy stochastic differential
equations is new and needs further investigations. In
this paper we proceed with a discussion on bipartite
fuzzy stochastic differential equation in its integral
form
x(t) ⊕ (−1) 
Z
t
0
f (s,x(s))ds (1.1)
⊕
D
(−1)
Z
t
0
g(s,x(s))dB(s)
E
= x
0
⊕
Z
t
0
˜
f (s,x(s))ds
⊕
D
Z
t
0
˜g(s,x(s))d
˜
B(s)
E
, t ∈ [0,T ]
which contains drift and diffusion parts at both sides
and is driven by m-dimensional and n-dimensional
Brownian motions B and
˜
B, respectively. A de-
tailed description of this form is included in Section 3.
Such the equations were introduced by (Malinowski,
2016a). A fundamental problem on existence of so-
lutions to such the equations was considered with
the Lipschitz type assumptions imposed on equations’
coefficients. In the current paper we intend to show
that these equations have solutions when one relax the
assumptions to a certain generalized condition involv-
ing a certain function instead of the Lipschitz con-
stant. We will use a sequence of approximations to
achieve it. We limit ourselves to prove the main re-
sults only, since there is a pages’ limitation for this
submission.
2 PRELIMINARIES
In this section, we give some definitions and use-
ful facts and introduce necessary notation which will
be used throughout the paper. Most of it can be
found, for example, in (Malinowski, 2014; Mali-
nowski, 2016b; Malinowski, 2016c).
Malinowski, M.
On Bipartite Fuzzy Stochastic Differential Equations.
DOI: 10.5220/0006079501090114
In Proceedings of the 8th International Joint Conference on Computational Intelligence (IJCCI 2016) - Volume 2: FCTA, pages 109-114
ISBN: 978-989-758-201-1
Copyright
c
 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
109

Let K (R
d
) be the family of all nonempty,
compact and convex subsets of R
d
. In K (R
d
)
we consider the Hausdorff metric d
H
which
is defined by d
H
(A,B) := max{sup
a∈A
inf
b∈B
ka −
bk,sup
b∈B
inf
a∈A
ka − bk}, where k · k denotes a norm
in R
d
. The addition and scalar multiplication in
K (R
d
) are defined as usual, i.e., for A,B ∈ K (R
d
),
λ ∈ R we have A+ B := {a + b : a ∈ A,b ∈ B}, λA :=
{λa : a ∈ A}.
Let (Ω, A,P) be a complete probability space
and M (Ω,A;K (R
d
)) denote the family of A-
measurable set-valued random variables. A set-
valued random variable F ∈ M (Ω,A; K (R
d
)) is said
to be L
p
-integrally bounded, p > 1, if there ex-
ists h ∈ L
p
(Ω,A,P;R) such that kak 6 h(ω) for
any a and ω with a ∈ F(ω). Let us denote
L
p
(Ω,A,P;K (R
d
)) :=

F ∈ M (Ω,A;K (R
d
)) :
ω 7→ d
H
(F(ω),{0}) is in L
p
(Ω,A,P;R)
o
.
A fuzzy set u in R
d
is characterized by its mem-
bership function (denoted by u again) u: R
d
→ [0,1]
and u(x) (for each x ∈ R
d
) is interpreted as the de-
gree of membership of x in the fuzzy set u. For
fuzzy set u : R
d
→ [0,1] one defines so-called α-
levels [u]
α
:= {a ∈ R
d
: u(a) > α} for α ∈ (0,1] and
[u]
0
:= cl{a ∈ R
d
: u(a) > 0}. Let F (R
d
) denote a set
of fuzzy sets u : R
d
→ [0,1] such that [u]
α
∈ K (R
d
)
for every α ∈ [0,1] and the mapping α 7→ [u]
α
is d
H
-
continuous on [0, 1]. By hri we mean the characteris-
tic function of the singleton {r}, r ∈ R
d
. Obviously,
hri ∈ F (R
d
). The addition u ⊕ v and scalar multi-
plication λ  u in F (R
d
) can be defined levelwise,
i.e. [u ⊕ v]
α
= [u]
α
+ [v]
α
, [λ  u]
α
= λ[u]
α
, where
u,v ∈ F (R
d
), λ ∈ R and α ∈ [0,1]. If for u,v ∈ F (R
d
)
there exists w ∈ F (R
d
) such that u = v ⊕ w then w is
said to be the fuzzy Hukuhara difference of u and v
and we denote it by u  v. In F (R
d
) we consider the
metric d
∞
(u,v) := sup
α∈[0,1]
d
H
([u]
α
,[v]
α
).
An x : Ω → F (R
d
) is called a fuzzy random
variable, if [x]
α
: Ω → K (R
d
) is a random set for
all α ∈ [0,1]. It is known that in the frame-
work considered here, this definition is equivalent
to A|B
d
∞
-measurability of x : Ω → F (R
d
), see (Joo
et al., 2006). A fuzzy random variable x: Ω →
F (R
d
) is said to be L
p
-integrably bounded, p ≥
1, if [x]
0
belongs to L
p
(Ω,A,P;K (R
d
)). By
L
p
(Ω,A,P;F (R
d
)) we denote the set of the all
L
p
-integrably bounded fuzzy random variables. In
the set L
2
(Ω,A,P;F (R
d
)) one can define a met-
ric ρ by ρ(x, y) :=

Ed
2
∞
(x,y)

1/2
. Then the met-
ric space

L
2
(Ω,A,P;F (R
d
)),ρ

is complete, see
(Feng, 1999).
Denote I := [0,T ]. We equip the probability space
with a filtration {A
t
}
t∈I
satisfying the usual hypothe-
ses. An x : I × Ω → F (R
d
) is called the fuzzy
stochastic process, if for every t ∈ I the mapping
x(t,·): Ω → F (R
d
) is a fuzzy random variable. It
is d
∞
-continuous, if almost all (with respect to the
probability measure P) its trajectories, i.e. the map-
pings x(·,ω): I → F (R
d
) are d
∞
-continuous func-
tions. A fuzzy stochastic process x is said to be
nonanticipating, if for every α ∈ [0,1] the mapping
[x(·,·)]
α
is measurable with respect to the σ-algebra
N , which is defined as follows N := {A ∈ B(I) ⊗ A :
A
t
∈ A
t
for every t ∈ I}, where A
t
= {ω : (t,ω) ∈ A}.
Let p ≥ 1 and L
p
(I × Ω, N ;R
d
) denote the set of all
nonanticipating stochastic processes h : I × Ω → R
d
such that E
R
I
kh(s)k
p
ds < ∞. A fuzzy stochastic
process x is called L
p
-integrably bounded (p ≥ 1),
if there exists a real-valued stochastic process h ∈
L
p
(I × Ω,N ;R) such that d
∞
(x(t,ω),h0i) ≤ h(t,ω)
for a.a. (t,ω) ∈ I × Ω. By L
p
(I × Ω,N ;F (R
d
)) we
denote the set of nonanticipating and L
p
-integrably
bounded fuzzy stochastic processes. For convenience,
from now on, the phrase “with P.1” stands for “with
probability one”. Also we will write x
P.1
= y instead
of P

x = y

= 1, where x, y are random elements.
Also we will write x(t)
I P.1
= y(t) instead of P

x(t) =
y(t) ∀ t ∈ I

= 1, where x, y are the stochastic pro-
cesses.
For τ,t ∈ I, τ < t, and x ∈ L
1
(I × Ω,N ; F (R
d
))
we can define, see (Malinowski, 2012a; Malinowski,
2012c), the fuzzy stochastic Lebesgue–Aumann inte-
gral Ω 3 ω 7→
R
t
τ
x(s,ω)ds ∈ F (R
d
) which is a fuzzy
random variable.
Lemma 2.1. Let p > 1. If x,y ∈ L
p
(I ×
Ω,N ;F (R
d
)) then
(i) I × Ω 3 (t,ω) 7→
R
t
0
x(s,ω)ds ∈ F (R
d
) belongs to
L
p
(I × Ω,N ;F (R
d
)),
(ii) the fuzzy process (t, ω) 7→
R
t
0
x(s,ω)ds is d
∞
-
continuous,
(iii)
sup
u∈[0,t]
d
p
∞

Z
u
0
x(s)ds,
Z
u
0
y(s)ds

I P.1
6 t
p−1
Z
t
0
d
p
∞

x(s),y(s)

ds,
(iv) for every t ∈ I it holds
Esup
u∈[0,t]
d
p
∞

Z
u
0
x(s)ds,
Z
u
0
y(s)ds

6 t
p−1
E
Z
t
0
d
p
∞

x(s),y(s)

ds.
As we mentioned e.g. in (Malinowski, 2012c; Ma-
linowski, 2013c) it is not possible to define fuzzy
stochastic integral of It
ˆ
o type such the integral in such
FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications
110

a fashion that it is not a crisp random variable. Hence,
we consider the diffusion part of the fuzzy stochastic
differential equation as the crisp stochastic It
ˆ
o integral
whose values are embedded into F (R
d
).
For convenience of the reader we give also formu-
lation of the Bihari inequality that we will useful in
the paper.
Lemma 2.2. (Bihari’s inequality, see e.g. Theorem
1.8.2 in (Mao, 2007)). Let T > 0 and c > 0. Let
κ: R
+
→ R
+
be a continuous nondecreasing function
such that κ(t) > 0 for every t > 0. Let u(·) be a Borel
measurable bounded nonnegative function on [0, T ],
and let v(·) be a nonnegative integrable function on
[0,T ]. If u(t) 6 c +
R
t
0
v(s)κ(u(s))ds for every t ∈
[0,T ], then u(t) 6 J
−1

J(c) +
R
t
0
v(s)ds

holds for
all such t ∈ [0,T ] that J(c) +
R
t
0
v(s)ds ∈ Dom(J
−1
),
where J(r) =
R
r
1
ds
κ(s)
, r > 0, and J
−1
is the inverse
function of J. Moreover, if c = 0 and
R
0+
ds
κ(s)
= ∞
then u(t) = 0 for every t ∈ [0,T ].
3 MAIN RESULTS
In (Malinowski, 2016a) we introduced a new type of
fuzzy stochastic differential equations (1.1) which is
by far the most general one. More precisely, we con-
sidered an expanded integral form of these equations
x(t) ⊕ (−1) 
Z
t
0
f (s,x(s))ds (3.1)
⊕
D
m
∑
i=1
Z
t
0
(−1)g
i
(s,x(s))dB
i
(s)
E
I P.1
= x
0
⊕
Z
t
0
˜
f (s,x(s))ds ⊕
D
n
∑
j=1
Z
t
0
˜g
j
(s,x(s))d
˜
B
j
(s)
E
,
where f ,
˜
f : I × Ω × F (R
d
) → F (R
d
), g : I × Ω ×
F (R
d
) → R
d
× R
m
, ˜g : I × Ω × F (R
d
) → R
d
×
R
n
, x
0
: Ω → F (R
d
) is a fuzzy random variable,
and B
1
,B
2
,...,B
m
,
˜
B
1
,
˜
B
2
,...,
˜
B
n
are the independent,
one-dimensional {A
t
}
t∈I
-Brownian motions.
It has been noticed that without loss of generality
one can study an equivalent form of (3.1), i.e.
x(t)
I P.1
=
h
x
0
⊕
Z
t
0
˜
f (s,x(s))ds

(3.2)


(−1) 
Z
t
0
f (s,x(s))ds
i
⊕
D
`
∑
k=1
Z
t
0
h
k
(s,x(s))dW
k
(s)
E
,
where ` = m + n, h
1
,h
2
,...,h
`
: I × Ω × F (R
d
) →
R
d
and W
1
,W
2
,...,W
`
are the independent one-
dimensional {A
t
}
t∈I
-Brownian motions, x
0
is a fuzzy
random variable. The existence of solutions to such
the equations is a fundamental issue. Below we ex-
plain what we mean by a solution to (3.2). Let
˜
T ∈
(0,T ],
˜
I = [0,
˜
T ].
Definition 3.1. A fuzzy stochastic process x :
˜
I ×Ω →
F (R
d
) is said to be the solution to (3.2) if it satisfies:
(i) x ∈ L
2
(
˜
I × Ω,N ;F (R
d
)), (ii) x is d
∞
-continuous,
(iii) it holds (3.2). If
˜
T < T then x is called a local
solution, and if
˜
T = T , then x is called the global
solution. A solution x :
˜
I × Ω → F (R
d
) to equa-
tion (3.2) is said to be unique, if x(t)
˜
I P.1
= y(t), where
y:
˜
I ×Ω → F (R
d
) is any other local solution to (3.2).
In what follows we begin our study with a first
and most important issue of existence and unique-
ness of solutions to (3.2). In the paper we require
that x
0
: Ω → F (R
d
), f ,
˜
f : I ×Ω×F (R
d
) → F (R
d
),
h
k
: I × Ω × F (R
d
) → R
d
(k = 1,2,...,`) satisfy:
(A0) x
0
∈ L
2
(Ω,A
0
,P;F (R
d
)),
(A1) the mappings f ,
˜
f : (I × Ω) × F (R
d
) →
F (R
d
) are N ⊗ B
d
∞
|B
d
∞
-measurable and
h
1
,h
2
,...,h
`
: (I × Ω) × F (R
d
) → R
d
are
N ⊗ B
d
∞
|B(R
d
)-measurable,
(A2) P-a.a. it holds
d
2
∞

f (t,ω,u), f (t,ω, v)

6 ξ(d
2
∞
(u,v)),
d
2
∞

˜
f (t,ω,u),
˜
f (t,ω,v)

6 ξ(d
2
∞
(u,v)),
for every t ∈ I and for any u, v ∈ F (R
d
), and
kh
k
(t, ω, u) − h
k
(t, ω, v)k
2
6 ξ(d
2
∞
(u,v)),
for every t ∈ I, for any u,v ∈ F (R
d
) and k =
1,2,...,`, where ξ: R
+
→ R
+
is a continuous,
concave, nondecreasing function such that ξ(0) =
0, ξ(u) > 0 for u > 0, and
R
0
+
du
ξ(u)
= ∞,
(A3) there exists a constant C > 0 such that P-a.a. it
holds: for every t ∈ I
d
2
∞

f (t,ω,h0i),h0i

∧ d
2
∞

˜
f (t,ω,h0i),h0i

6 C,
and for every t ∈ I and k = 1,2,...,`
kh
k
(t, ω, h0i)k
2
6 C,
(A4) there exists a constant
˜
T ∈ (0,T ] such that P-a.e.
the fuzzy Hukuhara differences
Z
t
τ
˜
f (s,ω,x(s,ω))ds 
Z
t
τ
f (s,ω,x(s,ω))ds
do exist, for every τ,t ∈
˜
I = [0,
˜
T ] and for every
d
∞
-continuous x ∈ L
2
(
˜
I × Ω,N ;F (R
d
)).
The condition (A2) is much more general than the
Lipschitz condition used in (Malinowski, 2016a). The
conditions (A0)-(A4) assure existence of a unique so-
lution to (3.2). This fact constitutes a main result of
the paper.
On Bipartite Fuzzy Stochastic Differential Equations
111

Theorem 3.2. Let x
0
: Ω → F (R
d
), f ,
˜
f : (I × Ω) ×
F (R
d
) → F (R
d
) and h
k
: (I × Ω) × F (R
d
) → R
d
(k = 1,2,. . . , `) satisfy (A0)-(A4). Then (3.2) pos-
sesses a unique solution x :
ˆ
I × Ω → F (R
d
).
To prove this result we will use a sequence of succes-
sive approximations {y
n
}
n∈N
defined as follows:
y
n
(t)
[−1,0] P.1
= x
0
,
y
n
(t)
˜
I P.1
=
h
x
0
⊕
Z
t
0
˜
f (s,y
n
(s −
1
n
))ds



(−1) 
Z
t
0
f (s,y
n
(s −
1
n
))ds
i
⊕
D
`
∑
k=1
Z
t
0
h
k
(s,y
n
(s −
1
n
))dW
k
(s)
E
.
Remark 3.3. Let x
0
, f ,
˜
f ,h
k
satisfy (A0)-(A4). Then
y
n
:
˜
I × Ω → F (R
d
) are d
∞
-continuous, nonanticipat-
ing fuzzy stochastic processes that belong to L
2
(
˜
I ×
Ω,N ; F (R
d
)).
It is intended to apply sequence {y
n
} to approach a
solution to (3.2). However firstly we need to ob-
serve some useful properties of the approximations
y
n
. They will be used later on. Below we state, as
a first observation, that {y
n
} is a bounded sequence.
Lemma 3.4. Let x
0
, f ,
˜
f ,h
k
satisfy (A0)-(A4). Then
there exists a positive constant C
1
such that for every
n ∈ N
Esup
t∈
˜
I
d
2
∞
(y
n
(t), h0i) 6 C
1
.
Lemma 3.5. Let the assumptions of Lemma 3.4 be
satisfied. Then there exists a positive constant C
2
such
that for every n ∈ N and every τ,t ∈
˜
I, τ 6 t
Ed
2
∞
(y
n
(t), y
n
(τ)) 6 C
2
(t − τ).
Lemma 3.6. Let the assumptions of Lemma 3.4 be
satisfied. Then
Esup
t∈
˜
I
d
2
∞
(y
n
(t), y
i
(τ)) −→ 0, as n,i → ∞.
Proof. Let us fix n,i ∈ N. Without loss of generality
we may assume that n > i. Observe that for t ∈
˜
I we
have, using the property d
∞
(u v, wz) 6 d
∞
(u,w)+
d
∞
(v,z) together with Lemma 2.1 and Doob’s inequal-
ity,
Esup
u∈[0,t]
d
2
∞
(y
n
(u),y
i
(u))
6 8tE
Z
t
0
d
2
∞

˜
f (s,y
n
(s −
1
n
)),
˜
f (s,y
i
(s −
1
n
))

+ 8tE
Z
t
0
d
2
∞

˜
f (s,y
i
(s −
1
n
)),
˜
f (s,y
i
(s −
1
i
))

+ 8tE
Z
t
0
d
2
∞

f (s,y
n
(s −
1
n
)), f (s,y
i
(s −
1
n
))

+ 8tE
Z
t
0
d
2
∞

f (s,y
i
(s −
1
n
)), f (s,y
i
(s −
1
i
))

+ 16`
`
∑
k=1
E
Z
t
0



h
k
(s,y
n
(s −
1
n
)) − h
k
(s,y
i
(s −
1
n
))



2
+ 16`
`
∑
k=1
E
Z
t
0



h
k
(s,y
i
(s −
1
n
)) − h
k
(s,y
i
(s −
1
i
))



2
,
Assumption (A2) lead us to
Esup
u∈[0,t]
d
2
∞
(y
n
(u),y
i
(u))
6 16(t + `
2
)
Z
t
0
ξ

Esup
u∈[0,s]
d
2
∞
(y
n
(u),y
i
(u))

ds
+ 16(t + `
2
)
Z
t
0
ξ

Ed
2
∞
(y
i
(s −
1
n
),y
i
(s −
1
i
))

ds.
By Lemma 3.5 we get
Esup
u∈[0,t]
d
2
∞
(y
n
(u),y
i
(u))
6 C
3
Z
t
0
ξ

Esup
u∈[0,s]
d
2
∞
(y
n
(u),y
i
(u))

ds
+ C
3
˜
T ξ

C
2
(
1
i
−
1
n
)

,
where C
3
= 16(
˜
T + `
2
). Application of Lemma 2.2
yields
Esup
u∈[0,t]
d
2
∞
(y
n
(u),y
i
(u))
6 J
−1
(J(C
3
˜
T ξ

C
2
(
1
i
−
1
n
)

+C
3
t)
for every t ∈
˜
I. Owing to Lemma 2.2 and properties
of function J from this lemma, we obtain
lim
n,i→∞
J
−1
(J(C
3
˜
T ξ

C
2
(
1
i
−
1
n
)

+C
3
˜
T ) = 0.
This allows us to infer that
lim
n,i→∞
Esup
t∈
˜
I
d
2
∞
(y
n
(t), y
i
(t)) = 0.

Now, we present a scheme of the proof of our main
result.
Proof of Theorem 3.2. Due to Lemma 3.6 we have
lim
n,i→∞
ρ
2
(y
n
(t), y
i
(t)) = 0 for every t ∈
˜
I,
FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications
112

where ρ(x,y) = [Ed
2
∞
(x,y)]
1/2
is a metric in
L
2
(Ω,A
t
,P;F (R
d
)). Since (L
2
(Ω,A
t
,P;F (R
d
)),ρ)
is a complete metric space we infer that for every
t ∈
˜
I there exists a unique fuzzy random variable x
t
∈
L
2
(Ω,A
t
,P;F (R
d
)) such that lim
n→∞
ρ(y
n
(t), x
t
) =
0. Let us define x :
˜
I ×Ω → F (R
d
) as x(t,ω) = x
t
(ω).
Then the fuzzy stochastic process x is {A
t
}-adapted.
Due to the Markov inequality we obtain that for every
ε > 0
lim
n,i→∞
P(sup
t∈
˜
I
d
∞
(y
n
(t), y
i
(t)) > ε) = 0.
Hence we can infer that there exists a subsequence
{y
n
`
(·,·)} of the sequence {y
n
(·,·)} such that
lim
`→∞
sup
t∈
˜
I
d
∞
(y
n
`
(t), x(t))
P.1
= 0.
Thus the process x is d
∞
-continuous and con-
sequently it is measurable. Since x is also
{A
t
}-adapted, it is nonanticipating. Also, since
x(t) ∈ L
2
(Ω,A
t
,P;F (R
d
)) for every t ∈
˜
I, we
have E
R
˜
I
d
2
∞
(x(t),h0i)dt 6
˜
T sup
t∈
˜
I
Ed
2
∞
(x(t),h0i) <
∞. This implies that x ∈ L
2
(
˜
I × Ω,N ; F (R
d
)).
Moreover, applying Lemma 3.4, we infer that
Esup
t∈
˜
I
d
2
∞
(x(t),h0i) 6 C
1
. We can also infer that
lim
`→∞
Esup
t∈
˜
I
d
2
∞
(y
n
`
(t), x(t)) = 0. (3.3)
In what follows we shall show that x is a solution to
(3.2). To this aim, let us observe that
Esup
u∈
˜
I
d
2
∞

x(u),
h
x
0
⊕
Z
t
0
˜
f (s,x(s))ds



(−1) 
Z
t
0
f (s,x(s))ds
i
⊕
D
`
∑
k=1
Z
t
0
h
k
(s,x(s))dW
k
(s)
E
6 2Q
`
+ 2P
`
,
where
Q
`
= Esup
u∈
˜
I
d
2
∞

y
n
`
(u),x(u)

and
P
`
= Esup
u∈
˜
I
d
2
∞
h
x
0
⊕
Z
t
0
˜
f (s,y
n
`
(s −
1
n
`
))ds



(−1) 
Z
t
0
f (s,y
n
`
(s −
1
n
`
))ds
i
⊕
D
`
∑
k=1
Z
t
0
h
k
(s,y
n
`
(s −
1
n
`
))dW
k
(s)
E
,
h
x
0
⊕
Z
t
0
˜
f (s,x(s))ds



(−1) 
Z
t
0
f (s,x(s))ds
i
⊕
D
`
∑
k=1
Z
t
0
h
k
(s,x(s))dW
k
(s)
E
.
By (3.3) the expression Q
`
converges to zero as ` goes
to infinity, and it can be verified that
P
`
6 C
4
E
Z
˜
T
0
ξ

d
2
∞
(y
n
`
(s −
1
n
`
),y
n
`
(s))

ds
+C
4
E
Z
˜
T
0
ξ

d
2
∞
(y
n
`
(s),x(s))

ds,
where C
4
= 16(
˜
T + 1). Thus
P
`
6 C
4
Z
˜
T
0
ξ

Ed
2
∞
(y
n
`
(s −
1
n
`
),y
n
`
(s))

ds
+C
4
Z
˜
T
0
ξ

Esup
u∈
˜
I
d
2
∞
(y
n
`
(u),x(u))

ds.
Applying Lemma 3.5 we obtain
P
`
6 C
4
˜
T ξ

C
2
n
`

+C
4
˜
T ξ

Esup
u∈
˜
I
d
2
∞
(y
n
`
(u),x(u))

.
By properties of ξ and in view of (3.3) the right-hand
side of the latter inequality converges to zero. Thus
Esup
u∈
˜
I
d
2
∞

x(u),
h
x
0
⊕
Z
t
0
˜
f (s,x(s))ds



(−1) 
Z
t
0
f (s,x(s))ds
i
⊕
D
`
∑
k=1
Z
t
0
h
k
(s,x(s))dW
k
(s)
E
= 0
which implies that
sup
u∈
˜
I
d
∞

x(u),
h
x
0
⊕
Z
t
0
˜
f (s,x(s))ds



(−1) 
Z
t
0
f (s,x(s))ds
i
⊕
D
`
∑
k=1
Z
t
0
h
k
(s,x(s))dW
k
(s)
E
P.1
= 0.
This shows that x is a solution to (3.2). Now we notice
that x is a unique solution. Indeed, assume that y :
˜
I ×
Ω → F (R
d
) is another solution to (3.2). Then for
t ∈
˜
I we have
Esup
u∈[0,t]
d
2
∞

x(u),y(u)

6 4tE
Z
t
0
d
2
∞
(
˜
f (s,x(s)),
˜
f (s,y(s)))ds
+ 4tE
Z
t
0
d
2
∞
( f (s, x(s)), f (s,y(s)))ds
+ 8`
`
∑
k=1
E
Z
t
0



h
k
(s,x(s)) − h
k
(s,y(s))



2
ds.
Hence
Esup
u∈[0,t]
d
2
∞

x(u),y(u)

6 8(
˜
T + `
2
)
Z
t
0
ξ

Esup
u∈[0,s]
d
2
∞
(x(s),y(s))

ds.
On Bipartite Fuzzy Stochastic Differential Equations
113

Invoking Lemma 2.2, we get
Esup
u∈[0,t]
d
2
∞

x(u),y(u)

6 0 for every t ∈
˜
I.
Therefore Esup
u∈
˜
I
d
2
∞

x(u),y(u)

= 0, which implies
that sup
u∈
˜
I
d
∞

x(u),y(u)

P.1
= 0. This proves unique-
ness of the solution x. The proof is completed. 
4 CONCLUSION
In the paper we consider bipartite fuzzy stochastic dif-
ferential equations. A main result treat on existence
of a unique solution to such the equations in the case
when coefficients satisfy a generalized Lipschitz con-
dition. A continuous dependence of the solution with
respect to initial value and drift and diffusion coeffi-
cients can be investigated in a future research.
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