Bipartite Fuzzy Stochastic Differential Equations with Global Lipschitz Condition

We introduce and analyze a new type of fuzzy stochastic differential equations. We consider equations with drift and diffusion terms occurring at both sides of equations. Therefore we call them the bipartite fuzzy stochastic differential equations. Under the Lipschitz and boundedness conditions imposed on drifts and diffusions coefficients we prove existence of a unique solution.Then, insensitivity of the solution under small changes of data of equation is examined. Finally, wemention that all results can be repeated for solutions to bipartite set-valued stochastic differential equations.


Introduction
Stochastic differential equations are often used in modelling dynamics of uncertain physical systems, where it is assumed that randomness and stochastic noises have an influence on a considered system.The theory of such equations involving stochastic integrals is well established (see, e.g., [1][2][3]).On the other hand in modelling of many real-world processes there appears uncertainty of different kind than randomness, namely, trying to describe a physical system one encounters, for instance, an imprecision of measurement equipment, imperfect human judgments, and opinions on parameters of such system.These are also symptoms of uncertainty but they do not locate in randomness or stochastic noises.This uncertainty is well treated by fuzzy set theory (c.f.[4][5][6]).Owing to this theory it is possible to handle mathematically such linguistic opinions, for example, "low pressure," "high temperature," and "about 7%."A usage of fuzzy sets gives ability to study deterministic fuzzy differential equations in modelling various phenomena which include imprecision [7][8][9][10].Moreover, some successful attempts of combining two kinds of uncertainties, that is, randomness and fuzziness, were undertaken for petroleum contamination [11], optimal tracking design of stochastic fuzzy systems [12], random fuzzy differential equations [13][14][15][16], stochastic fuzzy neural networks [17,18], civil engineering and mechanics [19], Markov chains with fuzzy states [20], fuzzy martingales [21], Petri nets [22], optimization [23], ballast water management [24], filtering of fuzzy stochastic systems [25], and fuzzy stochastic differential equations [26][27][28][29][30].
The latter topic on fuzzy stochastic differential equations is quite new and still developed.In papers [26][27][28] we considered such equations in their natural integral form generalizing one of the crisp stochastic differential equations, that is, where  is a random fuzzy set-valued drift coefficient,  is a random single-valued diffusion coefficient, and  0 is a fuzzy random variable.We investigated the problem of existence of a unique solution, since it is almost impossible to find explicit forms of solutions to such equations.This is very similar to the theory of crisp stochastic differential equations.However, unlike crisp equations, fuzzy set-valued equations exhibit new qualitative properties of their solutions.Namely, we mean here nondecreasing (in time) diameter of solution's values, which determines that uncertainty located in fuzziness cannot decrease as time increases.This could be an obstacle in some concrete situations, when an expert knows that fuzziness should be decreasing in his system.Therefore, in the works [29,30], we proposed to study fuzzy stochastic differential equations in integral form (2) If one would consider this equation in the crisp setting, then it would be no difference from previous equation.However, these two equations are not equivalent in fuzzy environment.Solutions of the second equation have nonincreasing fuzziness of their values.This property does not refer to solutions of crisp equations.Although some potential applications of fuzzy stochastic differential equations in finance, biology, control systems, and physics were studied, for example, in [26,28,30], there is still a need of a further development in this area to know better nature of these equations and properties of their solutions.
In this paper we propose to join two equations mentioned above in a one equation This way we introduce a new kind of fuzzy stochastic differential equations which are more general than those studied in our earlier works and mentioned above.Due to the new form of equations with integrals at both sides they will be called the bipartite fuzzy stochastic differential equations.Solutions to such equations may lose property of monotonicity of fuzziness.However, this can be an advantage, since it can allow for future examinations of periodic solutions.In current paper we initiate investigations of the bipartite fuzzy stochastic differential equations.Under the Lipschitz and boundedness conditions imposed on the drift and diffusion coefficients, existence of a unique solution is proved.It is also shown that the solution is stable with respect to small changes of equation's data; that is, the solution does not change much when the changes of drift and diffusion coefficients and initial value are small.This shows that the theory introduced in the paper is well-posed.We also indicate that parallel to bipartite fuzzy stochastic differential equations one can consider bipartite set-valued stochastic differential equations and all the results established for the first equations can be easily repeated for the second equations.
The subsequent part of the paper is organized as follows: in Section 2 we collect a prerequisite knowledge on setvalued random variables, set-valued stochastic processes, fuzzy sets, fuzzy random variables, and fuzzy stochastic Lebesgue-Aumann integral.This is done for convenience of the reader.Section 3 is a main part of the paper.The bipartite fuzzy stochastic differential equations are introduced here.We prove existence and uniqueness of solution to such equations and study properties of solutions.

Preliminaries
For a convenience of the reader we set up a framework which we work with.
Let K(R  ) be the set of all nonempty, compact, and convex subsets of R  .This set can be supplied with the Hausdorff metric   which is defined by where ‖ ⋅ ‖ denotes a norm in R  .Then the metric space (K(R  ),   ) is complete and separable (see [31]).Also, the addition and scalar multiplication in K(R  ) are defined as follows: for Let (Ω, A, ) be a complete probability space and M(Ω, A; K(R  )) denote the family of A-measurable setvalued mappings  : Ω → K(R  ) (set-valued random variable) such that A set-valued random variable  ∈ M(Ω, A; K(R  )) is called   -integrally bounded,  ⩾ 1, if there exists ℎ ∈   (Ω, A, ; R) such that ‖‖ ⩽ ℎ() for any  and  with  ∈ ().It is known (see [32]) that  is   -integrally bounded iff   →   ((), {0}) is in   (Ω, A, ; R), where   (Ω, A, ; R) is a space of equivalence classes (with respect to the equality -a.e.) of A-measurable random variables ℎ : The set-valued random variables ,  ∈ L  (Ω, A, ; K(R  )) are considered to be identical, if  =  holds -a.e.
Let  > 0, and denote  fl [0, ].Let the system (Ω, A, {A  } ∈ , ) be a complete, filtered probability space with a filtration {A  } ∈ satisfying the usual hypotheses; that is, {A  } ∈ is an increasing and right continuous family of sub--algebras of A, and A 0 contains all -null sets.We call  :  × Ω → K(R  ) a set-valued stochastic process, if for every  ∈  a mapping () : Ω → K(R  ) is a setvalued random variable.We say that a set-valued stochastic process  is   -continuous, if almost all (with respect to the probability measure ) its paths, that is, the mappings (⋅, ) :  → K(R  ), are the   -continuous functions.A set-valued stochastic process  is said to be {A  } ∈ -adapted, if for every  ∈  the set-valued random variable () : ) is a B()⊗A-measurable set-valued random variable, where B() denotes the Borel -algebra of subsets of .If  :  × Ω → K(R  ) is {A  } ∈ -adapted and measurable, then it will be called nonanticipating.Equivalently,  is nonanticipating iff  is measurable with respect to the algebra N which is defined as follows: where A fuzzy set  in R  (see [4]) is characterized by its membership function (denoted by  again)  : R  → [0, 1] and () (for each  ∈ R  ) is interpreted as the degree of membership of  in the fuzzy set .As the value () expresses "degree of membership of  in" or a "degree of satisfying by  a property," one can work with imprecise information.Obviously, every ordinary set  in R  is a fuzzy set, since then () = 1 if  ∈  and () = 0 if  ∉ .
Let , V ∈ F(R  ).If there exists  ∈ F(R  ) such that  = V +  then we call  the Hukuhara difference of  and V and we denote it by ⊖V.Note that ⊖V ̸ = +(−1)V.Also ⊖V may not exist, but if it exists it is unique.For , V ∈ F(R  ) and  1 ,  2 ∈ R  we have the following: The mapping  ∞ is a metric in F(R  ).It is known that (F(R  ),  ∞ ) is a complete metric space, but it is not separable and it is not locally compact.For every , V, ,  ∈ F(R  ),  ∈ R, one has (see, e.g., [14,34]) A mapping  : Ω → F(R  ) is said to be a fuzzy random variable (see [34] where   denotes the Skorohod metric in F(R  ) and B   denotes the -algebra generated by the topology induced by   .A fuzzy random variable  : Ω → F(R  ) is said to be   -integrally bounded,  ⩾ 1, if [] 0 belongs to L  (Ω, A, ; K(R  )).By L  (Ω, A, ; F(R  )) we denote the set of all   -integrally bounded fuzzy random variables, where we consider ,  ∈ L  (Ω, A, ; F(R  )) as identical if  =  holds -a.e.
We call  :  × Ω → F(R  ) a fuzzy stochastic process, if for every  ∈  the mapping (, ⋅) : Ω → F(R  ) is a fuzzy random variable.We say that a fuzzy stochastic process  is  ∞ -continuous, if almost all (with respect to the probability measure ) its trajectories, that is, the mappings we denote the set of nonanticipating and   -integrally bounded fuzzy stochastic processes.
In the whole paper, notation  .1 =  stands for abbreviation of ( = ) = 1, where ,  are some random elements.Also we will write () where ,  are some stochastic processes.Similar notations will be used for inequalities.
In this paper we establish a new kind of fuzzy stochastic differential equations ( 14) by joining (15) and (16).Therefore ( 14) is called the bipartite fuzzy stochastic differential equation.The solutions  to (14) can lose property that the mappings   → diam([(, )]  ) are monotone.Indeed, the fuzzy stochastic Lebesgue-Aumann integral on the left-hand side of ( 14) is an item which affects monotonicity of functions   → diam([(, )]  ).It makes them nonincreasing ones, but simultaneously the fuzzy stochastic Lebesgue-Aumann integral on the right-hand side of ( 14) forces that the functions   → diam([(, )]  ) do not decrease.However, the loss of monotonicity could be an advantage in the future, since it could open a gate for future studies of periodic solutions to fuzzy stochastic differential equations.
Below we write what we mean by a solution to bipartite fuzzy stochastic differential equation.Let T ∈ (0, ], Ĩ = [0, T].

Definition 2. Let a fuzzy stochastic process
and (iii) it holds (18).If T < , then  is said to be the local solution to bipartite fuzzy stochastic differential equation (18), and if T = , then  is called the global solution to (18).A local solution  : Ĩ × Ω → F(R  ) to ( 18) is said to be unique, if () = (), where  : Ĩ × Ω → F(R  ) is any other local solution to (18).The uniqueness of the global solution to ( 18) is defined similarly.
Since existence of Hukuhara differences in (18) depends on  0 , existence of solution to (18) cannot be independent of  0 .This fact differs bipartite fuzzy differential equations from crisp stochastic differential equations.
In what follows we begin our study with a first and most important issue of existence and uniqueness of solutions to (18).In the paper we require that  0 : Ω → F(R  ), , f : A2) there exists a constant  > 0 such that for  × -a.a.
is well defined; that is, in particular, the Hukuhara differences appearing above do exist.
A preliminary result is on the sequence {  } and it shows that {  } is uniformly bounded.As we intend to use {  } as the sequence of approximate solutions, we will be able to infer later on the fact that the exact solution is bounded as well.
We shall show that  is a solution to (18).Indeed, let us notice that  ( This shows that  : Ĩ × Ω → F(R  ) is a solution (possibly a local solution) to (18).What is left is to prove that the solution  is unique.Let us assume that ,  : Ĩ × Ω → F(R  ) are two solutions to the bipartite fuzzy stochastic differential equation (18).Denote () = E sup ∈[0,]  2 ∞ ((), ()) for  ∈ Ĩ.Let us notice that for every  ∈ Ĩ we have  = 0.This ends the proof.
As we mentioned earlier the sequence {  } can be treated as a sequence of approximate solutions.The next result presents an upper bound for the error of th approximation   .Proposition 6. Assume that (A0)-(A4) hold for  0 , , f, and ℎ  's.Then, for the approximations   defined in (22) and the exact solution  to the bipartite fuzzy stochastic differential equation (18) we have where  3 is like in (31).
Applying ( 31) we arrive at Now invoking Gronwall's inequality we can write for every  ∈ Ĩ. ( This leads us to the inequality ( T) ⩽ (2 As an immediate consequence of the assertion presented above, we have the following property E sup ∈ Ĩ 2 ∞ (  (), ()) → 0 as  → ∞, which, together with Proposition 4, allows us to find a bound for the expression E sup ∈ Ĩ 2 ∞ ((), ⟨0⟩).Another estimation is contained in the next claim.Proposition 7. Assume that conditions (A0)-(A4) are satisfied for  0 , , f, and ℎ  's.Let  : Ĩ × Ω → F(R  ) denote unique (possibly local) solution to (18).Then it holds (44) Further it can be verified that Hence, by Gronwall's inequality we can infer that The next part of this section is focused on well-posedness of the theory of bipartite fuzzy stochastic differential equations.We shall prove that the solutions are insensitive with respect to small changes of the equation's data.We start with insensitivity with respect to initial value  0 .

Application to Bipartite Set-Valued Stochastic Differential Equations
In this part of the paper we present some results concerning bipartite set-valued stochastic differential equations.We do this because set-valued analysis constitutes a branch of research also in context of set-valued differential equations [36].We discuss only main issues without including proofs.This is because the results presented here are parallel to those established in Section 3 for bipartite fuzzy stochastic differential equations.All the inference methods are similar to those contained in preceding section.By the bipartite set-valued stochastic differential equations written in their integral form we mean the following equations: = (), where  : Ĩ×Ω → K(R  ) is any other solution to (61).
is well defined.
Using the sequence {  } defined in (S4) and proceeding similarly like in the proof of Theorem 5 we are able to derive the following result.
The theory of bipartite set-valued stochastic differential equations is well-posed.Below, by stating two corollaries, we indicate that the set-valued solution to (61) possesses properties of continuous dependence on initial set-valued random variable and coefficients , F,   's.

Concluding Remarks
The paper introduces very first study on so-called bipartite fuzzy stochastic differential equations.Solutions of equations considered previously (cf.[26][27][28][29][30]) had a property that their trajectory values (the values are fuzzy sets) had either nondecreasing or nonincreasing diameter in time.Now, owing to new equations examined in this paper, we open a way to consider fuzzy stochastic differential equations with solutions that have trajectories of nonmonotone diameter of their values.Since seeking explicit solutions to such the equations is mostly without success, we provide a study on existence of a unique solution.This is achieved under conditions of Lipschitz coefficients of drift and diffusion.Then we indicate that solution is bounded and insensitive under small changes of coefficients and initial value.This confirms that the theory of the new equations investigated in this paper is well-posed.Finally, we show that all results achieved can be easily applied to bipartite set-valued stochastic differential equations.
The current study can be a starting point for some future investigations.For instance, from now on it is possible to speak on periodic diameter of solutions for fuzzy stochastic differential equations.Hence, a study in this direction would be interesting.Moreover, one can try to use some weaker assumptions (than Lipschitz conditions) imposed on coefficients to get existence of a unique solution; different kinds of stabilities of solutions are also of interest.