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Mathematical Problems in Engineering
Volume 2016, Article ID 3830529, 13 pages
http://dx.doi.org/10.1155/2016/3830529
Research Article

Bipartite Fuzzy Stochastic Differential Equations with Global Lipschitz Condition

Institute of Mathematics, Cracow University of Technology, ul. Warszawska 24, 31-155 Kraków, Poland

Received 27 April 2016; Revised 25 September 2016; Accepted 13 October 2016

Academic Editor: Leonid Shaikhet

Copyright © 2016 Marek T. Malinowski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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, we mention that all results can be repeated for solutions to bipartite set-valued stochastic differential equations.

1. 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., [13]). 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. [46]). 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 [710]. 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 [1316], 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 [2630].

The latter topic on fuzzy stochastic differential equations is quite new and still developed. In papers [2628] 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 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 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 set-valued 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.

2. Preliminaries

For a convenience of the reader we set up a framework which we work with.

Let be the set of all nonempty, compact, and convex subsets of . This set can be supplied with the Hausdorff metric which is defined by where denotes a norm in . Then the metric space is complete and separable (see [31]). Also, the addition and scalar multiplication in are defined as follows: for , ,

Let be a complete probability space and denote the family of -measurable set-valued mappings (set-valued random variable) such that A set-valued random variable is called -integrally bounded, , if there exists such that for any and with . It is known (see [32]) that is -integrally bounded iff is in , where is a space of equivalence classes (with respect to the equality -a.e.) of -measurable random variables such that . Let us denote The set-valued random variables are considered to be identical, if holds -a.e.

Let , and denote . Let the system be a complete, filtered probability space with a filtration satisfying the usual hypotheses; that is, is an increasing and right continuous family of sub--algebras of , and contains all -null sets. We call a set-valued stochastic process, if for every a mapping is a set-valued 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 , are the -continuous functions. A set-valued stochastic process is said to be -adapted, if for every the set-valued random variable is -measurable. It is called measurable, if is a -measurable set-valued random variable, where denotes the Borel -algebra of subsets of . If is -adapted and measurable, then it will be called nonanticipating. Equivalently, is nonanticipating iff is measurable with respect to the -algebra which is defined as follows: where . A set-valued nonanticipating stochastic process is called -integrally bounded, if there exists a measurable stochastic process such that and for a.a. . By we denote the set of all equivalence classes (with respect to the equality -a.e., denotes the Lebesgue measure) of nonanticipating and -integrally bounded set-valued stochastic processes.

A fuzzy set in (see [4]) is characterized by its membership function (denoted by again) and (for each ) 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 is a fuzzy set, since then if and if .

Let denote the fuzzy sets such that for every , where for and . Note that the set can be embedded into by the embedding defined as follows: for we have if , and if .

Addition and scalar multiplication in fuzzy set space can be defined levelwise (see [33]): where , , and .

Let . If there exists such that then we call the Hukuhara difference of and and we denote it by . Note that . Also may not exist, but if it exists it is unique. For and we have the following:(P1),(P2)the Hukuhara difference exists iff exists. Moreover, .

Define by the expression The mapping is a metric in . It is known that is a complete metric space, but it is not separable and it is not locally compact. For every , , one has (see, e.g., [14, 34]) (P3),(P4),(P5),(P6),(P7),(P8).

A mapping is said to be a fuzzy random variable (see [34]), if is an -measurable set-valued random variable for all . It is known from [35] that is the fuzzy random variable iff is -measurable, where denotes the Skorohod metric in and denotes the -algebra generated by the topology induced by . A fuzzy random variable is said to be -integrally bounded, , if belongs to . By we denote the set of all -integrally bounded fuzzy random variables, where we consider as identical if holds -a.e.

We call a fuzzy stochastic process, if for every the mapping 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 , are the -continuous functions. A fuzzy stochastic process is called -adapted, if for every the multifunction is -measurable for all . It is called measurable, if is a -measurable multifunction for all , where denotes the Borel -algebra of subsets of . If is -adapted and measurable, then it is called nonanticipating. Equivalently, is nonanticipating iff for every the set-valued random variable is measurable with respect to the -algebra . A fuzzy stochastic process is called -integrally bounded (), if there exists a measurable stochastic process such that and for a.a. . By we denote the set of nonanticipating and -integrally bounded fuzzy stochastic processes.

In the whole paper, notation stands for abbreviation of , where are some random elements. Also we will write instead of , where , are some stochastic processes. Similar notations will be used for inequalities.

Let , . For such process we can define (see, e.g., [26]) the fuzzy stochastic Lebesgue-Aumann integral which is a fuzzy random variable Then (from now on we do not write the argument ) is understood as . For the fuzzy stochastic Lebesgue-Aumann integral we have the following properties (see [26]).

Proposition 1. Let . If then (i) belongs to ,(ii)the fuzzy stochastic process is -continuous,(iii),(iv)for every it holds .

3. Main Results

Let denote an -dimensional -Brownian motion defined on , . The process is defined as follows: , where are the independent, one-dimensional -Brownian motions, and the symbol denotes transposition. Similarly stays for an -dimensional Brownian motion which is assumed to be independent of .

In the paper we make an examination of initial value problem for fuzzy stochastic differential equations of a new form with , , , and being a fuzzy random variable. Note that problem (12) can be reformulated as since we consider -dimensional Brownian motion , -dimensional Brownian motion , , , where for each and .

The way of writing fuzzy stochastic differential equations in differential forms (12) and (13) is symbolic only, because these equations are always considered as integral equations: where the first integrals on both sides are the fuzzy stochastic Lebesgue-Aumann integrals and the remaining integrals are the crisp stochastic Itô integrals.

One can observe that if and for , then (14) takes form Such equations were studied in [2628]. On the other hand if and for , then (14) reduces to and only investigations concerning these equations are presented in [29, 30]. If the data , , are the single-valued and singleton-defined mappings in (15) and (16), then we arrived at the same type of crisp stochastic differential equation. However, in fuzzy case, (15) and (16) are of different type, because fuzzy solutions to fuzzy equations (15) and (16) exhibit different geometric properties. The solutions to (15) have nondecreasing fuzziness in their values as increases; that is, with for every the mappings are nondecreasing (Theorem  3.8 [26]), but for the solutions to (16) the mappings are nonincreasing (Theorem  3.3 [30]).

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 are monotone. Indeed, the fuzzy stochastic Lebesgue-Aumann integral on the left-hand side of (14) is an item which affects monotonicity of functions . It makes them nonincreasing ones, but simultaneously the fuzzy stochastic Lebesgue-Aumann integral on the right-hand side of (14) forces that the functions 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.

Note that, using (P1) and (P2), (14) can be viewed as Hence, without loss of generality, we can consider bipartite fuzzy stochastic differential equations of the following integral form: where , , and are the independent one-dimensional -Brownian motions, and is a fuzzy random variable.

Below we write what we mean by a solution to bipartite fuzzy stochastic differential equation. Let , .

Definition 2. Let a fuzzy stochastic process satisfy the following: (i) , (ii) is -continuous, and (iii) it holds (18). If , then is said to be the local solution to bipartite fuzzy stochastic differential equation (18), and if , then is called the global solution to (18). A local solution to (18) is said to be unique, if , where 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 , existence of solution to (18) cannot be independent of . 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 , , () satisfy the following:(A0),(A1)the mappings are -measurable and are -measurable,(A2)there exists a constant such that for -a.a. and for every it holds (A3)there exists such that for -a.a. it holds: (A4)there exists a constant such that the sequence of the fuzzy mappings described as and for is well defined; that is, in particular, the Hukuhara differences appearing above do exist.

Remark 3. Assume that (A0)–(A4) are satisfied for , , , . Then the mappings described in (A4) are the fuzzy stochastic processes, and they are -continuous and belong to .

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.

Proposition 4. Assume that (A0)–(A4) are satisfied for , , , . Then, for the sequence defined like in (22) we have where and .

Proof. Denote for and . Applying properties (P4), (P6), and (P5) we arrive at Further, Invoking Proposition 1 and the Doob inequality we get By assumptions (A2) and (A3) we obtain Due to the last inequality we can infer that for . Hence for Applying Gronwall’s inequality we arrive at for . This allows us to infer that .

In what follows we formulate an existence and uniqueness theorem for solutions to the bipartite fuzzy stochastic differential equations.

Theorem 5. Assume that conditions (A0)–(A4) are satisfied. Then the bipartite fuzzy stochastic differential equation (18) has a unique local solution.

Proof. Denote for and . Then for applying properties (P4), (P8), and (P5) we can write By Proposition 1, Doob’s inequality, and assumptions (A2) and (A3) we obtain where . Moreover for we obtain . Thus one can infer that for Using Chebyshev’s inequality and (31) we arrive at Since the series is convergent, due to the Borel-Cantelli lemma we obtain Now, similarly like in [26], we infer that there exists a -continuous fuzzy stochastic process such that , as . It can also be verified that , .
We shall show that is a solution to (18). Indeed, let us notice that By Proposition 1 and Itô’s isometry, assumption (A2), and Lebesgue’s Dominated Convergence Theorem we get Hence for every Thus we can infer that for every Now, since the processes are -continuous, we get This shows that is a solution (possibly a local solution) to (18).
What is left is to prove that the solution is unique. Let us assume that are two solutions to the bipartite fuzzy stochastic differential equation (18). Denote for . Let us notice that for every we have Invoking Gronwall’s inequality we get for , which leads to the conclusion that . 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 , , , and ’s. Then, for the approximations defined in (22) and the exact solution to the bipartite fuzzy stochastic differential equation (18) we have where is like in (31).

Proof. Denote for . Notice that Applying (31) we arrive at Now invoking Gronwall’s inequality we can write This leads us to the inequality .

As an immediate consequence of the assertion presented above, we have the following property as , which, together with Proposition 4, allows us to find a bound for the expression . Another estimation is contained in the next claim.

Proposition 7. Assume that conditions (A0)–(A4) are satisfied for , , , and ’s. Let denote unique (possibly local) solution to (18). Then it holds

Proof. Denote for . Observe that 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 .

Let denote solutions to the bipartite fuzzy stochastic differential equation (18) and respectively. The initial values can differ on a set of positive probability and the remaining data are the same.

Theorem 8. Assume that satisfy condition (A0) and , , , , satisfy (A1)–(A3). Suppose that , , , ’s satisfy (A4) and , , , ’s fulfill (A4) and this happens for a common . Then, for the unique local (or global) solutions to the bipartite fuzzy stochastic differential equations (18) and (48) we have

Proof. The unique solutions to (18) and (48) exist owing to Theorem 5. For one gets