On solutions to set-valued and fuzzy stochastic differential equations

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On solutions to set-valued and fuzzy stochastic differential equations Marek T. Malinowskia,n, Ravi P. Agarwalb,c a

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland b Department of Mathematics, Texas A&M University - Kingsville, Kingsville, TX 78363-8202, USA c Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Received 30 March 2014; received in revised form 27 October 2014; accepted 12 November 2014

Abstract We study existence and uniqueness of solutions to nonlinear set-valued stochastic differential equations driven by multidimensional Brownian motion. The conditions imposed on the equation's coefficients are non-Lipschitz. The drift coefficient is set-valued and diffusion coefficient is single-valued, both coefficients are random. The approach used in this paper allows the solutions to be set-valued stochastic processes. The set-valued results are then extended for the parallel studies of nonlinear fuzzy stochastic differential equations with solutions being fuzzy stochastic processes. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Differential equations are mathematical tools useful in describing many nonlinear dynamical phenomena in e.g. physics, engineering, economics, or biology. Modeling with the ordinary differential equations x_ ðtÞ ¼ f ðt; xðtÞÞ is appropriate when the knowledge of the considered system is complete and the structure of the system is precisely described. In particular, all the parameters of the system, its initial values and the functional relationships governing the system's dynamics should n

Corresponding author. E-mail addresses: [email protected], [email protected] (M.T. Malinowski), [email protected] (R.P. Agarwal). http://dx.doi.org/10.1016/j.jfranklin.2014.11.010 0016-0032/& 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

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be given as single values of the states space and hence rid of all imprecision. However, in mathematical modeling one often encounters some level of uncertainty. For example, ordinary differential equations require a precise knowledge on the initial values, yet the measurement equipment used in practice is unlikely to yield such information with the required level of precision: the measurements are given within a given range and hence not presented as a single value but as a set of values. Imprecision can also appear in description of the functional relationships characterizing the considered nonlinear system. The dynamical system can have velocities that are not uniquely determined by its state and that can be described by sets of feasible velocities. Such an approach leads to the application of set-valued mappings in modeling nonlinear dynamics in the presence of uncertainty. Set-valued differential equations X 0 ðtÞ ¼ Fðt; XðtÞÞ, where both X and F are set-valued mappings, are very popular tools used in mathematical modeling of the nonlinear dynamical systems with imprecision or incomplete information, see [1–4]. These equations were formulated in [5] and studied in e.g. [6–21]. Currently this topic of research forms an independent branch of the set-valued analysis. The monograph [22] presents extensive studies concerning setvalued differential equations. The theory of fuzzy differential equations extends the theory of set-valued differential equations. It has been proposed in [23]. The states of nonlinear dynamical systems described with these equations are mathematically described in the form of the fuzzy sets. The notion of a fuzzy set was introduced as an extension of the notion of a set, see [24]. This allows the mathematical modeling of impreciseness, vagueness, ambiguity or fuzziness also in the context of control theory [25]. The fuzzy differential equations are mathematical tools used in modeling dynamical phenomena with ambiguous or fuzzy states. The monograph [26] presents an extensive study of these equations including two methods of defining them. The first definition, along the lines of [23], uses the notion of a fuzzy derivative. A solution to a fuzzy differential equation is then a differentiable mapping with values in the fuzzy sets of Rd . In the second definition, introduced in [27], the previously mentioned derivative is not used. Instead, the fuzzy differential equation is treated as a family of differential inclusions that are generated by the right-hand side of the fuzzy equation. There are many papers analyzing the fuzzy differential equations, notably [28–41] presenting the results related to these equations defined in any of the two described ways. The uncertainty considered above results from ambiguity, or more generally – fuzziness. It means nonstatistical inexactness that is due to subjectivity and imprecision of human knowledge rather than to the occurrence of random events. However, very often a second source of uncertainty should be taken into account in mathematical modeling real-world phenomena. This other uncertainty is produced by an influence of random factors and stochastic noises on the considered system. It is connected with uncertainty in prediction of the outcome of an experiment. It breaks the law of causality and the probabilistic methods are applied in its analysis. A need of incorporating two different types of uncertainties, i.e. fuzziness and randomness, into some mathematical models is explained in e.g. the studies of random fuzzy differential equations [42–45], stochastic fuzzy cellular neural networks [46], sliding mode control for stochastic systems [47], adaptive fuzzy output feedback control for stochastic nonlinear systems [48], nonlinear stochastic Takagi–Sugeno fuzzy systems [49], passivity-based resilient adaptive control scheme for Takagi–Sugeno fuzzy stochastic systems with Markovian switching [50], stochastic fuzzy Markovian jumping neural networks [51] and fuzzy stochastic differential equations [52–57]. The latter topic is very current. The fuzzy stochastic differential equations find their applications in terms of mathematical models of nonlinear dynamical systems subjected to two kinds of uncertainties: fuzziness and randomness. They were applied in modeling short-term interest rate, dynamics of stock price and in modeling population growth [54]. Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

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In this paper we consider the set-valued stochastic differential equations which are understood as the following integral equations: Z t Z t XðtÞ ¼ X 0 þ Fðs; XðsÞÞ ds þ Gðs; XðsÞÞ dBðsÞ; 0

0

where X 0 ; F; G are random, X 0 ; F are set-valued, G is single-valued, B denotes a multidimensional Brownian motion. These have solutions in the form of adapted set-valued stochastic processes and generalize the deterministic set-valued differential equations studied extensively in [22]. They could be of interest in the investigations of stochastic systems, where the quantities are not known precisely but they fluctuate within some intervals. Then we extend the set-valued stochastic differential equations to the fuzzy stochastic differential equations. They generalize the deterministic set-valued and fuzzy differential equations and the single-valued stochastic differential equations as well. There are some new mathematical tools in modeling nonlinear dynamics of systems subjected to two kinds of uncertainties: randomness (stochastic uncertainty), vagueness (nonstochastic uncertainty expressed in the language of sets or fuzzy sets), see [52–54]. Here, the random phenomena evolve in the phase space of fuzzy sets. These equations constitute an interdisciplinary approach in nonlinear dynamical systems and allow us to team up with set-valued, fuzzy and stochastic analysis. The fuzzy and set-valued stochastic differential equations have been studied with assumptions of Lipschitzian coefficients in [52–54]. The papers [58,59] followed them considering some nonLipschitz coefficients and using the Maruyama successive approximation scheme. In this paper we impose some weaker conditions than those in [58,59]. Also we exploit the Picard successive approximation sequence. In this way, the class of admissible coefficients in fuzzy and set-valued equations that possess unique solutions is extended. By a theoretical analysis, we show that fuzzy stochastic differential equations possess unique solutions and this is achieved with some nonLipschitz conditions which are by far the weakest ones used in this new theory of stochastic differential equations. The problem of existence and uniqueness of solutions belongs to the foundations of the theory. In the proofs we use method of successive approximations like in [52,54]. Then we prove a closeness of solutions to equations having coefficients which do not differ much. Finally, we present an illustration of modeling with a fuzzy stochastic differential equation of a control system. Some simulations of solution sample paths are also included. The paper is organized as follows. In Section 2 we collect a relevant material concerning setvalued random variables, set-valued stochastic processes and set-valued stochastic Lebesgue– Aumann integral to make the paper self-contained in these subjects. Section 3 is focused on setvalued stochastic differential equations while Section 4 on fuzzy stochastic differential equations. In Section 5 we present an example of application of the fuzzy stochastic differential equations in a real-world problem with control. Section 6 summarizes the contribution and discusses some future research directions. 2. Preliminaries Let KðRd Þ be the family of all nonempty, compact and convex subsets of Rd . In KðRd Þ we consider the Hausdorff metric dH which is defined by   dH ðA; BÞ≔max sup inf Ja  b J; sup inf Ja  bJ ; a A Ab A B

b A Ba A A

where J  J denotes a norm in Rd . Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

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The addition and scalar multiplication in KðRd Þ are defined as follows: for A; B A KðRd Þ, bA Rd , λ A R A þ B ¼ fa þ bja A A; b A Bg;

A þ b ¼ fa þ bja A Ag;

λA ¼ fλajaA Ag:

If A; B; C; D A KðRd Þ, we have (cf. [22]) d H ðA þ C; B þ CÞ ¼ dH ðA; BÞ;

dH ðA þ C; B þ DÞr dH ðA; BÞ þ dH ðC; DÞ:

It is known (cf. [22]) that KðRd Þ is a complete and separable metric space with respect to dH. Also if KðRd Þ is equipped with the algebraic operations of addition and non-negative scalar multiplication, then KðRd Þ becomes a semilinear metric space. Let ðΩ; A; PÞ be a complete probability space. By MðΩ; A; KðRd ÞÞ we denote the family of Ameasurable set-valued random variables (or set-valued random variables, for short, see e.g. [60,61]) with values in KðRd Þ, i.e. the mappings F : Ω-KðRd Þ such that   ω A Ω : FðωÞ \ O a ∅ A A for every open set O  Rd : It is known (see [60]) that F : Ω-KðRd Þ is a set-valued random variable if, and only if, F is a AjBdH measurable mapping (in classical sense), where BdH denotes the Borel σ-algebra generated by the topology induced by the metric dH in KðRd Þ. A set-valued random variable F AMðΩ; A; KðRd ÞÞ is said to be Lp integrally bounded, pZ 1, if there exists h A Lp ðΩ; A; P; RÞ such that Ja J r hðωÞ for any a and ω with a A FðωÞ. It is known (see [61]) that F A MðΩ; A; KðRd ÞÞ is Lp integrally bounded if, and only if, ω↦ JjFðωÞj J is in Lp ðΩ; A; P; RÞ, where JjAj J ≔dH ðA; f0gÞ ¼ sup Ja J aAA

for A A KðRd Þ

to the equality P-a.e.) of and Lp ðΩ; A; P; RÞ is a space of equivalence classes (with respect R Ameasurable random variables h : Ω-R such that Ejhjp ¼ Ω jhjp dPo1. Let us denote Lp ðΩ; A; P; KðRd ÞÞ≔fF A MðΩ; A; KðRd ÞÞ : F is Lp integrally boundedg;

p Z1:

The set-valued random variables F; G A L ðΩ; A; P; KðR ÞÞ are considered to be identical, if F ¼ G holds P-a.e. One can define a metric Δp in Lp ðΩ; A; P; KðRd ÞÞ as follows: p

d

Δp ðF; GÞ ¼ ðEdpH ðF; GÞÞ1=p : It is known (see [61]) that the space Lp ðΩ; A; P; KðRd ÞÞ is a complete metric space with respect to the metric Δp . Let T40, and denote I≔½0; T. Let ðΩ; A; fAt gt A I ; PÞ be a complete, filtered probability space with a filtration fAt gt A I satisfying the usual hypotheses, i.e. fAt gt A I is an increasing and right continuous family of sub-σ-algebras of A, and A0 contains all P-null sets. We call X : I  Ω-KðRd Þ a set-valued stochastic process, if for every t A I a mapping Xðt; Þ : Ω-KðRd Þ is a set-valued random variable. We say that a set-valued stochastic process X is dH-continuous, if almost all (with respect to the probability measure P) its paths, i.e. the mappings Xð; ωÞ : I-KðRd Þ are the dH-continuous functions. A set-valued stochastic process X is said to be fAt gt A I adapted if for every t A I the setvalued random variable Xðt; Þ : Ω-KðRd Þ is At measurable. It is called measurable, if X : I  Ω-KðRd Þ is a BðIÞ  Ameasurable set-valued random variable, where BðIÞ denotes the Borel σ-algebra of subsets of I. Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

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If X : I  Ω-KðRd Þ is fAt gt A I -adapted and measurable, then it will be called nonanticipating. Equivalently, X is nonanticipating if, and only if, Xð; Þ is measurable with respect to the σ-algebra N , which is defined as follows: N ≔fA A BðIÞ  A : At A At for every t A Ig; where At ¼ fω : ðt; ωÞA Ag. A set-valued nonanticipating stochastic process X : I  Ω-KðRd Þ is called Lp integrally R bounded, if there exists a measurable stochastic process h : I  Ω-R such that Eð I ‖h ðtÞ‖p dtÞo1 and J jXðt; ωÞj J r hðt; ωÞ for a:a: ðt; ωÞA I  Ω: By Lp ðI  Ω; N ; KðRd ÞÞ we denote the set of all equivalence classes (with respect to the equality dt  dP-a.e.) of nonanticipating and Lp integrally bounded set-valued stochastic processes. In the sequel we recall a notion of set-valued stochastic Lebesgue–Aumann integral. For X A Lp ðI  Ω; N ; KðRd ÞÞ, pZ 1, we can define the Aumann integral Z Xðs; ωÞ ds for ωA Ω\N X ; where N X A A and PðN X Þ ¼ 0 I

(ω appearing in this integral can be thought as a parameter). To be more precise, for fixed ω A Ω\N X Z  Z d Xðs; ωÞ ds ¼ jðsÞ dsjj : I-R is an integrable selection of Xð; ωÞ : I

I

R d Observe that for every ωA Ω\NRX the Aumann integral I Xðs; ωÞ ds belongs R to KðR Þ (see e.g. [22]). For complete definition of I Xðs; ωÞ ds on the whole Ω we can put I Xðs; ωÞ ds ¼ f0g for ω A NX . For X A Lp ðI  Ω; N ; KðRd ÞÞ, η; t A I, ηot, we denote Z t Z Xðs; ωÞ ds ¼ 1½η;t ðsÞXðs; ωÞ ds for ω A Ω\N X ; η I Z t and Xðs; ωÞ ds ¼ f0g for ω A Ω\N X ; η

=R½η; t. where 1½η;t ðsÞ ¼ 1 if s A ½η; t and 1½η;t ðsÞ ¼ 0 if s2 t It can be shown that the mapping Ω 3 ω↦ η Xðs; ωÞ ds A KðRd Þ is a set-valued random variable. This mapping is called a set-valued stochastic Lebesgue–Aumann integral of X A Lp ðI  Ω; N ; KðRd ÞÞ over the interval ½η; t, η; t AI, η r t. In the rest of the paper, we use the following notation as a convenient shorthand: the fact that P1 Pðfω A Ω : ξðωÞ ¼ ηðωÞgÞ ¼ 1, where ξ; η are random variables, we write as ξ ¼ η, and similarly for inequalities and other relations. Also if we have Pðfω A Ω : ξðt; ωÞ ¼ ηðt; ωÞ 8t A IgÞ ¼ 1, I P1 where ξ; η are the stochastic processes, then we write ξðtÞ ¼ ηðtÞ for short, similarly for the inequalities. The set-valued stochastic Lebesgue–Aumann integral possesses the following properties (cf. [52,53]). Proposition 2.1. Let X A Lp ðI  Ω; N ; KðRd ÞÞ, p Z 1. Then the mapping I  Ω 3 Rt ðt; ωÞ↦ 0 Xðs; ωÞ dsA KðRd Þ is a set-valued stochastic process that belongs to Lp ðI Ω; N ; KðRd ÞÞ. Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

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1 d Proposition R t 2.2. Assume that X A L ðI  Ω; N ; KðR ÞÞ. Then the set-valued stochastic process ðt; ωÞ↦ 0 Xðs; ωÞ ds is dH-continuous.

Proposition 2.3. Let pZ 1. Assume that X; Y A Lp ðI  Ω; N ; KðRd ÞÞ. Then it holds Z u  Z u Z t I P1 XðsÞ ds; YðsÞ ds r t p  1 dpH ðXðsÞ; YðsÞÞ ds: sup dpH u A ½0;t

0

0

0

Corollary 2.4. Under assumptions of Proposition 2.3 it holds Z u  Z u Z t XðsÞ ds; YðsÞ ds r t p  1 E dpH ðXðsÞ; YðsÞÞ ds E sup d pH u A ½0;t

0

0

0

for every t A I. 3. Stochastic differential equations with set-valued solutions Let 0oTo1, I ¼ ½0; T and let ðΩ; A; PÞ be a complete probability space with a filtration fAt gt A I satisfying usual conditions. By B ¼ fBðtÞgt A I we denote an m-dimensional fAt g-Brownian motion defined on ðΩ; A; fAt gt A I ; PÞ, m A N. The process B is defined as follows B ¼ ðB1 ; B2 ; …; Bm Þ0 , where B1 ¼ fB1 ðtÞgt A I , B2 ¼ fB2 ðtÞgt A I ; …; Bm ¼ fBm ðtÞgt A I are the independent, one-dimensional fAt gt A I -Brownian motions, and the symbol 0 denotes transposition. In this section we shall consider the set-valued stochastic differential equations driven by mdimensional Brownian motion and with random coefficients. Such equations can be written in a symbolic differential form as I P1

dXðtÞ ¼ Fðt; XðtÞÞ dt þ Gðt; XðtÞÞ dBðtÞ;

P1

Xð0Þ ¼ X 0 ;

ð3:1Þ

with F : I  Ω  KðRd Þ-KðRd Þ, G : I  Ω  KðRd Þ-Rd  Rm and X 0 : Ω-KðRd Þ being a set-valued random variable. Since G ¼ ðG1 ; G2 ; …; Gm Þ where Gk : I  Ω  KðRd Þ-Rd , we can rewrite Eq. (3.1) as follows: m

I P1

dXðtÞ ¼ Fðt; XðtÞÞ dt þ ∑ Gk ðt; XðtÞÞ dBk ðtÞ; k¼1

P1

Xð0Þ ¼ X 0 :

ð3:2Þ

If X0, F, Gk's were single-valued and single-defined mappings then equations of the form (3.1) would be the classical stochastic differential equations whose solutions would be Rd valued stochastic processes. In this sense the theory of set-valued stochastic differential equations extends the theory of classical single-valued stochastic differential equations. Definition 3.1. By a solution to Eq. (3.1) we mean a set-valued stochastic process X : I  Ω-KðRd Þ such that (i) X A L2 ðI  Ω; N ; KðRd ÞÞ, (ii) X is dH-continuous, (iii) it holds Z t Z t m I P1 Fðs; XðsÞÞ ds þ ∑ Gk ðs; XðsÞÞ dBk ðsÞ: ð3:3Þ XðtÞ ¼ X 0 þ 0

k¼1

0

The second term of the right-hand side of Eq. (3.3) is the set-valued stochastic Lebesgue– Aumann integral, while the third one is the sum of classical Rd valued stochastic Itô integrals [62,63]. Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

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Definition 3.2. A solution X : I  Ω-KðRd Þ to Eq. (3.1) is said to be unique if I P1 dH ðXðtÞ; YðtÞÞ ¼ 0, where Y : I  Ω-KðRd Þ is any solution of Eq. (3.1). A main goal of this section is to accomplish existence and uniqueness of solutions to Eq. (3.1) under conditions which are weaker than global Lipschitz and linear growth conditions. The global Lipschitz condition was used in [52,54]. In this paper we propose to consider the following conditions: (C0) X 0 A L2 ðΩ; A0 ; P; KðRd ÞÞ, (C1) the mapping F : ðI  ΩÞ  KðRd Þ-KðRd Þ is N  BdH jBdH measurable and each Gk : ðI  ΩÞ  KðRd Þ-Rd is N  BdH jBðRd Þmeasurable, (C2) there exists a function L : I  Rþ -Rþ such that (i) Lð; rÞ is integrable for every r A Rþ , (ii) Lðt; Þ is continuous, nondecreasing and concave for every t A I, (iii) Lðt; 0Þ ¼ 0 for every t A I, (iv) if for g : I-Rþ it holds gð0Þ ¼ 0 and Z t gðtÞr a Lðs; gðsÞÞ ds; t A I; 0

where a is a positive constant, then gðtÞ ¼ 0 for t A I, (v) P-a.e. it holds 8 t A I 8 A; B A KðRd Þ d 2H ðFðt; ω; AÞ; Fðt; ω; BÞÞ r Lðt; d2H ðA; BÞÞ; 8 k A f1; 2; …; mg 8t A I 8 A; B A KðRd Þ ‖Gk ðt; ω; AÞ Gk ðt; ω; BÞ‖2 r Lðt; d2H ðA; BÞÞ; there exists a function C : I  Rþ -Rþ such that (i) Cðt; Þ is nondecreasing and concave for every t A I, (C3) (ii) P-a.e. it holds 8 t AI 8 AA KðRd Þ d2H ðFðt; ω; AÞ; f0gÞ r Cðt; d2H ðA; f0gÞÞ; 8 k A f1; 2; …; mg 8 t A I 8A A KðRd Þ ‖Gk ðt; ω; AÞ‖2 r Cðt; d 2H ðA; f0gÞÞ: We recall a sufficient condition under which the condition (C2)(iv) is satisfied, see [64]. Remark 3.3 (Taniguchi [64]). Assume that

    

the function Lð; rÞ : I-Rþ is integrable for every r A Rþ , the function Lðt; Þ : Rþ -Rþ is continuous, nondecreasing for every t A I, Lðt; 0Þ ¼ 0 for every t A I, rðÞ is a solution of the differential equation r_ ðtÞ ¼ Lðt; rðtÞÞ, if there exists t n A ½0; TÞ such that rðt n Þ ¼ 0, then rðtÞ ¼ 0 for every t A ½t n ; T.

Then, if a continuous function g : I-Rþ satisfies gð0Þ ¼ 0 and Z t Lðs; gðsÞÞ ds for every t A I; gðtÞr 0

then gðtÞ ¼ 0 for every t A I. Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

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Remark 3.4 (Taniguchi [64]). An example of the function L which satisfies (C2)(iv) could be Lðt; rÞ ¼ λðtÞαðrÞ, R where λ : I-Rþ is integrable and α : Rþ -Rþ is continuous, non-decreasing, αð0Þ ¼ 0 and 0þ 1=αðrÞ dr ¼ 1. Note that if λ c, where c40, and α is like in Remark 3.4 and concave, then one obtains a condition used in [59]. Also if λ c, where c40, and αðrÞ ¼ r, then we arrive at Lipschitz condition in the assumption (C2)(v). The following functions α1, α2 are known (cf. [65]) as examples of function α appearing in Remark 3.4. They are defined with δA ð0; 1Þ being sufficiently small ( 0r r r δ; rlog ðr  1 Þ; α1 ðrÞ ¼ δlog ðδ  1 Þ þ α01 ðδ Þðr  δÞ; r4δ; ( 0r r r δ; rlog ðr  1 Þlog log ðr  1 Þ; α2 ðrÞ ¼ 1 1 0 δlog ðδ Þlog log ðδ Þ þ α2 ðδ Þðr  δÞ; r4δ; where α0i ðδ Þ (i ¼ 1,2) denotes left-sided derivative of αi at δ. In a derivation of the existence of solution to set-valued stochastic differential equation (3.1) we will use Picard's successive approximations. Therefore we will consider a sequence of setvalued stochastic processes X n : I  Ω-KðRd Þ, n ¼ 0; 1; …, defined as follows: I P1

X 0 ðtÞ ¼ X 0 ; and for n ¼ 1; 2; … I P1

ð3:4Þ Z

t

X n ðtÞ ¼ X 0 þ

m

Z

Fðs; X n  1 ðsÞÞ ds þ ∑

k¼1

0

t

Gk ðs; X n  1 ðsÞÞ dBk ðsÞ:

ð3:5Þ

0

Before formulation of the main result of this paper we provide some useful assertions. Lemma 3.5. Let conditions (C0), (C1), (C3) be satisfied. Then Xn's described in Eqs. (3.4) and (3.5) are well defined dH-continuous set-valued stochastic processes from L2 ðI  Ω; N ; KðRd ÞÞ. Proposition 3.6. Let conditions (C0), (C1), (C3) be satisfied. Then there exists a constant M40 such that for every n AN it holds E supd 2H ðX n ðtÞ; f0gÞ r M: tAI

Proof. Let us fix n A N and t A I. Then "

Z

E sup d 2H ðX n ðuÞ; f0gÞ r ðm þ 2Þ Ed 2H ðX 0 ; f0gÞ þ E sup d2H u A ½0;t

m

þ ∑ E k¼1

u A ½0;t

Z sup d2H u A ½0;t

u

u

 Fðs; X n  1 ðsÞÞ ds; f0g

0

# G ðs; X n  1 ðsÞÞ dB ðsÞ ; f0g : k



k

0

Applying Corollary 2.4 and the Doob inequality, we get  Z t 2 E sup d H ðX n ðuÞ; f0gÞ r ðm þ 2Þ Ed2H ðX 0 ; f0gÞ þ tE d2H ðFðs; X n  1 ðsÞÞ; f0gÞ ds u A ½0;t

0

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]] m

Z

þ4 ∑ E k¼1

t

0

d2H ðfGk ðs; X n  1 ðsÞÞg; f0gÞ

9

ds :

Due to assumption (C3) and Jensen's inequality we have  2 E sup dH ðX n ðuÞ; f0gÞ r ðm þ 2Þ Ed 2H ðX 0 ; f0gÞ u A ½0;t

Z þðt þ 4mÞE "

t 0

Cðs; d2H ðX n  1 ðsÞ; f0gÞÞ ds

r ðm þ 2Þ Ed2H ðX 0 ; f0gÞ Z

!

t

þðt þ 4mÞE

C 0

"

s; sup d2H ðX n  1 ðuÞ; f0gÞ u A ½0;s

# ds

r ðm þ 2Þ Ed2H ðX 0 ; f0gÞ Z þðt þ 4mÞ

t

! C s; E

0

sup d2H ðX n  1 ðuÞ; f0gÞ u A ½0;s

# ds :

Since the function Cðs; Þ is concave for any fixed sA I, we can find positive constants a(s), b(s) such that Cðs; rÞr aðsÞ þ bðsÞr for r A Rþ . Denoting a ¼ sups A I aðsÞ, b ¼ sups A I bðsÞ we can write Cðs; rÞ r a þ br

for every ðs; rÞA I  Rþ :

Therefore E sup d2H ðX n ðuÞ; f0gÞ r ðm þ 2Þ½Ed 2H ðX 0 ; f0gÞ þ ðT þ 4mÞTa u A ½0;t Z t þðm þ 2ÞðT þ 4mÞb E sup d 2H ðX n  1 ðuÞ; f0gÞ ds: 0

u A ½0;s

Hence for k A N we obtain max E sup d2H ðX n ðuÞ; f0gÞ r ðm þ 2Þ½Ed2H ðX 0 ; f0gÞ þ ðT þ 4mÞTa Z t þðm þ 2ÞðT þ 4mÞb max E sup d2H ðX n  1 ðuÞ; f0gÞ ds:

1 r n r k u A ½0;t

0

1 r n r k u A ½0;s

Since max E sup d2H ðX n  1 ðuÞ; f0gÞ rEd2H ðX 0 ; f0gÞ þ max E sup d2H ðX n ðuÞ; f0gÞ;

1 r n r k u A ½0;t

1 r n r k u A ½0;t

we have max E sup d2H ðX n ðuÞ; f0gÞ r ðm þ 2Þ½Ed2H ðX 0 ; f0gÞ þ ðT þ 4mÞTa

1 r n r k u A ½0;t

þðm þ 2ÞðT þ 4mÞbTEd2H ðX 0 ; f0gÞ Z t max E sup d2H ðX n ðuÞ; f0gÞ ds: þðm þ 2ÞðT þ 4mÞb 0

1 r n r k u A ½0;s

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

10

Thus, by Gronwall's inequality, max E sup d2H ðX n ðuÞ; f0gÞ r M 1 expfM 2 tg;

t A I;

1 r n r k u A ½0;t

where M 1 ¼ ðm þ 2Þ½Ed 2H ðX 0 ; f0gÞ þ ðT þ 4mÞTa þ ðm þ 2ÞðT þ 4mÞbTEd2H ðX 0 ; f0gÞ, M 2 ¼ ðm þ 2ÞðT þ 4mÞb. Hence we infer that for every n A N it holds Esupd2H ðX n ðuÞ; f0gÞ r M 1 expfM 2 Tg; uAI

which ends the proof.

t A I;

□

Theorem 3.7. Assume that conditions (C0)–(C3) are satisfied. Then Eq. (3.1) possesses a unique solution. Proof. To prove the existence of the solution to Eq. (3.1) we shall use the Picard sequence fX n g1 n ¼ 0 . Due to Lemma 3.5 the nth approximation Xn is well defined for every nA N. Let us fix t A I. Note that for every n; ℓ A N [ f0g it holds E sup d 2H ðX nþ1 ðuÞ; X ℓþ1 ðuÞÞ u A ½0;t

Z

u

m

Z

u

¼ E sup Fðs; X n ðsÞÞ ds þ ∑ Gk ðs; X n ðsÞÞ dBk ðsÞ; u A ½0;t 0 k¼1 0  Z u Z u m k Fðs; X ℓ ðsÞÞ ds þ ∑ G ðs; X ℓ ðsÞÞ dBk ðsÞ 0 k¼1 0 " Z  Z d2H

r ðm þ 1Þ E sup d2H u A ½0;t

u

u

Fðs; X n ðsÞÞ ds;

0

Fðs; X ℓ ðsÞÞ ds

0

Z u Z u 2 þ ∑ E sup d H Gk ðs; X n ðsÞÞ dBk ðsÞ; Gk ðs; X ℓ ðsÞÞ 0 0 k ¼ 1 u A ½0;t  Z t r ðm þ 1Þ tE d2H ðFðs; X n ðsÞÞ; Fðs; X ℓ ðsÞÞÞ ds Z 0t m k k 2 þ4 ∑ E ‖G ðs; X n ðsÞÞ G ðs; X ℓ ðsÞÞ‖ ds : 0 k¼1 m

# dB ðsÞ k

Using assumption (C2) and applying Jensen's inequality we obtain E sup d 2H ðX nþ1 ðuÞ; X ℓþ1 ðuÞÞ u A ½0;t

Z

t

r ðm þ 1Þðt þ 4mÞE 0 t

Z r ðm þ 1Þðt þ 4mÞ 0

Z r ðm þ 1ÞðT þ 4mÞ

0

Lðs; d2H ðX n ðsÞ; X ℓ ðsÞÞÞ ds

!

EL s; sup d2H ðX n ðuÞ; X ℓ ðuÞÞ u A ½0;s

t

ds !

L s; E sup d 2H ðX n ðuÞ; X ℓ ðuÞÞ u A ½0;s

ds:

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

11

Due to Proposition 3.6 we can define a function g : I-Rþ as ! gðtÞ ¼ lim sup E sup d 2H ðX n ðuÞ; X ℓ ðuÞÞ ; n;ℓ-1

u A ½0;t

t A I:

Applying Fatou's lemma and assumption on continuity of Lðs; Þ we get Z t Lðs; gðsÞÞ ds for t A I; gðtÞr ðm þ 1ÞðT þ 4mÞ 0

which implies (by assumption (C2)(iv)) that gðtÞ ¼ 0 for every t A I. Hence for every t A I lim Ed2H ðX n ðtÞ; X ℓ ðtÞÞ ¼ 0:

n;ℓ-1

Since for every t A I the metric space ðL2 ðΩ; At ; P; KðRd ÞÞ; Δ2 Þ is complete, for every t A I there exists X t A L2 ðΩ; At ; P; KðRd ÞÞ such that Ed2H ðX n ðtÞ; X t Þ⟶0

as n-1:

Hence, by putting Xðt; ωÞ ¼ X t ðωÞ we can define a set-valued stochastic process X : I  Ω-KðRd Þ which is fAt g-adapted. In the sequel, we shall show that process X satisfies conditions of Definition 3.1. Indeed, since   lim E supd 2H ðX n ðuÞ; X ℓ ðuÞÞ ¼ 0; n;ℓ-1

uAI

by Chebyshev's inequality we obtain: for every ε40   P supd 2H ðX n ðuÞ; X ℓ ðuÞÞ4ε ⟶0 as n; ℓ-1: uAI

Thus we infer that there exists a subsequence fX nk ð; Þg of the sequence fX n ð; Þg such that P1

supdH ðX nk ðuÞ; XðuÞÞ⟶0 uAI

as k-1:

Therefore process X is dH-continuous and consequently BðIÞ  Ameasurable. Since X is also fAt gadapted, we obtain that X is N measurable. Now, having that XðtÞA L2 ðΩ; A; P; KðRd ÞÞ for every t AI, we can write Z E ‖jXðtÞj‖2 dt r T supE‖jXðtÞj‖2 o1 tAI

I

which implies that X A L ðI  Ω; N ; KðRd ÞÞ. R Rt k I P1 t k What is left is to prove that XðtÞ ¼ X 0 þ 0 Fðs; XðsÞÞ ds þ ∑m k ¼ 1 0 G ðs; XðsÞÞ dB ðsÞ. To this end let us note that for every t A I   Z t Z t m Ed2H XðtÞ; X 0 þ Fðs; XðsÞÞ ds þ ∑ Gk ðs; XðsÞÞ dBk ðsÞ r 3I 1n ðtÞ þ 3I 2n ðtÞ þ 3I 3n ðtÞ; 2

k¼1

0

0

where I 1n ðtÞ ¼ Ed 2H ðX n ðtÞ; XðtÞÞ;   Z t Z t m I 2n ðtÞ ¼ Ed 2H X n ; X 0 þ Fðs; X n  1 ðsÞÞ ds þ ∑ Gk ðs; X n  1 ðsÞÞ dBk ðsÞ ; 0

k¼1

0

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

12

Z

Z t m Fðs; X n  1 ðsÞÞ ds þ ∑ Gk ðs; X n  1 ðsÞÞ dBk ðsÞ; 0 k¼1 0  Z t Z t m k k Fðs; XðsÞÞ ds þ ∑ G ðs; XðsÞÞ dB ðsÞ :

I 3n ðtÞ ¼ Ed2H

t

k¼1

0

Note that

n-1 I 1n ðtÞ ⟶ 0

0

I 2n ðtÞ ¼ 0 for every t A I. Further, t d 2H ðFðs; X n  1 ðsÞÞ; Fðs; XðsÞÞÞ ds

and observe that Z I 3n ðtÞ r ðm þ 1Þ tE 0  Z t m þ ∑ E ‖Gk ðs; X n  1 ðsÞÞ  Gk ðs; XðsÞÞ‖2 ds 0 k¼1 Z t r ðm þ 1Þðt þ mÞE Lðs; d2H ðX n  1 ðsÞ; XðsÞÞÞ ds 0 Z t r ðm þ 1Þðt þ mÞ Lðs; Ed2H ðX n  1 ðsÞ; XðsÞÞÞ ds: 

0

Applying the Lebesgue dominated convergence theorem together with continuity of Lðs; Þ and n-1 assumption Lðs; 0Þ ¼ 0 we obtain I 3n ðtÞ ⟶ 0 for t A I. Hence for every t A I   Z t Z t m Ed2H XðtÞ; X 0 þ Fðs; XðsÞÞ ds þ ∑ Gk ðs; XðsÞÞ dBk ðsÞ ¼ 0: k¼1

0

0

This fact together with continuity of X allows us to conclude that   Z t Z t m I P1 Fðs; XðsÞÞ ds þ ∑ Gk ðs; XðsÞÞ dBk ðsÞ ¼ 0: d H XðtÞ; X 0 þ k¼1

0

0

Thus X is a solution to Eq. (3.1). Finally, we shall show that X is unique. To this end let X; Y denote two solutions to Eq. (3.1). Then, for every t A I we have  Z t d2H ðFðs; XðsÞÞ; Fðs; YðsÞÞÞ ds E sup d 2H ðXðuÞ; YðuÞÞ r ðm þ 1Þ tE u A ½0;t

m

0

Z



t

þ4 ∑ E

‖G ðs; XðsÞÞ  G ðs; YðsÞÞ‖ ds Z t r ðm þ 1Þðt þ 4mÞ ELðs; d 2H ðXðsÞ; YðsÞÞÞ ds k¼1

k

k

2

0

0

Z r ðm þ 1ÞðT þ 4mÞ

0

t

!

L s; E sup d2H ðXðsÞ; YðsÞÞ u A ½0;s

ds:

Using assumption (C2)(iv) we infer that Esupu A I d2H ðXðuÞ; YðuÞÞ ¼ 0. This yields I P1

d H ðXðuÞ; YðuÞÞ ¼ 0: Thus uniqueness of solution follows.

□

In the formulation of the set-valued stochastic differential equations we use the single-valued diffusion term in the form of single-valued Itô stochastic integral. There is an obstacle to extend this term to set-valued one. Although a notion of a set-valued Itô stochastic integral has been proposed in [66], this set-valued integral is not useful in the studies of set-valued stochastic Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

13

differential equations. The set-valued stochastic Itô integral defined in [66] may not be integrally bounded and may not satisfy a set-valued counterpart of Itô' isometry (cf. [67]). Let us consider Eq. (3.2) and the following equations: m

I P1

dX n ðtÞ ¼ F n ðt; X n ðtÞÞ dt þ ∑ Gkn ðt; X n ðtÞÞ dBk ðtÞ; k¼1

P1

X n ð0Þ ¼ X 0;n ;

ð3:6Þ

for n A N. Theorem 3.8. Let X 0 ; X 0;n : Ω-KðRd Þ, nA N, satisfy (C0). Let F : I  Ω  KðRd Þ-KðRd Þ and Gk : I  Ω  KðRd Þ-Rd , k ¼ 1; 2; …; m, satisfy (C1)–(C3). Assume that for every nA N the mappings F n : I  Ω  KðRd Þ-KðRd Þ and Gkn : I  Ω  KðRd Þ-Rd , k ¼ 1; 2; …; m, satisfy (C1)–(C3) with the functions L and C which do not depend on n. Suppose that Ed2H ðX 0;n ; X 0 Þ⟶0 as n-1 1

2

m

G G and there are sequences fM Fn g; fM G n g; fM n g; …; fM n g  ½0; 1Þ converging to zero such that

Ed2H ðF n ðt; AÞ; Fðt; AÞÞ r M Fn ;

k

Ed2H ðGkn ðt; AÞ; Gk ðt; AÞÞ r M G n ; k ¼ 1; 2; …; m;

for every ðt; AÞA I  KðRd Þ. Then the solution X to Eq. (3.2) and the solution Xn to Eq. (3.6) satisfy E supd2H ðX n ðtÞ; XðtÞÞ⟶0 as n-1: tAI

Proof. Let t A I be fixed. We notice that E sup d2H ðX n ðuÞ; XðuÞÞ u A ½0;t  Z t 2 r ðm þ 2Þ EdH ðX 0;n ; X 0 Þ þ tE d2H ðF n ðs; X n ðsÞÞ; Fðs; XðsÞÞÞ ds 0 Z t m þ4 ∑ E d2H ðGkn ðs; X n ðsÞÞ; Gk ðs; XðsÞÞÞ ds k¼1 0  r ðm þ 2Þ Ed2H ðX 0;n ; X 0 Þ Z t þ2tE fd2H ðF n ðs; X n ðsÞÞ; F n ðs; XðsÞÞÞ þ d 2H ðF n ðs; XðsÞÞ; Fðs; XðsÞÞÞg ds 0 Z t m þ8 ∑ E fd2H ðGkn ðs; X n ðsÞÞ; Gkn ðs; XðsÞÞÞ þ d2H ðGkn ðs; XðsÞÞ; Gk ðs; XðsÞÞÞg ds : k¼1

0

Applying (C2) we arrive at E sup d2H ðX n ðuÞ; XðuÞÞ u A ½0;t  Z t r ðm þ 2Þ Ed2H ðX 0;n ; X 0 Þ þ 2t Ed2H ðF n ðs; XðsÞÞ; Fðs; XðsÞÞÞ ds 0 Z t m Ed2H ðGkn ðs; XðsÞÞ; Gkn ðs; XðsÞÞÞ ds þ8 ∑ k¼1 0  Z t  L s; E sup d 2H ðX n ðuÞ; XðuÞÞ ds : þ2ðt þ 4mÞ 0

ð3:7Þ

u A ½0;s

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

14

For the function L we obtain that there are some non-negative constants c; d such that Lðs; rÞr c þ dr, ðs; rÞA I  Rd . Therefore using this and the assumptions we get E sup d 2H ðX n ðuÞ; XðuÞÞ u A ½0;t

h 1 G2 Gm r ðm þ 2Þ Ed 2H ðX 0;n ; X 0 Þ þ 2t 2 M Fn þ 8tðM G n þ Mn þ ⋯ þ Mn Þ # Z t þ2ctðt þ 4mÞ þ 2dðt þ 4mÞ E sup d2H ðX n ðuÞ; XðuÞÞ ds : u A ½0;s

0

1

2

m

Since there are constants M X 0 ; M F ; M G ; M G ; …; M G such that Ed 2H ðX 0;n ; X 0 Þr M X 0 , M Fn r M F , 1 G1 G2 G2 Gm Gm MG for every n A N, we have n r M , M n r M , …, M n r M E sup d 2H ðX n ðuÞ; XðuÞÞ u A ½0;t

1

2

m

r ðm þ 2Þ½M X 0 þ 2T 2 M F þ 8TðM G þ M G þ ⋯ þ M G Þ Z t þ2cTðT þ 4mÞ þ 2dðm þ 2ÞðT þ 4mÞ E sup d2H ðX n ðuÞ; XðuÞÞ ds: 0

u A ½0;s

Hence, by the Gronwall lemma, 1

2

m

E sup d 2H ðX n ðuÞ; XðuÞÞ r ðm þ 2Þ½M X 0 þ 2T 2 M F þ 8TðM G þ M G þ ⋯ þ M G Þ u A ½0;t

þ2cTðT þ 4mÞexpf2dðm þ 2ÞðT þ 4mÞtg: Esupu A ½0;t d 2H ðX n ðuÞ; XðuÞÞ

Thus the expression is bounded for every t A I. Therefore the expressions Lðt; Esupu A ½0;t d2H ðX n ðuÞ; XðuÞÞÞ, t A I, are well defined and the function t↦Lðt; Esupu A I d 2H ðX n ðuÞ; XðuÞÞÞ is integrable. Also Lðt; Esupu A ½0;t d2H ðX n ðuÞ; XðuÞÞÞ r Lðt; Esupu A I d2H ðX n ðuÞ; XðuÞÞÞ. Now using Eq. (3.7), the Lebesgue dominated convergence theorem and the assumptions we infer that lim E sup d2H ðX n ðuÞ; XðuÞÞ

n-1

u A ½0;t

Z

!

t

r 2ðm þ 2ÞðT þ 4mÞ

L 0

s; lim E sup d 2H ðX n ðuÞ; XðuÞÞ n-1 u A ½0;s

ds:

Thus lim E sup d2H ðX n ðuÞ; XðuÞÞ ¼ 0

n-1

u A ½0;t

which ends the proof.

for every t A I;

□

The results presented in this part of the paper have immediate consequences for the deterministic set-valued differential equations which constitute an independent branch of setvalued analysis [22]. Note that in the case when X 0 A KðRd Þ, F : I R KðRd Þ-KðRd Þ and Gk 0 t for k ¼ 1; 2; …; m, the equality (3.3) reduces to XðtÞ ¼ X 0 þ 0 Fðs; XðsÞÞ ds, t A I. Such deterministic set-valued integral equation is equivalent to the initial value problem for setvalued differential equations of the form X 0 ðtÞ ¼ Fðt; XðtÞÞ

for t A I; Xð0Þ ¼ X 0 ;

ð3:8Þ

where 0 denotes the set-valued Hukuhara derivative for set-valued mappings, see [22]. Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

15

In what follows we rewrite the conditions appropriate to the deterministic framework and state the main result on existence and uniqueness of solution. They are as follows: (H1) the mapping F : I  KðRd Þ-KðRd Þ is βðIÞ  BdH jBdH measurable, where βðIÞ denotes the Borel σ-algebra of subsets of I, (H2) there exists a function L : I  Rþ -Rþ such that Lð; rÞ is integrable for every r A Rþ , Lðt; Þ is continuous, nondecreasing and concave forRevery t A I, Lðt; 0Þ ¼ 0 for every t A I, t ○ if for g : I-Rþ it holds gð0Þ ¼ 0 and gðtÞr a 0 Lðs; gðsÞÞ ds for t A I, where a is a positive constant, then gðtÞ ¼ 0 for t A I, ○ for every t A I and for any A; BA KðRd Þ d 2H ðFðt; AÞ; Fðt; BÞÞ r Lðt; d2H ðA; BÞÞ; (H3) there exists a function C : I  Rþ -Rþ with the property that Cðt; Þ is nondecreasing and concave for every t A I and such that for every t A I and for every A A KðRd Þ d 2H ðFðt; AÞ; f0gÞ r Cðt; d2H ðA; f0gÞÞ: Corollary 3.9. Let X 0 A KðRd Þ. Assume that F : I  KðRd Þ-KðRd Þ satisfies (H1)–(H3). Then deterministic set-valued differential equation (3.8) has a unique solution. Note that the assertions, collected in [22], which guarantee existence of a unique solution are formulated with an assumption of continuity of F in the first variable t. Investigations presented in this paper show that this can be weakened to a measurability condition. Beside Corollary 3.9, the remaining deterministic counterparts of the set-valued stochastic results can also be stated for deterministic set-valued differential equations. 4. Stochastic differential equations with fuzzy-set-valued solutions In this section we show that the obtained set-valued results can be extended to a fuzzy case. To consider fuzzy stochastic differential equations we need some background concerning fuzzy sets, fuzzy stochastic processes and fuzzy stochastic integrals. For convenience of the reader, we recall a needed material (cf. [52–54]). A concept of a fuzzy set generalizes notion of ordinary set (cf. [24]). A fuzzy set u in Rd is characterized by its membership function (denoted by u again) u : Rd -½0; 1 and u(x) (for each x A Rd ) is interpreted as the degree of membership of x in the fuzzy set u. As the value u(x) expresses “degree of membership of x in” or a “degree of satisfying by x a property”, one can work with imprecise information. Every ordinary set u in Rd is a fuzzy set, since then uðxÞ ¼ 1 if x A u and uðxÞ ¼ 0 if x= 2 u. By F ðRd Þ we denote a set of fuzzy sets u : Rd -½0; 1 such that ½uα A KðRd Þ for every α A ½0; 1, where ½uα ≔fa A Rd : uðaÞZ αg for α A ð0; 1 and ½u0 ≔clfa A Rd : uðaÞ40g. If α A ð0; 1, the set ½uα is called α-cut or α-level set. It is known that for uA F ðRd Þ the mapping ½0; 1 3 α↦½uα A KðRd Þ is left continuous on ð0; 1, right continuous at 0 and has right-limits on ½0; 1Þ. Addition u v of fuzzy sets u; vA F ðRd Þ and scalar multiplication λ  u, where λ A R, u A F ðRd Þ, can be defined levelwise (cf. [26]): ½u vα ¼ ½uα þ ½vα ;

½λ  uα ¼ λ½uα :

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

16

A generalized Hausdorff metric d1 : F ðRd Þ  F ðRd Þ-½0; 1Þ is defined by the expression d 1 ðu; vÞ ¼ sup d H ð½uα ; ½vα Þ; α A ½0;1

where dH is the Hausdorff metric in KðRd Þ. In fact ðF ðRd Þ; d1 Þ is a complete metric space, and for every u; v; w; z A F ðRd Þ, λ A R one has d 1 ðu w; v wÞ ¼ d1 ðu; vÞ, d1 ðu v; w zÞr d1 ðu; wÞ þ d1 ðv; zÞ, d 1 ðλ  u; λ  vÞ ¼ jλjd1 ðu; vÞ (see e.g. [68]). It is also known that ðF ðRd Þ; d1 Þ is not separable and is not locally compact as distinct from ðKðRd Þ; dH Þ. The set Rd can be embedded into F ðRd Þ by an embedding 〈  〉 : Rd -F ðRd Þ defined as follows: for r A Rd ( 1 if a ¼ r; 〈r〉ðaÞ ¼ 0 if a ARd \frg: Let ðΩ; A; PÞ be a probability space. A mapping x : Ω-F ðRd Þ is said to be a fuzzy random variable (cf. [68]), if ½xα : Ω-KðRd Þ is an Ameasurable set-valued random variable for all α A ½0; 1. We will say that a fuzzy random variable x : Ω-F ðRd Þ is said to be Lp integrally bounded, pZ 1, if ½xα A Lp ðΩ; A; P; KðRd ÞÞ for every α A ½0; 1. Let Lp ðΩ; A; P; F ðRd ÞÞ denote the set of all the Lp integrally bounded fuzzy random variables, where we consider x; yA Lp ðΩ; A; P; F ðRd ÞÞ as identical if Pð½xα ¼ ½yα ; 8 α A ½0; 1Þ ¼ 1. Let us mention that the definition given above is one of the possibilities to be considered for fuzzy random variables. Namely, having a metric d1 in the set F ðRd Þ one can consider σ-algebra Bd1 generated by the topology induced by d 1 . Then one can consider measurable (in the classical sense) mappings x (called fuzzy random variables, too) acting between two measurable spaces, namely ðΩ; AÞ and ðF ðRd Þ; Bd1 Þ. Using the classical notation, we write this fact as x is AjBd1 measurable. As distinct from the set-valued case, here, there is no equivalence of these two definitions of fuzzy random variables. It is known (see [69]) that for a mapping x : Ω-F ðRd Þ, where ðΩ; A; PÞ is a given probability space, it holds:

 

if x is AjBd1 measurable (i.e. x is a fuzzy random variable in the second sense), then it is a fuzzy random variable understood in the first sense described above, the opposite implication is not true. Denote F c ðRd Þ ¼ fu A F ðRd Þjα↦½uα is a dH continuous mapping on ½0; 1g:

Then ðF c ðRd Þ; d1 Þ is a complete and separable metric space and the two notions of fuzzy random variables coincide for F c ðRd Þvalued random mappings (cf. [70]). Denote L2 ðΩ; A; P; F c ðRd ÞÞ≔fx : Ω-F c ðRd Þjx is an L2 integrally bounded fuzzy random variableg: Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

17

One can define a metric δ2 in this set by δ2 ðx; yÞ ¼ ðEd 21 ðx; yÞÞ1=2 for x; y A L2 ðΩ; A; P; F c ðRd ÞÞ: Then the couple ðL2 ðΩ; A; P; F c ðRd ÞÞ; δ2 Þ is a complete metric space (cf. [71]). We call x : I  Ω-F c ðRd Þ a fuzzy stochastic process, if for every t A I the mapping xðt; Þ : Ω-F c ðRd Þ is a fuzzy random variable. We say that a fuzzy stochastic process x is d 1 continuous, if almost all (with respect to the probability measure P) its paths, i.e. the mappings xð; ωÞ : I-F c ðRd Þ are d 1 continuous functions. A fuzzy stochastic process x is said to be fAt gt A I -adapted, if for every α A ½0; 1 the set-valued random variable ½xðtÞα : Ω-KðRd Þ is At measurable for all t A I. It is called measurable, if ½xα : I  Ω-KðRd Þ is a BðIÞ  Ameasurable set-valued random variable for all α A ½0; 1. If x : I  Ω-F c ðRd Þ is fAt gt A I -adapted and measurable, then it will be called nonanticipating. Equivalently, x is nonanticipating if and only if for every α A ½0; 1 the set-valued random variable ½xα is measurable with respect to the σ-algebra N . A nonanticipating fuzzy stochastic process x is called Lp integrally bounded, if there exists a real-valued stochastic process hA Lp ðI  Ω; N ; RÞ such that J j½xðt; ωÞ0 j J r hðt; ωÞ

for a:a: ðt; ωÞ A I  Ω:

By L ðI  Ω; N ; F c ðR ÞÞ we denote the set of nonanticipating and Lp integrally bounded fuzzy stochastic processes x : I  Ω-F c ðRd Þ. Let x A Lp ðI  Ω; N ; F c ðRd ÞÞ, pZ 1. For such x and any fixed t A I we can consider a fuzzy stochastic integral as a fuzzy random variable Z t xðs; ωÞ ds A F c ðRd Þ Ω 3 ω↦ p

d

0

satisfying Z t

α Z t xðs; ωÞ ds ¼ ½xðs; ωÞα ds for every α A ½0; 1;

0

0

where integral on the right-hand side is the set-valued stochastic Lebesgue–Aumann integral described earlier. In [53] one can find a definition of this type of integral for a wider class than Lp ðI  Ω; N ; F c ðRd ÞÞ. The following properties have been established in [52,53]. Proposition 4.1. Let pZ 1. If x A Lp ðI  Ω; N ; F c ðRd ÞÞ then fuzzy stochastic process Rt ðt; ωÞ↦ 0 xðs; ωÞ ds belongs to Lp ðI  Ω; N ; F c ðRd ÞÞ. Proposition 4.2. Let x A L1 ðI  Ω; N ; F c ðRd ÞÞ. Rt ðt; ωÞ↦ 0 xðs; ωÞ ds is d1 continuous.

Then

the

fuzzy

stochastic

process

Proposition 4.3. Let p Z 1. Assume that x; y A Lp ðI  Ω; N ; F c ðRd ÞÞ. Then it holds Z u  Z t Z u I P1 p p1 xðsÞ ds; yðsÞ r t dp1 ðxðsÞ; yðsÞÞ ds: sup d 1 u A ½0;t

0

0

0

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

18

Corollary 4.4. Under assumptions of Proposition 4.3 we have for every t A I Z u  Z u Z t E sup d p1 xðsÞ ds; yðsÞ ds r t p  1 E dp1 ðxðsÞ; yðsÞÞ ds: u A ½0;t

0

0

0

In the framework of fuzzy stochastic differential equations we need also a concept of a fuzzy Itô integral. A lack of boundedness properties of set-valued stochastic Itô integral (being a setvalued stochastic process cf. [13]) causes that the definition of fuzzy stochastic Itô integral used in a formulation of fuzzy stochastic differential equation should be restricted. Therefore we exploit (cf. [52–54]) the single-valued Itô integral that is embedded into the fuzzy sets space. Let fBðtÞgt A I be a one-dimensional fAt gt A I Brownian motion defined on a complete 2 probability space R T ðΩ; A; PÞ with a filtration fAt gt A I satisfying usual hypotheses. For xA L ðI  d Ω; N ; R Þ let 0 xðsÞ dBðsÞ denote the classical stochastic Itô integral (see e.g. [62,63]). By a fuzzy stochastic Itô integral we mean the fuzzy random variable

Z T  Ω 3 ω↦ xðsÞ dBðsÞðωÞ A F c ðRd Þ: 0

For every t A I one can consider the fuzzy stochastic Itô integral 〈 understood in the sense:

Z t  Z T  xðsÞ dBðsÞ ≔ 1½0;t ðsÞxðsÞ dBðsÞ ; 0

Rt 0

xðsÞ dBðsÞ〉, which is

0

where 1½0;t ðsÞ ¼ 1 if sA ½0; t and 1½0;t ðsÞ ¼ 0 if sA ðt; T. The properties of this fuzzy Itô integral are inherited from the classical Itô integral (cf. [52]). Proposition 4.5. Let xA L2 ðI  Ω; N ; Rd Þ. Then the mapping

Z t  I  Ω 3 ðt; ωÞ↦ xðsÞ dBðsÞðωÞ A F c ðRd Þ 0

belongs to L ðI  Ω; N ; F c ðRd ÞÞ. 2

Proposition 4.6. Let xA L2 ðI  Ω; N ; Rd Þ. Then P-a.a. the sample paths of the process Rt f〈 0 xðsÞ dBðsÞ〉gt A I , i.e. the mappings

Z t  I 3 t↦ xðsÞ dBðsÞðωÞ A F c ðRd Þ 0

are d1 continuous. Proposition 4.7. Let x; y A L2 ðI  Ω; N ; Rd Þ. Then for every t A I  Z u  Z u  Z t E sup d 21 xðsÞ dBðsÞ ; yðsÞ dBðsÞ r 4E d 21 ð〈xðsÞ〉; 〈yðsÞ〉Þ ds: u A ½0;t

0

0

0

In the rest of the paper we shall present a study of fuzzy stochastic differential equations driven by m-dimensional Brownian motion and with random and non-Lipschitz coefficients. Such equations can be written as I P1

dxðtÞ ¼ f ðt; xðtÞÞ dt 〈gðt; xðtÞÞ dBðtÞ〉;

P1

xð0Þ ¼ x0 ;

ð4:1Þ

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

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with f : I  Ω  F c ðRd Þ-F c ðRd Þ, g : I  Ω  F c ðRd Þ-Rd  Rm , and x0 : Ω-F c ðRd Þ being a fuzzy random variable. Since g ¼ ðg1 ; g2 ; …; gm Þ, where gk : I  Ω  F c ðRd Þ-Rd (k ¼ 1; 2; …; m), we can rewrite Eq. (4.1) as follows: m

I P1

P1

dxðtÞ ¼ f ðt; xðtÞÞ dt 〈gk ðt; xðtÞÞ dBk ðtÞ〉;

xð0Þ ¼ x0 :

k¼1

ð4:2Þ

Definition 4.8. By a solution to Eq. (4.1) we mean a fuzzy stochastic process x : I  Ω-F c ðRd Þ such that (i) xA L2 ðI  Ω; N ; F c ðRd ÞÞ, (ii) x is d1 -continuous, (iii) it holds

Z t  Z t m I P1 xðtÞ ¼ x0

f ðs; xðsÞÞ ds

gk ðs; xðsÞÞ dBk ðsÞ : ð4:3Þ k¼1

0

0

The right-hand side of Eq. (4.3) is understood in the meaning described earlier, i.e. the second term is the fuzzy stochastic Lebesgue–Aumann integral, while the third one is a sum of the Rd valued stochastic Itô integrals which is embedded into F c ðRd Þ. Definition 4.9. A solution x : I  Ω-F c ðRd Þ to Eq. (4.1) is said to be unique, if I P1 d1 ðxðtÞ; yðtÞÞ ¼ 0, where y : I  Ω-F c ðRd Þ is any solution of Eq. (4.1). Although the form of the fuzzy stochastic differential equation considered in this paper is very similar to that of the set-valued stochastic differential equation, it does not mean that the fuzzy stochastic equations can be omitted in examinations. Firstly, they extend the notion of set-valued stochastic differential equations, deterministic set-valued differential equations and deterministic fuzzy differential equations. Secondly, the fuzzy-set-valued solutions to fuzzy stochastic differential equations (4.1) might not yield the set-valued stochastic processes that would be solutions to Eq. (3.1). This can be seen in the following justifications. Note that if x is a solution to Eq. (4.1) then according to Definition 4.8 we have   Z t Z t m P ½xðtÞα ¼ ½x0 α þ ½f ðs; xðsÞÞα ds þ ∑ gk ðs; xðsÞÞ dBk ðsÞ 8t A I 8 α A ½0; 1 ¼ 1 k¼1

0

0

ð4:4Þ and here the set-valued stochastic differential equations appear in a natural way. However, in general, the set-valued equations in Eq. (4.4) are not of the type (3.3). In general, the drift and diffusion coefficients in Eq. (4.4) do not depend on ½xðsÞα only. They depend on x(s), i.e. whole family f½xðsÞα ; α A½0; 1g. Thus the solution of fuzzy equation (4.1) does not have to lead to the solution of set-valued equations (3.1), although the form of Eq. (4.1) is reducible to Eq. (3.1) by replacement F c ðRd Þ with KðRd Þ. Starting from the set-valued equations one could consider for every α A ½0; 1 the following setvalued stochastic differential equation Z t Z t m I P1 X α ðtÞ ¼ X 0;α þ F α ðs; X α ðsÞÞ ds þ ∑ Gk ðs; X α ðsÞÞ dBk ðsÞ ð4:5Þ 0

k¼1

0

where X 0;α : Ω-KðR Þ, F α : I  Ω  KðR Þ-KðRd Þ and Gk : I  Ω  KðRd Þ-Rd . Let X α : I  Ω-KðRd Þ denote a unique solution to Eq. (4.5). In general, the family of set-valued stochastic processes fX α : α A½0; 1g may not constitute family of α-levels of any fuzzy stochastic process x : I  Ω-F ðRd Þ. However, if so, then does exist any fuzzy stochastic differential d

d

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

20

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

equation (with data x0 ; f ; gk ) whose solution is x? What relationships should be between F α and f, Gk and gk? There are some obstacles. Namely, how to generate gk which acts from I  Ω  F c ðRd Þ having as a base the mapping Gk with the domain I  Ω  KðRd Þ? The same concerns f and the family fF α : α A ½0; 1g. It seems that F α should coincide with ½f α , but there are still problems in domains of these mappings. A last issue, which is also an obstacle, is that solving the set-valued stochastic differential equations (4.5) separately for each α A½0; 1, we associate a measurable set N α with X α . This set is such that PðN α Þ ¼ 0 and the equality in Eq. (4.5) holds for every t A I and every ω A Ω\N α . If the set-valued stochastic differential equations with solutions X α were useful for the fuzzy stochastic differential equation with solution x, it would be expected that PðX α ðtÞ ¼ ½xðtÞα

8 t A I 8 α A ½0; 1Þ ¼ 1:

This is difficult and could be impossible, since one could only have X α ðt; ωÞ ¼ ½xðt; ωÞα 8 t A I 8 ωA Ω\N α for each α separately and the set ⋃α A ½0;1 N α could be nonmeasurable. So, the knowledge of set-valued solutions to set-valued stochastic differential equations (3.1) is not sufficient to built a fuzzy stochastic process that would be a solution to Eq. (4.1). Hence the fuzzy stochastic differential equations should be studied. We shall prove the existence and uniqueness of solution to Eq. (4.1) under the following conditions: (c0) x0 A L2 ðΩ; A0 ; P; F c ðRd ÞÞ, (c1) the mapping f : ðI  ΩÞ  F c ðRd Þ-F c ðRd Þ is N  Bd1 jBd1 measurable and gk : ðI  ΩÞ  F c ðRd Þ-Rd is N  Bd1 jBðRd Þmeasurable, (c2) there exists a function L : I  Rþ -Rþ such that (i) Lð; rÞ is integrable for every r A Rþ , (ii) Lðt; Þ is continuous, nondecreasing and concave for every t A I, (iii) Lðt; 0Þ ¼ 0 for every t A I, (iv) if for g : I-Rþ it holds gð0Þ ¼ 0 and Z t gðtÞ r a Lðs; gðsÞÞ ds; t A I; 0

where a is a positive constant, then gðtÞ ¼ 0 for t A I, (v) P-a.e. it holds 8 t AI 8 u; vA F c ðRd Þ d 21 ðf ðt; ω; uÞ; f ðt; ω; vÞÞ r Lðt; d 21 ðu; vÞÞ; 8 k A f1; 2; …; mg 8 t A I 8u; vA F c ðRd Þ ‖gk ðt; ω; uÞ gk ðt; ω; vÞ‖2 r Lðt; d21 ðu; vÞÞ; (c3) there exists a function C : I  Rþ -Rþ such that (i) Cðt; Þ is nondecreasing and concave for every t A I, (ii) P-a.e. it holds 8 t A I 8 uA F c ðRd Þ d 21 ðf ðt; ω; uÞ; 〈0〉Þr Cðt; d21 ðu; 〈0〉ÞÞ; 8 k A f1; 2; …; mg 8 t A I 8 uA F c ðRd Þ ‖gk ðt; ω; uÞ‖2 r Cðt; d21 ðu; 〈0〉ÞÞ: Notice that the condition (c2) is weaker than the Lipschitz condition exploited in [52,54] and more general than non-Lipschitz condition formulated in [58]. Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

21

In a derivation of existence of solution we will use the successive approximations xn : I  Ω-F c ðRd Þ, n ¼ 0; 1; …, defined as I P1

x0 ðtÞ ¼ x0 ; and for n ¼ 1; 2; … I P1

ð4:6Þ Z

xn ðtÞ ¼ x0

t

m

Z

f ðs; xn  1 ðsÞÞ ds

k¼1

0

t

 g ðs; xn  1 ðsÞÞ dB ðsÞ : k

ð4:7Þ

k

0

Under conditions (c0), (c1) and (c3), fuzzy stochastic processes xn, n A N [ f0g, are well defined, d1 continuous and belong to L2 ðI  Ω; N ; F c ðRd ÞÞ. The results presented below are the fuzzy counterparts of Proposition 3.6 and Theorem 3.7. Their proofs are very similar to those presented in Section 3. We include their schemes for completeness. Proposition 4.10. Assume that conditions (c0), (c1) and (c3) holds. Then there exists a constant ~ 40 such that for every n A N M ~: E supd21 ðxn ðtÞ; 〈0〉Þr M tAI

Proof. For fixed n AN and t A I we obtain "

Z

E sup d21 ðxn ðuÞ; 〈0〉Þr ðm þ 2Þ Ed21 ðx0 ; 〈0〉Þ þ E sup d21 u A ½0;t

u A ½0;t

 Z

m

þ ∑ E sup k¼1

r ðm þ 2Þ

u A ½0;t

u

d21

#

 g ðs; xn  1 ðsÞÞ dB ðsÞ ; 〈0〉

0

Ed21 ðx0 ; 〈0〉Þ

 f ðs; xn  1 ðsÞÞ ds; 〈0〉

0

k



u

k

Z

t

þ tE

d21 ðf ðs; xn  1 ðsÞÞ; 〈0〉Þ ds Z t m þ4 ∑ E d21 ð〈gk ðs; xn  1 ðsÞÞ〉; 〈0〉Þ ds k¼1  0 Z t 2 2 r ðm þ 2Þ Ed1 ðx0 ; 〈0〉Þ þ ðt þ 4mÞE Cðs; d1 ðxn  1 ðsÞ; 〈0〉ÞÞ ds 0

0

" r ðm þ 2Þ Ed21 ðx0 ; 〈0〉Þ Z

t

þðt þ 4mÞ

! C s; E sup d 21 ðxn  1 ðuÞ; 〈0〉Þ

0

u A ½0;s

# ds :

Using concavity of the function Cðs; Þ, we infer that there exist positive constants a, b such that Cðs; rÞr a þ br for every ðs; rÞA I  Rþ . Exploiting this property we can write E sup d21 ðxn ðuÞ; 〈0〉Þr ðm þ 2Þ½Ed21 ðx0 ; 〈0〉Þ þ ðT þ 4mÞTa u A ½0;t

Z

t

þðm þ 2ÞðT þ 4mÞb 0

E sup d21 ðxn  1 ðuÞ; 〈0〉Þ ds: u A ½0;s

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

22

Further, for k A N we have



max E sup d21 ðxn ðuÞ; 〈0〉Þr ðm þ 2Þ Ed21 ðx0 ; 〈0〉Þ þ ðT þ 4mÞTa

1 r n r k u A ½0;t

þðm þ 2ÞðT þ 4mÞbTEd21 ðx0 ; 〈0〉Þ Z t max E sup d 21 ðxn ðuÞ; 〈0〉Þ ds: þðm þ 2ÞðT þ 4mÞb 0

1 r n r k u A ½0;s

Application of Gronwall's inequality leads to the end of the proof.

□

Theorem 4.11. Let x0 : Ω-F c ðRd Þ satisfy (c0) and let f : I  Ω  F c ðRd Þ-F c ðRd Þ, gk : I  Ω  F c ðRd Þ-Rd , k ¼ 1; 2; …; m, satisfy (c1)–(c3). Then Eq. (4.1) has a unique solution. Proof. We will proceed in a very similar manner as in the proof of Theorem 3.7. In fact we have, for n; ℓ A N [ f0g and t A I, E sup d 21 ðxnþ1 ðuÞ; xℓþ1 ðuÞÞ u A ½0;t

 Z t d21 ðf ðs; xn ðsÞÞ; f ðs; xℓ ðsÞÞÞ ds r ðm þ 1Þ tE 0 Z t m þ4 ∑ E ‖gk ðs; xn ðsÞÞ  gk ðs; xℓ ðsÞÞ‖2 ds 0 k¼1 Z t r ðm þ 1Þðt þ 4mÞ ELðs; d21 ðxn ðsÞ; xℓ ðsÞÞÞ ds 0

Z

t

r ðm þ 1ÞðT þ 4mÞ

L s; E

0

sup d 21 ðxn ðuÞ; xℓ ðuÞÞ u A ½0;s

! ds:

Defining a function g : I-Rþ as gðtÞ ¼ lim supn;ℓ-1 ðEsupu A ½0;t d21 ðxn ðuÞ; xℓ ðuÞÞÞ for t A I we obtain Z t gðtÞr ðm þ 1ÞðT þ 4mÞ Lðs; gðsÞÞ ds for t A I: 0

Hence g 0, and consequently limn;ℓ-1 Ed 21 ðxn ðtÞ; xℓ ðtÞÞ ¼ 0 for every t A I. Since metric space ðL2 ðΩ; At ; P; F c ðRd ÞÞ; δ2 Þ is complete, we infer that for every t A I there exists xt A L2 ðΩ; At ; P; F c ðRd ÞÞ such that Ed21 ðxn ðtÞ; xt Þ⟶0

as n-1:

Hence we can define a fuzzy stochastic process x : I  Ω-F c ðRd Þ by the formula xðt; ωÞ ¼ xt ðωÞ and this process is fAt g-adapted. Since limn;ℓ-1 ðEsupu A I d21 ðxn ðuÞ; xℓ ðuÞÞÞ ¼ 0, using Chebyshev's inequality we obtain: for every ε40   P supd21 ðxn ðuÞ; xℓ ðuÞÞ4ε ⟶0 as n; ℓ-1: uAI

P1

This implies that there exists a subsequence fxnk ð; Þg such that supu A I d1 ðxnk ðuÞ; xðuÞÞ⟶0 as k-1. Therefore x is d1 continuous and measurable. Hence x is also nonanticipating. Since R E I d21 ðxðtÞ; 〈0〉Þ dt r Tsupt A I Ed21 ðxðtÞ; 〈0〉Þo1, we obtain that X A L2 ðI  Ω; N ; KðRd ÞÞ. Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

23

In the similar way as in the proof of Theorem 3.7 we can show that 

Z t  Z t m Ed21 xðtÞ; x0

f ðs; xðsÞÞ ds

gk ðs; xðsÞÞ dBk ðsÞ ¼0 k¼1

0

0

for t A I. Using this and d 1 continuity of x we get 

Z t  Z t m I P1 k k d1 xðtÞ; x0

f ðs; xðsÞÞ ds

g ðs; xðsÞÞ dB ðsÞ ¼ 0: k¼1

0

0

Finally, we note that x is a unique solution. If x; y were two solutions to Eq. (4.1), then we would have (for t A I) ! Z t 2 2 L s; E sup d 1 ðxðsÞ; yðsÞÞ ds: E sup d1 ðxðuÞ; yðuÞÞ r ðm þ 1ÞðT þ 4mÞ u A ½0;t

u A ½0;s

0

Due to assumption (c2)(iv) we obtain Esupu A I d 21 ðxðuÞ; yðuÞÞ ¼ 0 and the uniqueness of solution x follows. □ Let us consider Eq. (4.2) and the following equations: m

I P1

dxn ðtÞ ¼ f n ðt; xn ðtÞÞ dt 〈gkn ðt; xn ðtÞÞ dBk ðtÞ〉; k¼1

P1

xn ð0Þ ¼ x0;n ;

ð4:8Þ

for n A N. Theorem 4.12. Let x0 ; x0;n : Ω-F c ðRd Þ, nA N, satisfy (c0). Let f : I  Ω  F c ðRd Þ-F c ðRd Þ and gk : I  Ω  F c ðRd Þ-Rd , k ¼ 1; 2; …; m, satisfy (c1)–(c3). Assume that for every nA N the mappings f n : I  Ω  F c ðRd Þ-F c ðRd Þ and gkn : I  Ω  F ðRd Þ-Rd , k ¼ 1; 2; …; m, satisfy (c1)–(c3) with the functions L and C which do not depend on n. Suppose that Ed21 ðx0;n ; x0 Þ⟶0

as n-1 1

2

m

and there are sequences fM fn g; fM gn g; fM gn g; …; fM gn g  ½0; 1Þ converging to zero such that Ed21 ðf n ðt; uÞ; f ðt; uÞÞ r M fn ;

k

Ed21 ðgkn ðt; uÞ; gk ðt; uÞÞ r M gn ; k ¼ 1; 2; …; m;

for every ðt; uÞA I  F c ðRd Þ. Then the solution x to Eq. (4.2) and the solution xn to Eq. (4.8) satisfy E supd21 ðxn ðtÞ; xðtÞÞ⟶0 tAI

as n-1:

Proof. Similar to the proof of Theorem 3.8 we obtain " E sup d21 ðxn ðuÞ; xðuÞÞ r ðm þ 2Þ Ed21 ðx0;n ; x0 Þ u A ½0;t

Z þ2t

t

Ed21 ðf n ðs; xðsÞÞ; f ðs; xðsÞÞÞ ds Z t þ8 ∑ Ed21 ðgkn ðs; xðsÞÞ; gkn ðs; xðsÞÞÞ ds 0 m

k¼1

0

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

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Z þ2ðt þ 4mÞ

t

! L s; E

0

sup d 21 ðxn ðuÞ; xðuÞÞ u A ½0;s

# ds :

ð4:9Þ

Therefore, like in the set-valued case, we can infer that Esupu A ½0;t d 21 ðxn ðuÞ; xðuÞÞ is bounded for every t A I and the expression Lðt; Esupu A ½0;t d21 ðxn ðuÞ; xðuÞÞÞ is well defined and the function t↦Lðt; Esupu A I d21 ðxn ðuÞ; xðuÞÞÞ is integrable. Due to Eq. (4.9), the Lebesgue dominated convergence theorem and the assumptions we obtain lim E sup d21 ðxn ðuÞ; xðuÞÞ

n-1

u A ½0;t

Z

r 2ðm þ 2ÞðT þ 4mÞ

!

t

L 0

s; lim E sup d 21 ðxn ðuÞ; xðuÞÞ n-1 u A ½0;s

ds:

In view of (c2)(iv), we get lim E sup d21 ðxn ðuÞ; xðuÞÞ ¼ 0

n-1

u A ½0;t

for every t A I:

□

Similarly like Corollary 3.9 is a consequence of set-valued stochastic results for deterministic set-valued differential equations, we can infer about the existence and uniqueness of a solution to the initial value problem of deterministic fuzzy differential equations x0 ðtÞ ¼ f ðt; xðtÞÞ for t A I;

xð0Þ ¼ x0 ;

ð4:10Þ

0

where denotes the fuzzy-set-valued Hukuhara derivative for fuzzy mappings, see [26], f : I  F c ðRd Þ-F c ðRd Þ, x0 A F c ðRd Þ. This result for Eq. (4.10) can be established with measurability of the mapping f instead of continuity. This section contains the theoretical results which show a collection of conditions under which the fuzzy stochastic differential equations possess unique solutions. Such issue is very important, since usually it is impossible to determine the solutions in closed, explicit forms. The next section is aimed in presentation of an application of the fuzzy differential equations. 5. An example of applications Let us consider a situation of a biologist that grows a colony of a microbe. The size of this population can be controlled by the biologist by steering amount of nourishment which the population receives. Assume that the feeding takes place instantly at every moment of time and let the control function c : I-R reflects relative changes in the feeding. If the control c takes on positive values, this corresponds to the increasing amount of nourishment. Also, the negative values of c symbolize that the biologist took a decision on a curtailment of the food. Taking into account that some stochastic noises can affect the number of individuals, one of the first models of population density y can be formulated as a crisp stochastic differential I P1 P1 equation dyðtÞ ¼ cðtÞyðtÞ dt þ νdBðtÞ, yð0Þ ¼ y0 , where ν is a positive constant and y0 is a realvalued random variable. However, no lab assistant is able to determine the precise values of the initial density y0. The technician observes the population under the microscope and the microbes move and change their positions. Some results of his perception can be communicated as linguistic expressions only, since he is not able to count the number of individuals precisely. He is able to say that the density is, for instance, “small”, “large”, “huge”, etc. Such information formulated in terms of the Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

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linguistic variables can be modelled by using fuzzy sets. Hence instead of crisp equation mentioned above we propose to consider the following fuzzy stochastic differential equation: I P1

dxðtÞ ¼ cðtÞxðtÞ dt 〈ν dBðtÞ〉;

P1

xð0Þ ¼ x0 ;

ð5:1Þ

where x0 : Ω-F c ðR Þ is a fuzzy random variable. Since the initial density is fuzzy-set-valued, it is clear that the density x at the instant t should also be fuzzy-set-valued. In fact, the solution to Eq. (5.1) will be a fuzzy stochastic process x : I  Ω-F c ðR1 Þ. Notice that Eq. (5.1) fits better in a realistic framework of the considered situation, since it does not neglect a presence of imprecision. In what follows we shall find solution to Eq. (5.1), solution in closed form. As it will be showed it is possible to give such form in this case. However, one should be aware that in general it is impossible to find explicit form of the solutions to fuzzy stochastic differential equations. Hence in the future, some approximation schemes will be of a great interest. It is easy to check that the coefficients of Eq. (5.1) satisfy assumptions of Theorem 4.11. Hence we can conclude that a unique solution x to Eq. (5.1) exists. To deduce its form we will determine the boundaries of the α-levels of x. To this end for every α A ½0; 1 we define the   þ random variables x0;α ; xþ 0;α : Ω-R and the stochastic processes xα ; xα : I  Ω-R by 1

α  ; xþ ½x0;α 0;α  ¼ ½x0  ; α ½xα ðtÞ; xþ α ðtÞ ¼ ½xðtÞ :

The sign of control function will have an influence on the solution. Therefore we present two particular considerations for positive and negative function c. Firstly, we assume that the biologist, who is a controller, has observed that the density is “small” and decided to make an adjustment to increase the nourishment according to the function cðtÞ ¼ t=2, t A I. Since x is the solution to Eq. (5.1) we infer that Z t Z t s þ I P1 þ x xþ ð t Þ ¼ x þ ð s Þ ds þ ν dBðsÞ; α 0;α 2 α Z0 t Z0 t s  I P1  xα ðsÞ ds þ xα ðt Þ ¼ x0;α þ ν dBðsÞ: 2 0 0 The solutions of these crisp stochastic differential equations are the processes   Z t I P1 t 2 =4 þ  s2 =4 xþ ðtÞ ¼ e x þ ν e dBðsÞ α 0;α 0

and I P1 2 xα ðtÞ ¼ et =4

  Z t   s2 =4 x0;α þ ν e dBðsÞ 0

This enables us to deduce that the solution x to Eq. (5.1) with cðtÞ ¼ t=2 is of the form

 Z t I P1 t 2 =4 t 2 =4  s2 =4 xðtÞ ¼ e  x0 νe e dBðsÞ : 0

To simulate a path of the solution established, we assume that for a chosen ωn the initial density x0 ðωn Þ is a fuzzy set with the support ½x0 ðωn Þ0 ¼ ½100; 200. The graph of support of solution is illustrated in Fig. 1. Now assume that the controller has seen under the microscope that the density is “huge” and taken an action of decline in the nourishment according to the function cðtÞ ¼  t=2, t A I. In this Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

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case we arrive at þ xþ α ðt Þ ¼ x0;α  I P1

I P1   xα ðt Þ ¼ x0;α

Z

t

s  x ðsÞ ds þ 2 α

0

Z

t

0

s þ x ðsÞ ds þ 2 α

Z

t

ν dBðsÞ;

0

Z

t

ν dBðsÞ

0

and this system of equations possesses solution given by the stochastic processes  2  2 Z t t t 2 I P1  þ  t 2 =4 xþ ð t Þ ¼ sinh þ cosh þ νe es =4 dBðsÞ x x α 0;α 0;α 4 4 0 and xα ðt Þ ¼ cosh I P1



  2 Z t  t2  t 2  t 2 =4 þ νe es =4 dBðsÞ: x0;α þ sinh xþ 0;α 4 4 0

Hence one can infer that the solution x to Eq. (5.1) with cðtÞ ¼  t=2 is as follows:   

 2  2 Z t t t 2 2 I P1 es =4 dBðsÞ : xðt Þ ¼ cosh  x0  sinh  x0 νe  t =4 4 4 0 Since coshðt 2 =4Þ40 and ð sinhðt 2 =4ÞÞo0 R t 2 for positive t, one cannot write this solution in the 2 2 I P1 form xðtÞ ¼ e  t =4  x0 〈νe  t =4 0 es =4 dBðsÞ〉. Hence the solution does depend on the sign of the control function c. A simulated path of the support of solution x to Eq. (5.1) with cðtÞ ¼  t=2 is drawn in Fig. 2. For this simulation we assume that the initial value of the density is a fuzzy set with the support ½x0 ðωnn Þ0 ¼ ½7000; 8000, where ωnn is fixed. The presented example is a basic one. However it can serve as a foundation to more complex models in the future. By this example we show a method of finding the closed, explicit form of the fuzzy solutions. Also we indicate that examinations of fuzzy stochastic differential equations require carefulness, since, for instance, a change of the sign of a parameter can lead to different forms of the solution.

Fig. 1. Support of solution to Eq. (5.1) with cðtÞ ¼ t=2. Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

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Fig. 2. Support of solution to Eq. (5.1) with cðtÞ ¼  t=2.

6. Concluding remarks The paper is concerned with nonlinear set-valued and fuzzy stochastic differential equations. Such equations involve two sources of uncertainties that appear in real-world problems, i.e. ambiguity driven by set-valued and fuzzy-set-valued mappings and randomness caused by stochastic noises. The main aim of the paper is to derive some new results for the existence and uniqueness of solutions to the equations under considerations. They are achieved using a successive approximation approach. We impose a non-Lipschitz condition on the coefficients of the equations. Up to now, this condition is the weakest one. An insensitivity of the solution with respect to small changes of the coefficients is showed as well. To illustrate a potential applicability of the considered theory, a control system in biology is modelled in terms of a fuzzy stochastic differential equation. The model is chosen so that we are able to find its explicit solution in a closed form. However, in general the explicit forms of solutions do not exist or it is a very hard task to find them. Therefore some methods of establishing approximate solutions will be needed in the future. Such approximate solutions should converge to the exact solutions. A rate of convergence could be investigated to reflect how fast the method is. The solutions considered in this paper have continuous sample paths. One can study the fuzzy stochastic differential equations with the Poisson jump process with discontinuous solutions. They could be applied to phenomena that are subjected to shocks and breakdowns. Acknowledgements The authors would like to thank the anonymous referees for their comments that improved presentation of the paper.

References [1] Ju.I. Alimov, On the application of Lyapunov's direct method to differential equations with ambiguous right sides, Autom. Remote Control 22 (1961) 713–725. Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

28

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

[2] E.A. Barbashin, Ju.I. Alimov, On the theory of dynamical systems with multi-valued and discontinuous characteristics, Dokl. Akad. Nauk SSSR 140 (1961) 9–11. [3] E.A. Barbashin, Ju.I. Alimov, On the theory of rely differential equations, Izv. Vyss. Ucebn. Zaved. Mat. 1 (26) (1962) 3–13. [4] E.A. Barbashin, Introduction to the Theory of Stability, Wolters-Noordhoff Publishing, Groningen, 1970. [5] F.S. De Blasi, F. Iervolino, Equazioni differenziali con soluzioni a valore compatto convesso, Boll. Unione Mat. Ital. 4 (1969) 491–501. [6] R.P. Agarwal, D. O'Regan, Existence for set differential equations via multivalued operator equations, Differential equations and applications, Nova Science Publishers, New York, 2007, pp. 1–5. [7] B. Ahmad, S. Sivasundaram, Dynamics and stability of impulsive hybrid setvalued integro-differential equations with delay, Nonlinear Anal. Theory Methods Appl. 65 (2006) 2082–2093. [8] B. Ahmad, S. Sivasundaram, The monotone iterative technique for impulsive hybrid set valued integro-differential equations, Nonlinear Anal. Theory Methods Appl. 65 (2006) 2260–2276. [9] A.I. Brandão Lopes Pinto, F.S. De Blasi, F. Iervolino, Uniqueness and existence theorems for differential equations with convex valued solutions, Boll. Unione Mat. Ital. 4 (1970) 1–12. [10] Z. Drici, F.A. McRae, J. Vasundhara Devi, Stability results for set differential equations with causal maps, Dyn. Syst. Appl. 15 (2006) 451–463. [11] G.N. Galanis, T. Gnana Bhaskar, V. Lakshmikantham, P.K. Palamides, Set valued functions in Fréchet spaces: continuity, Hukuhara differentiability and applications to set differential equations, Nonlinear Anal. Theory Methods Appl. 61 (2005) 559–575. [12] T. Gnana Bhaskar, V. Lakshmikantham, J. Vasundhara Devi, Nonlinear variation of parameters formula for set differential equations in a metric space, Nonlinear Anal. Theory Methods Appl. 63 (2005) 735–744. [13] M. Kisielewicz, Description of a class of differential equations with set-valued solutions, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975) 158–162. [14] V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Stability theory for set differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 11 (2004) 181–189. [15] V. Lakshmikantham, A.S. Vatsala, Set differential equations and monotone flows, Nonlinear Dyn. Stability Theory 3 (2003) 33–43. [16] M.T. Malinowski, Interval differential equations with a second type Hukuhara derivative, Appl. Math. Lett. 24 (2011) 2118–2123. [17] M.T. Malinowski, Interval Cauchy problem with a second type Hukuhara derivative, Inf. Sci. 213 (2012) 94–105. [18] M.T. Malinowski, Second type Hukuhara differentiable solutions to the delay set-valued differential equations, Appl. Math. Comput. 218 (2012) 9427–9437. [19] M.T. Malinowski, On set differential equations in Banach spaces a second type Hukuhara differentiability approach, Appl. Math. Comput. 219 (2012) 289–305. [20] F.A. McRae, J. Vasundhara Devi, Impulsive set differential equations with delay, Appl. Anal. 84 (2005) 329–341. [21] J. Vasundhara Devi, A.S. Vatsala, A study of set differential equations with delay, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 11 (2004) 287–300. [22] V. Lakshmikantham, T. Gnana Bhaskar, J. Vasundhara Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge, 2006. [23] O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst. 24 (1987) 301–317. [24] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353. [25] H. Zhang, D. Liu, Fuzzy Modelling and Fuzzy Control, Birkhäuser, Boston, 2006. [26] V. Lakshmikantham, R.N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, London, 2003. [27] E. Hülleremeier, An approach to modelling and simulation of uncertain dynamical systems, Int. J. Uncertain. Fuzziness Knowledge-Based Syst. 5 (1997) 117–137. [28] R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal. Theory Methods Appl. 72 (2010) 2859–2862. [29] R.P. Agarwal, D. O'Regan, V. Lakshmikantham, A stacking theorem approach for fuzzy differential equations, Nonlinear Anal. Theory Methods Appl. 55 (2003) 299–312. [30] R.P. Agarwal, D. O'Regan, V. Lakshmikantham, Viability theory and fuzzy differential equations, Fuzzy Sets Syst. 151 (2005) 563–580. [31] L.C. Barros, L.T. Gomes, P.A. Tonelli, Fuzzy differential equations: an approach via fuzzification of the derivative operator, Fuzzy Sets Syst. 230 (2013) 39–52. Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

29

[32] B. Bede, S.G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets Syst. 151 (2005) 581–599. [33] P. Diamond, Brief note on the variation of constants formula for fuzzy differential equations, Fuzzy Sets Syst. 129 (2002) 65–71. [34] A. Khastan, J.J. Nieto, R. Rodríguez-López, Periodic boundary value problems for first-order linear differential equations with uncertainty under generalized differentiability, Inf. Sci. 222 (2013) 544–558. [35] V. Lakshmikantham, Uncertain systems and fuzzy differential equations, J. Math. Anal. Appl. 251 (2000) 805–817. [36] V. Lakshmikantham, Set differential equations versus fuzzy differential equations, Appl. Math. Comput. 164 (2005) 277–294. [37] V. Lakshmikantham, S. Leela, A.S. Vatsala, Interconnection between set and fuzzy differential equations, Nonlinear Anal. Theory Methods Appl. 54 (2003) 351–360. [38] V. Lakshmikantham, A.A. Tolstonogov, Existence and interrelation between set and fuzzy differential equations, Nonlinear Anal. Theory Methods Appl. 55 (2003) 255–268. [39] J.J. Nieto, The Cauchy problem for continuous fuzzy differential equations, Fuzzy Sets Syst. 102 (1999) 259–262. [40] J.J. Nieto, R. Rodríguez-López, Some results on boundary value problems for fuzzy differential equations with functional dependence, Fuzzy Sets Syst. 230 (2013) 92–118. [41] S. Pederson, M. Sambandham, Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem, Inf. Sci. 179 (2009) 319–328. [42] M.T. Malinowski, On random fuzzy differential equations, Fuzzy Sets Syst. 160 (2009) 3152–3165. [43] M.T. Malinowski, Existence theorems for solutions to random fuzzy differential equations, Nonlinear Anal. Theory Methods Appl. 73 (2010) 1515–1532. [44] M.T. Malinowski, Random fuzzy differential equations under generalized Lipschitz condition, Nonlinear Anal. Real World Appl. 13 (2012) 860–881. [45] M.T. Malinowski, Random fuzzy fractional integral equations—theoretical foundations, Fuzzy Sets Syst., http://dx. doi.org/10.1016/j.fss.2014.09.019, in press. [46] L. Chen, R. Wu, D. Pan, Mean square exponential stability of impulsive stochastic fuzzy cellular neural networks with distributed delays, Expert Syst. Appl. 38 (2011) 6294–6299. [47] Y.G. Niu, D.W. Ho, J. Lam, Robust integral sliding mode control for uncertain stochastic systems with time-varying delay, Automatica 41 (2005) 873–880. [48] T. Wang, S. Tong, Y. Li, Robust adaptive fuzzy control for a class of stochastic nonlinear systems with dynamical uncertainties, J. Frankl. Inst. 349 (2012) 3121–3141. [49] L.G. Wu, Z.D. Wang, Fuzzy filtering of nonlinear fuzzy stochastic systems with time-varying delay, Signal Process. 89 (2009) 1739–1753. [50] Z. Xia, J. Li, J. Li, Passivity-based resilient adaptive control for fuzzy stochastic delay systems with Markovian switching, J. Frankl. Inst. 351 (2014) 3818–3836. [51] C.-D. Zheng, Y. Wang, Z. Wang, Stability of stochastic fuzzy Markovian jumping neural networks with leakage delay under impulsive perturbations, J. Frankl. Inst. 351 (2014) 1728–1755. [52] M.T. Malinowski, Strong solutions to stochastic fuzzy differential equations of Itô type, Math. Comput. Model. 55 (2012) 918–928. [53] M.T. Malinowski, Itô type stochastic fuzzy differential equations with delay, Syst. Control Lett. 61 (2012) 692–701. [54] M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inf. Sci. 252 (2013) 62–80. [55] M.T. Malinowski, Approximation schemes for fuzzy stochastic integral equations, Appl. Math. Comput. 219 (2013) 11278–11290. [56] M.T. Malinowski, On a new set-valued stochastic integral with respect to semimartingales and its applications, J. Math. Anal. Appl. 408 (2013) 669–680. [57] M.T. Malinowski, Fuzzy and set-valued stochastic differential equations with local Lipschitz condition, IEEE Trans. Fuzzy Syst., http://dx.doi.org/10.1109/TFUZZ.2014.2372072, in press. [58] W. Fei, Existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non-Lipschitzian condition, Nonlinear Anal. Theory Methods Appl. 76 (2013) 202–214. [59] W. Fei, D. Xia, On solutions to stochastic set differential equations of Itô type under the non-Lipschitzian condition, Dyn. Syst. Appl. 22 (2013) 137–156. [60] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin, 1977. [61] F. Hiai, H. Umegaki, Integrals, conditional expectation, and martingales of multivalued functions, J. Multivar. Anal. 7 (1977) 149–182. [62] I.I. Gihman, A.V. Skorohod, Stochastic Differential Equations, Springer-Verlag, Berlin, 1972. [63] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chichester, 2007. Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010

30

M.T. Malinowski, R.P. Agarwal / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

[64] T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differ. Equ. 96 (1992) 152–169. [65] T. Yamada, On the successive approximations of solutions of stochastic differential equations, J. Math. Kyoto Univ. 21 (1981) 501–515. [66] E.J. Jung, J.H. Kim, On set-valued stochastic integrals, Stoch. Anal. Appl. 21 (2003) 401–418. [67] M. Kisielewicz, Some properties of set-valued stochastic integrals, J. Math. Anal. Appl. 388 (2012) 984–995. [68] M.L. Puri, D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409–422. [69] Y.K. Kim, Measurability for fuzzy valued functions, Fuzzy Sets Syst. 129 (2002) 105–109. [70] S.Y. Joo, Y.K. Kim, J.S. Kwon, G.S. Choi, Convergence in distribution for level-continuous fuzzy random sets, Fuzzy Sets Syst. 157 (2006) 243–255. [71] Y. Feng, Mean-square integral and differential of fuzzy stochastic processes, Fuzzy Sets Syst. 102 (1999) 271–280.

Please cite this article as: M.T. Malinowski, R.P. Agarwal, On solutions to set-valued and fuzzy stochastic differential equations, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.11.010