Some properties of strong solutions to stochastic fuzzy differential equations

Information Sciences xxx (2013) xxx–xxx

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Some properties of strong solutions to stochastic fuzzy differential equations Marek T. Malinowski ⇑ Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

a r t i c l e

i n f o

Article history: Received 22 September 2011 Received in revised form 19 February 2013 Accepted 23 February 2013 Available online xxxx Keywords: Epistemic and random uncertainty Stochastic fuzzy differential equation Set-valued stochastic differential equation Stochastic fuzzy integral equation Fuzzy stochastic Lebesgue–Aumann integral Fuzzy stochastic Itô integral

a b s t r a c t We consider stochastic fuzzy differential equations driven by m-dimensional Brownian motion. Such equations can be useful in modeling of hybrid dynamic systems, where the phenomena are subjected to two kinds of uncertainties: randomness and fuzziness, simultaneously. Under a boundedness condition, which is weaker than linear growth condition, and the Lipschitz condition we obtain existence and uniqueness of solution to stochastic fuzzy differential equations. Solutions, which are fuzzy stochastic processes, and their uniqueness are considered to be in a strong sense. An estimation of error of the Picard approximate solution is established. We give a boundedness type result for the solution defined on finite time interval. Also the stabilities of solution on initial condition and coefficients of the equation are shown. The existence and uniqueness of a solution defined on infinite time interval is proven. Finally, some applications of fuzzy stochastic differential equations are considered. All the results presented in this paper apply to set-valued stochastic differential equations.  2013 Elsevier Inc. All rights reserved.

1. Introduction Stochastic differential equations are used in numerous applications to model classical problems in control theory, physics, biology, economics and engineering. The theory of such equations and their solutions being stochastic processes can be found in [1,5,31]. In such studies, random disturbances are the only source of uncertainty. To handle these situations, methods of stochastic analysis are used. However, in the real world problems we encounter a second source of uncertainty: vagueness (sometimes called imprecision, fuzziness, ambiguity, softness). This is mostly observed when a state of a considered system is described by linguistic variables (for example ‘‘around 5’’, ‘‘cold’’, ‘‘large’’ and so on). It is known that the fuzzy set theory plays an appropriate role to deal with such type of uncertainty (see, e.g. [34,35,37]). Having imprecise, incomplete or vague informations on parameters of a considered system, we have to reflect impreciseness in a model of the system. Therefore in this paper we present the studies on stochastic fuzzy differential equations, which can be useful in modeling of hybrid dynamic systems, where the phenomena are subjected to two kinds of uncertainties: randomness and fuzziness, simultaneously. In integral form of fuzzy stochastic differential equations we exploit the fuzzy stochastic integrals of Lebesgue–Aumann and Itô types, which are the fuzzy random variables. The notion of fuzzy random variable has been proposed to handle the linguistic and imprecisely valued random variables. A research in this topic can be found, e.g. in [2,6,33]. In [3,4,13,14,16] one can find a research concerning random fuzzy differential equations, which can be appropriate in modeling of the dynamic systems where two kinds of uncertainty, i.e. randomness and fuzziness, are incorporated. Fuzzy ⇑ Tel.: +48 68 3282826. E-mail address: [email protected] 0020-0255/$ - see front matter  2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2013.02.053

Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

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M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

random variables and fuzzy stochastic processes play important role in such studies. The fuzzy stochastic processes being the solutions to random fuzzy differential equations have trajectories which are differentiable in sense of a fuzzy derivative involved in a formulation of a differential problem. In more complex systems, where so-called ‘‘white noise’’ acts, stochastic set differential equations (see [25–27,29]) and stochastic fuzzy differential equations (see [9,17,18,22–24,30]) could be some of the best tools in modeling phenomena with uncertainties. The papers devoted to stochastic fuzzy differential equations offer a few views on formulating such equations and defining their solutions. A main thing which causes this variety of approaches is diversity in a concept of fuzzy stochastic Itô integral. Its crisp counterpart, being a random variable, is a fundamental notion of stochastic analysis. In [9] one can find a definition of fuzzy Itô integral. However, in [36] it is stated that the intersection property of a set-valued Itô integral (a crucial one in defining fuzzy stochastic Itô integral as in [9]) may not hold true in general. Thus the definition of fuzzy stochastic Itô integral proposed in [9] seems to be questionable. On the other hand, in [30] a presented approach in the treatment of fuzzy stochastic differential equations does not contain any notion of fuzzy stochastic Itô integral. A method utilized there is based on selections sets. Therefore, in [17,18,23,24,26] some new ways of defining the notion of fuzzy stochastic Itô integral are proposed. In the papers [23,26], the fuzzy stochastic Itô integral is defined as a fuzzy set of the space of square integrable random vectors and is not a fuzzy random variable. The next method, contained in [17,18,24], deals with fuzzy stochastic Itô integral as the embedding of classical d-dimensional Itô integral into fuzzy sets space, and then the fuzzy Itô integral is a fuzzy random variable. In the papers [17,23,24] the authors consider stochastic fuzzy differential equations with corresponding fuzzy Itô integrals. In each paper we propose to consider a different type of solution. Thus there appear the stochastic solutions of a strong type [17], a mean-square type [24], and the solutions which incorporate randomness in the set of values [23]. In this paper we will use the definition of fuzzy Itô integral given in [17,18,24], as we want to work with fuzzy random variables and fuzzy stochastic processes. We present the studies on solutions to stochastic fuzzy differential equations driven by m-dimensional Brownian motion. All the equation coefficients are considered to be random. In this work we deal with strong solutions. Also, their uniqueness is proposed to be in a strong sense, i.e. we consider pathwise uniqueness as in [17]. To obtain a result on existence of strong solution we impose a boundedness condition on the equation coefficients and this assumption is weaker than a linear growth condition used in [17]. Then we examine some properties of strong solutions. We would like to mention that all the results established in this paper oriented towards stochastic fuzzy differential equations apply straightforwardly to set-valued stochastic differential equations. This is because every ordinary set is also a fuzzy set. The paper is organized as follows: in Section 2 we give some preliminaries on measurable multifunctions, fuzzy random variables, fuzzy stochastic processes, fuzzy stochastic integrals of Lebesgue–Aumann type and Itô type to make the paper self-contained in these subjects. In Section 3 we prove an existence and uniqueness theorem for stochastic fuzzy differential equations. This part of the paper concerns the strong solutions existing on a finite interval [0, T]. An estimation of the error of the Picard approximate solution is obtained. We establish a boundedness type result for the solution and prove the stabilities of solution on initial condition and coefficients of the equation. In Section 4 we show the existence and uniqueness of strong solution defined on the whole half-line [0, 1). Then in Section 5 we apply stochastic fuzzy differential equations in a model of population evolution and in a model of financial interest rate. Finally, Section 6 contains some concluding remarks and some potential directions in a further future research.

2. Preliminaries To study stochastic fuzzy differential equations we need some preliminaries. Since the topic is new, for convenience of the reader we summarize the relevant material on measurable multifunctions, fuzzy random variables, fuzzy stochastic processes and fuzzy stochastic integrals, thus making our exposition self-contained. 2.1. Measurable multifunctions 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 :¼ max sup inf ka  bk; sup inf ka  bk ; a2A b2B

b2B a2A

where kk denotes a norm in Rd . It is known that KðRd Þ is a complete and separable metric space with respect to dH. If A; B; C 2 KðRd Þ, we have dH(A + C, B + C) = dH(A, B) (see, e.g. [12]). Let ðX; A; PÞ be a complete probability space and MðX; A; KðRd ÞÞ denote the family of A-measurable multifunctions with values in KðRd Þ, i.e. the mappings F : X ! KðRd Þ such that

fx 2 X : FðxÞ \ O – ;g 2 A for every open set O  Rd :

Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

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A multifunction F 2 MðX; A; KðRd ÞÞ is said to be Lp-integrably bounded, p P 1, if there exists h 2 Lp ðX; A; P; RÞ such that kak 6 h(x) for any a and x with a 2 F(x). It is known (see [7]) that F is Lp-integrably bounded iff x ´ kjF(x)kj is in Lp ðX; A; P; RÞ, where

kjAjk :¼ dH ðA; f0gÞ ¼ supkak for A 2 KðRd Þ a2A

and Lp ðX; A; P; RÞ is a space of equivalence classes (with respect to the equality P-a.e.) of A-measurable random variables R h : X ! R such that Ejhjp ¼ X jhjp dP < 1. Let us denote

Lp ðX; A; P; KðRd ÞÞ :¼ fF 2 MðX; A; KðRd ÞÞ : F is Lp  integrably boundedg: The multifunctions F; G 2 Lp ðX; A; P; KðRd ÞÞ are considered to be identical, if F = G holds P-a.e. 2.2. Fuzzy sets and fuzzy random variables 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 2 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. Let us remark that every ordinary set u in Rd is a fuzzy set, since then u(x) = 1 if x 2 u and u(x) = 0 if x R u. Let F ðRd Þ denote the fuzzy sets u : Rd ! ½0; 1 such that ½ua 2 KðRd Þ for every a 2 [0, 1], where ½ua :¼ fa 2 Rd : uðaÞ P ag for a 2 (0, 1] and ½u0 :¼ clfa 2 Rd : uðaÞ > 0g. Addition and scalar multiplication in fuzzy set space F ðRd Þ can be defined levelwise:

½u þ v a ¼ ½ua þ ½v a ;

½kua ¼ k½ua ;

where u; v 2 F ðRd Þ; k 2 R and a 2 [0, 1]. Define d1 : F ðRd Þ  F ðRd Þ ! ½0; 1Þ by the expression

d1 ðu; v Þ :¼ sup dH ð½ua ; ½v a Þ; a2½0;1

where dH is the Hausdorff metric in KðRd Þ. It is easy to see that d1 is a metric in F ðRd Þ. In fact ðF ðRd Þ; d1 Þ is a complete metric space, and for every u; v ; w; z 2 F ðRd Þ; k 2 R one has (see, e.g. [32])

d1 ðu þ w; v þ wÞ ¼ d1 ðu; v Þ; d1 ðu þ v ; w þ zÞ 6 d1 ðu; wÞ þ d1 ðv ; zÞ; d1 ðku; kv Þ ¼ jkjd1 ðu; v Þ: It is also known that the metric space ðF ðRd Þ; d1 Þ is not separable and is not locally compact. We define h0i 2 F ðRd Þ as h0i :¼ 1{0}, where for y 2 Rd we have 1{y}(x) = 1 if x = y and 1{y}(x) = 0 if x – y. Definition 2.1 [32]. Let ðX; A; PÞ be a probability space. A mapping x : X ! F ðRd Þ is said to be a fuzzy random variable, if ½xa : X ! KðRd Þ is an A-measurable multifunction for all a 2 [0, 1]. This definition is one of the possibilities considered for fuzzy random variables. Generally, having a metric q in the set F ðRd Þ one can consider r-algebra Bq generated by the topology induced by q. Then a fuzzy random variable can be viewed as a measurable (in the classical sense) mapping between two measurable spaces, namely ðX; AÞ and ðF ðRd Þ; Bq Þ. Using the classical notation, we write this fact as: x is AjBq -measurable. Beside the metric d1 in the set F ðRd Þ the following Skorohod metric is also used

( dS ðu; v Þ :¼ inf max k2K

) sup jkðtÞ  tj; sup dH ðxu ðtÞ; xv ðkðtÞÞÞ ;

t2½0;1

t2½0;1

where K denotes the set of strictly increasing continuous functions k:[0, 1] ? [0, 1] such that k(0) = 0, k(1) = 1, and xu ; xv : ½0; 1 ! KðRd Þ are the càdlàg representations for the fuzzy sets u; v 2 F ðRd Þ, see [2,10] for details. The space ðF ðRd Þ; d1 Þ is complete and non-separable, and the space ðF ðRd Þ; dS Þ is a Polish metric space. It is known (see [2,10]) that for a mapping x : X ! F ðRd Þ, where ðX; A; PÞ is a given probability space, it holds: — x is a fuzzy random variable if and only if x is AjBdS -measurable, — if x is AjBd1 -measurable, then it is a fuzzy random variable; the opposite implication is not true. Definition 2.2. A fuzzy random variable x : X ! F ðRd Þ is said to be Lp-integrably bounded, p P 1, if ½xa 2 Lp ðX; A; P; KðRd ÞÞ for every a 2 [0, 1]. Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

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Let Lp ðX; A; P; F ðRd ÞÞ denote the set of all Lp-integrably bounded fuzzy random variables, where we consider x; y 2 Lp ðX; A; P; F ðRd ÞÞ as identical if P(d1(x, y) = 0) = 1. It is easy to see that for x : X ! F ðRd Þ being a fuzzy random variable and p P 1, the following conditions are equivalent: (a) x 2 Lp ðX; A; P; F ðRd ÞÞ, (b) ½x0 2 Lp ðX; A; P; KðRd ÞÞ, (c) x ´ kj[x(x)]0kj is in Lp ðX; A; P; RÞ. 2.3. Fuzzy stochastic processes Let T 2 (0, 1) and denote I:¼[0, T]. Let ðX; A; fAt gt2I ; PÞ be a complete, filtered probability space with a filtration fAt gt2I satisfying the usual hypotheses, i.e. fAt gt2I is an increasing and right continuous family of sub-r-algebras of A, and A0 contains all P-null sets. Definition 2.3. We call x : I  X ! F ðRd Þ a fuzzy stochastic process, if for every t 2 I a mapping xðt; Þ ¼ xðtÞ : X ! F ðRd Þ is a fuzzy random variable in the sense of Definition 2.1. Hence x can be thought as a family {x(t)}t2I of fuzzy random variables. Definition 2.4. We say that a fuzzy stochastic process x is d1-continuous, if almost all (with respect to the probability measure P) its trajectories, i.e. the mappings xð; xÞ : I ! F ðRd Þ are d1-continuous functions. A fuzzy stochastic process x is said to be fAt gt2I -adapted, if for every a 2 [0, 1] the multifunction ½xðtÞa : X ! KðRd Þ is At measurable for all t 2 I. It is called measurable, if ½xa : I  X ! KðRd Þ is a BðIÞ  A-measurable multifunction for all a 2 [0, 1], where BðIÞ denotes the Borel r-algebra of subsets of I. Definition 2.5. If x : I  X ! F ðRd Þ is fAt gt2I -adapted and measurable, then it will be called nonanticipating. Equivalently, x is nonanticipating if and only if for every a 2 [0, 1] the multifunction [x]a is measurable with respect to the

r-algebra N , which is defined as follows N :¼ fA 2 BðIÞ  A : At 2 At for every t 2 Ig; where At = {x: (t, x) 2 A}. p d d RLet p P 1 and  L ðI  X; N ; R Þ denote the set of all nonanticipating R -valued stochastic processes {h(t)}t2I such that T E 0 khðsÞkp ds < 1. Definition 2.6. A fuzzy stochastic process x is called Lp-integrably bounded (p P 1), if there exists a real-valued stochastic process h 2 Lp ðI  X; N ; RÞ such that

kj½xðt; xÞ0 jk 6 hðt; xÞ for a:a: ðt; xÞ 2 I  X: By Lp ðI  X; N ; F ðRd ÞÞ we denote the set of nonanticipating and Lp-integrably bounded fuzzy stochastic processes. For convenience, in the rest of the article, the fact that P(x = y) = 1, where x, y are random elements, we will often write as P:1 x ¼ y, and similarly for inequalities. Also if we will have: P(x(t) = y(t), "t 2 I) = 1, where x, y are the stochastic processes, then I P:1 we will write xðtÞ ¼ yðtÞ for short, similarly for the inequalities. 2.4. Fuzzy stochastic integrals of Lebesgue–Aumann and Itô types Let ðX; A; fAt gt2I ; PÞ be a complete, filtered probability space with a filtration fAt gt2I satisfying the usual hypotheses. Let x 2 L1 ðI  X; N ; F ðRd ÞÞ. For such x, by the Fubini Theorem, we can define the fuzzy integral

Z

T

xðs; xÞds for x 2 X n Nx ;

0

RT where N x 2 A and P(Nx) = 0. Let us recall that the fuzzy integral 0 xðs; xÞds is defined levelwise, i.e. the a-level sets of this integral are the set-valued integrals of a-level sets of x(, x) in the sense of Aumann. For the details and properties of such a RT fuzzy integral we refer to [8]. For every a 2 [0, 1] and every x 2 X n Nx the Aumann integral 0 ½xðs; xÞa ds belongs to KðRd Þ RT (see, e.g. [11]). We can show (see [18] for details) that X 3 x # 0 xðs; xÞds 2 F ðRd Þ is a fuzzy random variable. Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

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Definition 2.7. By the fuzzy stochastic Lebesgue–Aumann integral of x 2 L1 ðI  X; N ; F ðRd ÞÞ on the interval [0, t], t 2 I, we mean a fuzzy random variable Lx(t, ) expressed as

Lx ðt; xÞ ¼

(R

T 0

1½0;t ðsÞxðs; xÞds for x 2 X n N x ; for x 2 Nx :

h0i

Let us remark that such the integral can be defined for every fuzzy stochastic process x 2 Lp ðI  X; N ; F ðRd ÞÞ; p P 1. For this definition we could also relax the N -measurability of x to the measurability of x (see [18]). For the integral Lx we have the following properties (see [17,18]). Proposition 2.8. Let p P 1. If x 2 Lp ðI  X; N ; F ðRd ÞÞ then Lx ð; Þ 2 Lp ðI  X; N ; F ðRd ÞÞ. Proposition 2.9. Let x 2 L1 ðI  X; N ; F ðRd ÞÞ. Then the fuzzy stochastic process {Lx(t)}t2I is d1-continuous. Proposition 2.10. Let p P 1. Assume that x; y 2 Lp ðI  X; N ; F ðRd ÞÞ. Then it holds I P:1

p

sup d1 ðLx ðuÞ; Ly ðuÞÞ 6 tp1

u2½0;t

Z

t

0

p

d1 ðxðsÞ; yðsÞÞds:

Corollary 2.11. Under assumptions of Proposition 2.10 we have p

E sup d1 ðLx ðuÞ; Ly ðuÞÞ 6 t p1 E u2½0;t

Z 0

t

p

d1 ðxðsÞ; yðsÞÞds

for every t 2 I. To consider strong solutions of Itô type stochastic fuzzy differential equations we need to exploit a concept of fuzzy stochastic Itô integral being a fuzzy random variable. As we mentioned in Introduction we will use the definition of fuzzy stochastic Itô integral given in [17,18,24]. Let us recall this definition. Let hi : Rd ! F ðRd Þ denote the embedding of Rd into F ðRd Þ, i.e. for r 2 Rd we have

 hriðaÞ ¼

1; if a ¼ r; 0; if a 2 Rd n frg:

It is easy to see that if x : X ! Rd is an Rd -valued random variable defined on a probability space ðX; A; PÞ, then hxi : X ! F ðRd Þ is a fuzzy random variable. For stochastic processes we have a similar property. Remark 2.12. Let x : I  X ! Rd be an Rd -valued stochastic process which is fAt gt2I -adapted (measurable, respectively). Then hxi : I  X ! F ðRd Þ is an fAt gt2I -adapted (measurable, respectively) fuzzy stochastic process. Let {B(t)}t2I be a one-dimensional fAt gt2I -Brownian motion defined on a complete probability space ðX; A; PÞ with a filRT tration fAt gt2I satisfying usual hypotheses. For x 2 L2 ðI  X; N ; Rd Þ let 0 xðsÞdBðsÞ denote the classical stochastic Itô integral (see, e.g. [1,5,31]). Definition 2.13. By a fuzzy stochastic Itô integral we mean the fuzzy random variable

X3x#

Z

T

 xðsÞdBðsÞðxÞ 2 F ðRd Þ:

0

For every t 2 I one can consider the fuzzy stochastic Itô integral

Z



t

xðsÞdBðsÞ

Z

T

:¼

0

 1½0;t ðsÞxðsÞdBðsÞ ;

DR t 0

E xðsÞdBðsÞ , which is understood in the sense:

0

where 1[0, t](s) = 1 if s 2 [0, t] and 1[0, t](s) = 0 if s 2 (t, T]. Also, the following properties hold (see [17,18]) and will be useful in the context of fuzzy stochastic differential equations. Proposition 2.14. Let x 2 L2 ðI  X; N ; Rd Þ. Then the mapping

I  X 3 ðt; xÞ #

Z

t

xðsÞdBðsÞðxÞ



2 F ðRd Þ

0

belongs to L2 ðI  X; N ; F ðRd ÞÞ. Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

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M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

Proposition 2.15. Let x 2 L2 ðI  X; N ; Rd Þ. Then the fuzzy stochastic process

nDR

t 0

Eo xðsÞdBðsÞ

t2I

is d1-continuous.

Proposition 2.16. Let x; y 2 L2 ðI  X; N ; Rd Þ. Then for every t 2 I 2

E sup d1

Z

u2½0;t

u

 Z xðsÞdBðsÞ ;

0



u

yðsÞdBðsÞ

6 4E

Z

0

0

t

2

d1 ðhxðsÞi; hyðsÞiÞds:

3. Stochastic fuzzy differential equations with solutions on [0, T] Let 0 < T < 1, I = [0, T], and let ðX; A; PÞ be a complete probability space with a filtration fAt gt2I satisfying usual conditions. By B = {B(t)}t2I we denote an m-dimensional fAt g-Brownian motion defined on ðX; A; fAt gt2I ; PÞ; m 2 N. The process B is defined as follows

B ¼ ðB1 ; B2 ; . . . ; Bm Þ0 ; 0

where {B1(t)}t2I, {B2(t)}t2I, . . . , {Bm(t)}t2I are the independent, one-dimensional fAt gt2I -Brownian motions, and the symbol denotes transposition. In [17] we have started the studies on strong solutions to stochastic fuzzy differential equations driven by m-dimensional Rt Brownian motion. For convenience, the Lebesgue–Aumann integral Lx(t) we have denoted by 0 xðsÞds. In this paper we continue this research. Thus, let us recall that such the equations can be written in a symbolic form as: I P:1

dxðtÞ ¼ f ðt; xðtÞÞdt þ hgðt; xðtÞÞdBðtÞi;

P:1

xð0Þ ¼ x0 ;

ð3:1Þ

with

f : I  X  F ðRd Þ ! F ðRd Þ; g : I  X  F ðRd Þ ! Rd  Rm ;

and

x0 : X ! F ðRd Þ being a fuzzy random variable: Since hr1 + r2i = hr1i + hr2i for r1 ; r2 2 Rd and g = (g1, g2, . . . , gm) where g k : I  X  F ðRd Þ ! Rd ðk ¼ 1; 2; . . . ; mÞ, we can rewrite (3.1) as follows I P:1

dxðtÞ ¼ f ðt; xðtÞÞdt þ

m X P:1 k hg k ðt; xðtÞÞdB ðtÞi; xð0Þ¼ x0 ;

ð3:2Þ

k¼1

P where denotes the addition of fuzzy sets. One can observe that such equations generalize the classical crisp stochastic differential equations, since Rd can be embedded into F ðRd Þ. Definition 3.1. By a strong solution to Eq. (3.1) we mean a fuzzy stochastic process x : I  X ! F ðRd Þ such that (i) x 2 L2 ðI  X; N ; F ðRd ÞÞ, (ii) x is a d1-continuous fuzzy stochastic process, (iii) it holds I P:1

xðtÞ ¼ x0 þ

Z 0

t

f ðs; xðsÞÞds þ

m Z X k¼1

t

 k g k ðs; xðsÞÞdB ðsÞ :

ð3:3Þ

0

Definition 3.2. A strong solution x : I  X ! F ðRd Þ to Eq. (3.1) is said to be strongly unique, if I P:1

xðtÞ ¼ yðtÞ; where y : I  X ! F ðRd Þ is any strong solution of (3.1). Here the concepts of solution to (3.1) and its uniqueness are in a strong sense as in [17]. This type of solution is different from a mean-square type proposed in [24]. In [17] we have stated the existence and uniqueness of strong solution to (3.1) under assumptions that the coefficients f, g satisfy an uniform Lipschitz condition and a linear growth condition. Here, as a first result, we present an existence and uniqueness theorem which is obtained under a boundedness condition and this assumption is weaker than standard linear growth condition. Firstly, let us write down conditions imposed on the coefficients of Eq. (3.1). We will assume that f : I  X  F ðRd Þ ! F ðRd Þ; g k : I  X  F ðRd Þ ! Rd ðk ¼ 1; 2; . . . ; mÞ satisfy: (c1) the mapping f : ðI  XÞ  F ðRd Þ ! F ðRd Þ is N  BdS jBdS -measurable and g k : ðI  XÞ  F ðRd Þ ! Rd is N  BdS jBðRd Þmeasurable, Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

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M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

(c2) there exists a constant L > 0 such that P-a.e. for every t 2 I, and for every u; v 2 F ðRd Þ it holds 2

2

maxfd1 ðf ðt; x; uÞ; f ðt; x; v ÞÞ; kg k ðt; x; uÞ  g k ðt; x; v Þk2 g 6 Ld1 ðu; v Þ;

k ¼ 1; 2; . . . ; m;

(c3) there exists a constant C > 0 such that P-a.e. for every t 2 I it holds: 2

maxfd1 ðf ðt; x; h0iÞ; h0iÞ; kg k ðt; x; h0iÞk2 g 6 C;

k ¼ 1; 2; . . . ; m:

Now we are in a position to formulate the existence and uniqueness theorem. Theorem 3.3. Let x0 2 L2 ðX; A0 ; P; F ðRd ÞÞ. Suppose that f : I  X  F ðRd Þ ! F ðRd Þ; g k : I  X  F ðRd Þ ! Rd ; k ¼ 1; 2; . . . ; m, satisfy (c1)–(c3). Then Eq. (3.1) possesses a strong solution which is strongly unique. Before we prove this assertion we present some important observations. To show the existence of a solution to (3.1) we will use the method of successive approximations. Therefore, let us define a Picard type sequence xn : I  X ! F ðRd Þ; n ¼ 0; 1; . . . as follows: I P:1

x0 ðtÞ :¼ x0 ;

ð3:4Þ

and for n = 1, 2, . . . I P:1

xn ðtÞ :¼ x0 þ

Z

t

f ðs; xn1 ðsÞÞds þ

0

m Z X k¼1

t

 k g k ðs; xn1 ðsÞÞdB ðsÞ :

ð3:5Þ

0

Note that xn’s are the well defined d1-continuous fuzzy stochastic processes from L2 ðI  X; N ; F ðRd ÞÞ. Indeed, it is easy to see that x0 2 L2 ðI  X; N ; F ðRd ÞÞ and x0 is d1-continuous. Further assuming that xn1 2 L2 ðI  X; N ; F ðRd ÞÞ we obtain: — due to (c1) that the composition f ð; ; xn1 ð; ÞÞ : I  X ! F ðRd Þ is a nonanticipating fuzzy stochastic process, — due to (c1) that the composition g k ð; ; xn1 ð; ÞÞ : I  X ! Rd is a nonanticipating Rd -valued stochastic process (k = 1, 2, . . . , m), — due to (c2) and (c3) that f ð; ; xn1 ð; ÞÞ 2 L2 ðI  X; N ; F ðRd ÞÞ and g k ð; ; xn1 ð; ÞÞ 2 L2 ðI  X; N ; Rd Þðk ¼ 1; 2; . . . ; mÞ, — due to Propositions 2.8 and 2.14 that the process xn defined as in (3.5) belongs to L2 ðI  X; N ; F ðRd ÞÞ, — due to Propositions 2.9 and 2.15 that the process xn is d1-continuous. The Picard approximations xn defined by (3.4) and (3.5) are the fuzzy stochastic processes which are nonanticipating and RT 2 L2-integrably bounded (i.e. E 0 d1 ðxn ðtÞ; h0iÞdt < 1). In the next result we show that xn’s satisfy also the following stronger property. Proposition 3.4. Let x0 : X ! F ðRd Þ; f : I  X  F ðRd Þ ! F ðRd Þ; g k : I  X  F ðRd Þ ! Rd ; k ¼ 1; 2; . . . ; m, satisfy assumptions of Theorem 3.3. Then, for the sequence fxn g1 n¼1 defined as in (3.5) it holds

  2 2 E sup d1 ðxn ðtÞ; h0iÞ 6 C 1 þ C 2 TEd1 ðx0 ; h0iÞ eC 2 T ;

n 2 N;

t2I

h i 2 where C 1 ¼ ðm þ 2Þ Ed1 ðx0 ; h0iÞ þ 2CT 2 þ 8CTm and C2 = 2L(m + 2)(T + 4m). Proof. Let us denote 2

en ðtÞ ¼ E sup d1 ðxn ðuÞ; h0iÞ for n 2 N and t 2 I: u2½0;t

Then we have

" en ðtÞ 6 ðm þ 2Þ

2 Ed1 ðx0 ; h0iÞ

þ E sup u2½0;t

2 d1

Z 0

u



Z m X 2 f ðs; xn1 ðsÞÞds; h0i þ E sup d1 k¼1

u2½0;t

u

 # g ðs; xn1 ðsÞÞdB ðsÞ ; h0i : k

k

0

Due to Propositions 2.10 and 2.16 and triangle inequality we obtain

Z tn o 2 2 d1 ðf ðs; xn1 ðsÞÞ; f ðs; h0iÞÞ þ d1 ðf ðs; h0iÞ; h0iÞ ds 0 # Z tn m o X 2 2 k k k d1 ðhg ðs; xn1 ðsÞÞi; hg ðs; h0iÞiÞ þ d1 ðhg ðs; h0iÞi; h0iÞ ds : þ8 E

h 2 en ðtÞ 6 ðm þ 2Þ Ed1 ðx0 ; h0iÞ

k¼1

þ 2tE

0

Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

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M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

Using assumptions (c2) and (c3) we get

 Z t Z t 2 2 en ðtÞ 6 ðm þ 2Þ Ed1 ðx0 ; h0iÞ þ 2Ct 2 þ 8Ctm þ 2Lðt þ 4mÞE d1 ðxn1 ðsÞ; h0iÞds 6 C 1 þ C 2 en1 ðsÞds: 0

0

By the last inequality we obtain

Z

max en ðtÞ 6 C 1 þ C 2

16n6k

t

max en1 ðsÞds for k 2 N:

0 16n6k

Since 2

max en1 ðtÞ 6 e0 ðtÞ þ max en ðtÞ ¼ Ed1 ðx0 ; h0iÞ þ max en ðtÞ;

16n6k

16n6k

16n6k

we get further that for k 2 N it holds 2

max en ðtÞ 6 C 1 þ C 2 TEd1 ðx0 ; h0iÞ þ C 2

Z

16n6k

t

max en ðsÞds;

t 2 I:

0 16n6k

By the Gronwall inequality it follows that

h i 2 max en ðtÞ 6 C 1 þ C 2 TEd1 ðx0 ; h0iÞ eC 2 t ;

t 2 I:

16n6k

  2 Hence en ðTÞ 6 C 1 þ C 2 TEd1 ðx0 ; h0iÞ eC 2 T . h Now we prove Theorem 3.3. Proof of Theorem 3.3. Let us denote 2

jn ðtÞ ¼ E sup d1 ðxn ðuÞ; xn1 ðuÞÞ for n 2 N and t 2 I: u2½0;t

Then, for t 2 I we have

j1 ðtÞ ¼ E sup

u2½0;t

Z

2 d1

f ðs; x0 Þds þ

0

k¼1

"

Z

2

6 quadðm þ 1ÞE sup d1 u2½0;t



2

2ðm þ 1ÞE sup d1

6

2 d1

þ

Z

u

f ðs; x0 ðsÞÞds; h0i þ

2

f ðs; x0 Þds;

u2½0;t

Z



u

0

2

f ðs; h0iÞds þ d1

u



k

k

2 d1

Z

u

f ðs; x0 Þds;

0

 # k g k ðs; x0 ðsÞÞdB ðsÞ ; h0i

u

0

k¼1

u 0

Z

u2½0;t

E sup

Z m X 2 d1

g ðs; h0iÞdB ðsÞ

þ

Z

u

f ðs; h0iÞds; h0i 2 d1

Z

0

u

Z 2 f ðs; h0iÞds þ E sup d1 u2½0;t

 Z u k g k ðs; x0 ÞdB ðsÞ ;

0

u

k

k



g ðs; h0iÞdB ðsÞ

0

u

 # g ðs; h0iÞdB ðsÞ ; h0i k

k

0

k¼1

Z



0 m X

0

6 2ðm þ 1Þ E sup d1

k¼1

u

 Z k g k ðs; x0 ÞdB ðsÞ ;

"

m X

!  g ðs; x0 ÞdB ðsÞ ; h0i k

k

0

0

k¼1

þ

u

0

Z

u2½0;t

m X

m Z X

u

u

f ðs; h0iÞds; h0i

0

Z m X 2 þ E sup d1 k¼1

u2½0;t

u

 # g ðs; h0iÞdB ðsÞ ; h0i : k

k

0

Due to Propositions 2.10 and 2.16 we infer that

" j1 ðtÞ 6 2ðm þ 1Þ tE

Z

t 0

2

d1 ðf ðs; x0 Þ; f ðs; h0iÞÞds þ tE

# Z t m X 2 k d1 ðhg ðs; h0iÞi; h0iÞds : þ4 E k¼1

Z 0

t

2

d1 ðf ðs; h0iÞ; h0iÞds þ 4

Z t m X 2 E d1 ðhg k ðs; x0 Þi; hg k ðs; h0iÞiÞds k¼1

0

0

Next, using the assumptions (c2) and (c3) we obtain

j1 ðtÞ 6 K 1 t; for every t 2 I; h i 2 2 where K 1 ¼ 2ðm þ 1Þ TLEd1 ðx0 ; h0iÞ þ TC þ 4mLEd1 ðx0 ; h0iÞ þ 4mC < 1. Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

9

M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

Proceeding similarly we obtain

jnþ1 ðtÞ 6 ðm þ 1Þðt þ 4mÞLE

Z

t

6 ðm þ 1ÞðT þ 4mÞL

Z

t

2

E sup d1 ðxn ðuÞ; xn1 ðuÞÞds u2½0;s

0

¼ ðm þ 1ÞðT þ 4mÞL

2

d1 ðxn ðsÞ; xn1 ðsÞÞds

0

Z

t

jn ðsÞds:

0

Hence one can infer that

K 1 ðK 2 tÞn  ; K2 n!

jn ðtÞ 6

t 2 I; n 2 N;

ð3:6Þ

where K2 = (m + 1)(T + 4m)L. Now by the Chebyshev inequality and (3.6) we obtain

 1 K 1 ð4K 2 TÞn 2 P sup d1 ðxn ðuÞ; xn1 ðuÞÞ > n 6 4n jn ðTÞ 6  : K2 n! 4 u2I Since the series

P1

n¼1

ð4K 2 TÞn n!

is convergent, due to the Borel–Cantelli lemma we obtain

 1 P sup d1 ðxn ðuÞ; xn1 ðuÞÞ > n infinitely often ¼ 0: 2 u2I Hence we have: for almost all x 2 X there exists n0 = n0(x) such that

sup d1 ðxn ðu; xÞ; xn1 ðu; xÞÞ 6 u2I

1 ; 2n

if n P n0 :

Thus the sequence of trajectories {xn(, x)} is uniformly convergent to a d1-continuous function ~ xð; xÞ : I ! F ðRd Þ for every x 2 Xc, where Xc 2 A and P(Xc) = 1. Let us define now the mapping x : I  X ! F ðRd Þ as follows: xð; xÞ ¼ ~ xð; xÞ if x 2 Xc, and x(, x) as freely chosen fuzzy function in the case x 2 X n Xc. Observing that P-a.e. for every a 2 [0, 1] and every t 2 I it holds

dH ð½xn ðt; xÞa ; ½xðt; xÞa Þ ! 0; a

as n ! 1;

we infer that ½xðt; Þ : X ! KðR Þ is an At -measurable multifunction. Therefore x is a continuous fAt gt2I -adapted fuzzy stochastic process, and hence nonanticipating. Since xn 2 L2 ðI  X; N ; F ðRd ÞÞ, we have that for every fixed t 2 I the fuzzy random variable xn ðtÞ 2 L2 ðX; A; P; F ðRd ÞÞ. This RT 2 2 2 implies that Ed1 ðxðtÞ; h0iÞ < 1 for every t 2 I. Then E 0 d1 ðxðtÞ; h0iÞdt 6 Tsupt2I Ed1 ðxðtÞ; h0iÞ < 1, which means that 2 x 2 L ðI  X; N ; F ðRd ÞÞ. In what follows we shall show that x is a solution to (3.1). Note that for every t 2 I 2 Ed1

xðtÞ; x0 þ

d

Z

m Z X

t

f ðs; xðsÞÞds þ

0

k¼1

t

! g ðs; xðsÞÞdB ðsÞ ¼ 0: k

k

ð3:7Þ

0

Indeed, one can observe that 2 Ed1

xðtÞ; x0 þ

Z

m Z X

t

f ðs; xðsÞÞds þ

0 m Z t X

þ

k¼1

t

0

k

0

 !# Z t m Z t X k k k k g ðs; xn1 ðsÞÞdB ðsÞ ; x0 þ f ðs; xðsÞÞds þ g ðs; xðsÞÞdB ðsÞ :

0

k¼1

!

 Z t 2 2 g ðs; xðsÞÞdB ðsÞ f ðs; xn1 ðsÞÞds 6 2 Ed1 ðxðtÞ; xn ðtÞÞ þ Ed1 x0 þ k

0

k¼1

0

2 Ed1 ðxn ðtÞ; xðtÞÞ

Since ! 0, it remains to note that the second summand on right-hand side above converges to zero. For the second summand, according to Propositions 2.10 and 2.16 and assumption (c2) we obtain 2

Ed1

Z 0

t

f ðs; xn1 ðsÞÞds þ

m Z X k¼1

" 2

6 ðm þ 1Þ Ed1

Z

t

 Z t ! m Z t X k k g k ðs; xn1 ðsÞÞdB ðsÞ ; f ðs; xðsÞÞds þ g k ðs; xðsÞÞdB ðsÞ

0

t

f ðs; xn1 ðsÞÞds; 0

6 ðm þ 1Þðt þ mÞLE

0

Z 0

Z 0

t

2

t

k¼1

0

X Z t  Z t  # m 2 k k k k f ðs; xðsÞÞds þ Ed1 g ðs; xn1 ðsÞÞdB ðsÞ ; g ðs; xðsÞÞdB ðsÞ k¼1

d1 ðxn1 ðsÞ; xðsÞÞds 6 ðm þ 1ÞðT þ mÞL

0

Z 0

T

0

2

Ed1 ðxn1 ðsÞ; xðsÞÞds ! 0;

as n ! 1:

Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

10

M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

Hence (3.7) follows, which implies that for every t 2 I

d1 xðtÞ; x0 þ

Z

t

f ðs; xðsÞÞds þ

m Z X

0

t

k

g k ðs; xðsÞÞdB ðsÞ

Z

k¼1

t

f ðs; xðsÞÞds þ

m Z X

0

P:1

¼ 0:

0

Due to d1-continuity of considered processes we get

d1 xðtÞ; x0 þ

!

t

k

g k ðs; xðsÞÞdB ðsÞ

!

I P:1

¼ 0:

0

k¼1

Thus x satisfies (3.1). What is left is to prove that x is strongly unique. Let us assume that x; y : I  X ! F ðRd Þ are strong solutions to Eq. (3.1). 2 For t 2 I we denote jðtÞ :¼ E supu2½0;t d1 ðxðuÞ; yðuÞÞ. Observe that for every t 2 I it holds

jðtÞ 6 ðm þ 1Þðt þ 4mÞLE

Z

t

2

d1 ðxðsÞ; yðsÞÞds 6 ðm þ 1ÞðT þ 4mÞL

0

Z

t

jðsÞds:

0

Application of the Gronwall inequality yields j(t) = 0 for t 2 I. This lead us to the conclusion P:1

sup d1 ðxðtÞ; yðtÞÞ¼ 0; t2I

which completes the proof.

h

Using the sequence (3.5) we can establish some approximate solutions of Eq. (3.1). It is desirable to estimate an error of nth approximation. Proposition 3.5. Let x0 : X ! F ðRd Þ; f : I  X  F ðRd Þ ! F ðRd Þ; g k : I  X  F ðRd Þ ! Rd ; k ¼ 1; 2; . . . ; m, satisfy assumptions of Theorem 3.3. Then, for the sequence fxn g1 n¼1 of Picard approximations defined as in (3.5) and the exact solution x to (3.1) it holds 2

E sup d1 ðxn ðtÞ; xðtÞÞ 6 t2I

2K 1 ½TK 2 nþ1 2TK 2  ; e K 2 ðn þ 1Þ!

ð3:8Þ

where K1, K2 are as in the proof of Theorem 3.3. Proof. Let us denote 2

rðtÞ ¼ E sup d1 ðxn ðuÞ; xðuÞÞ for t 2 I; u2½0;t

and observe that due to Propositions 2.10 and 2.16 we have

" 2

rðtÞ 6 ðm þ 1Þ E sup d1

Z

u2½0;t

m X 2 þ E sup d1 k¼1

f ðs; xn1 ðsÞÞds; 0

Z

u2½0;t

u

Z

f ðs; xðsÞÞds

0

 Z u k g k ðs; xn1 ðsÞÞdB ðsÞ ;

0

6 ðm þ 1Þðt þ 4mÞL

u

u k

k

 #

g ðs; xðsÞÞdB ðsÞ

0

Z

t

0

2

E sup d1 ðxn1 ðuÞ; xðuÞÞds 6 2ðm þ 1Þðt þ 4mÞL u2½0;s

# Z t" 2 2 E sup d1 ðxn1 ðuÞ; xn ðuÞÞ þ E sup d1 ðxn ðuÞ; xðuÞÞ ds:  0

u2½0;s

u2½0;s

Using (3.6) we obtain

rðtÞ 6 2K 1 K n2

T nþ1 þ 2K 2 ðn þ 1Þ!

Z

t

rðsÞds:

0

Hence, by the Gronwall inequality 2

E sup d1 ðxn ðuÞ; xðuÞÞ 6 u2½0;t

2K 1 ½TK 2 nþ1 2tK 2  for every t 2 I; e K 2 ðn þ 1Þ!

and (3.8) follows easily. h

Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

11

Applying this result we get the convergence 2

n!1

E sup d1 ðxn ðtÞ; xðtÞÞ ! 0: t2I

Now by Proposition 3.4, we can infer that

  2 2 E sup d1 ðxðtÞ; h0iÞ < C 1 þ C 2 TEd1 ðx0 ; h0iÞ eC 2 T ; t2I

where x is the solution to (3.1) and the constants C1, C2 are defined as earlier. In fact, by direct calculations (similar to those contained in the proof of Proposition 3.4) we get 2

E sup d1 ðxðtÞ; h0iÞ 6 C 1 eC 2 T : t2I

In the sequel we investigate stability of solution to (3.1) with respect to initial value. This kind of stability is a desired property. It ensures that in the case of replacement of x0 by its approximate value y0, the solution of equation with initial value y0 does not differ much from the solution of equation with initial value x0. We show that such the property holds for strong solutions of fuzzy stochastic differential equations. Let x, y denote strong solutions to the fuzzy stochastic differential equations I P:1

dxðtÞ ¼ f ðt; xðtÞÞdt þ hgðt; xðtÞÞdBðtÞi; I P:1

dyðtÞ ¼ f ðt; yðtÞÞdt þ hgðt; yðtÞÞdBðtÞi;

P:1

xð0Þ ¼ x0 ;

ð3:9Þ

P:1

yð0Þ ¼ y0 ;

ð3:10Þ

respectively. Theorem 3.6. Assume that x0 ; y0 2 L2 ðX; A0 ; P; F ðRd ÞÞ, and f : I  X  F ðRd Þ ! F ðRd Þ; g k : I  X  F ðRd Þ ! Rd ; k ¼ 1; 2; . . . ; m, satisfy (c1)–(c3). Then, for the solutions x; y : I  X ! F ðRd Þ of Eqs. (3.9) and (3.10) it holds 2

2

E sup d1 ðxðtÞ; yðtÞÞ 6 ðm þ 2ÞEd1 ðx0 ; y0 ÞeLTðmþ2ÞðTþ4mÞ : t2I

Proof. Due to Theorem 3.3, there exist the unique strong solutions x, y to (3.9) and (3.10), respectively. For t 2 I we obtain 2

2

E sup d1 ðxðuÞ; yðuÞÞ 6 ðm þ 2ÞEd1 ðx0 ; y0 Þ þ Lðm þ 2ÞðT þ 4mÞ u2½0;t

Z 0

t

2

E sup d1 ðxðuÞ; yðuÞÞds; u2½0;s

and by the Gronwall inequality we end the proof. h The second kind of stability for strong solutions to stochastic fuzzy differential equations is stability with respect to the equation coefficients f, gk, k = 1, 2, . . . , m. We will show that if approximations fn ; g 1n ; g 2n ; . . . ; g m n of the coefficients f, g1, g2, . . . , gm converge to the exact coefficients, then approximate solutions converge to the solution of the equation with exact coefficients, too. Therefore, let x, xn denote strong solutions of the following stochastic fuzzy differential equations I P:1

dxðtÞ ¼ f ðt; xðtÞÞdt þ hgðt; xðtÞÞdBðtÞi; I P:1

P:1

xð0Þ ¼ x0 ;

dxn ðtÞ ¼ fn ðt; xn ðtÞÞdt þ hg n ðt; xn ðtÞÞdBðtÞi;

ð3:11Þ P:1

xn ð0Þ ¼ x0 ;

ð3:12Þ

respectively. Theorem 3.7. Let x0 2 L2 ðX; A0 ; P; F ðRd ÞÞ, and f ; fn : I  X  F ðRd Þ ! F ðRd Þ; g k ; g kn : I  X  F ðRd Þ ! Rd ðn 2 N; k ¼ 1; 2; . . . ; mÞ satisfy (c1)–(c3). Assume that for every u 2 F ðRd Þ it holds

Z

T

2

d1 ðfn ðt; uÞ; f ðt; uÞÞdt ! 0

E 0

ð3:13Þ

and

Z

T

E 0

 2  d1 g kn ðt; uÞ ; hg k ðt; uÞi dt ! 0;

k ¼ 1; 2; . . . ; m;

ð3:14Þ

as n ? 1. Then, for the solutions x; xn : I  X ! F ðRd Þ of Eqs. (3.11) and (3.12) we have 2

E sup d1 ðxn ðtÞ; xðtÞÞ ! 0;

as n ! 1:

t2I

Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

12

M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

Proof. The strong solutions x to (3.11) and xn to (3.12) exist and are unique due to Theorem 3.3. For t 2 I we have

" E sup u2½0;t

2 d1 ðxn ðuÞ; xðuÞÞ

6 ðm þ 1Þ tE

Z

t

2 d1 ðfn ðs; xn ðsÞÞ; f ðs; xðsÞÞÞds

0

# Z t m X

k  2  k þ4 E d1 g n ðs; xn ðsÞÞ ; hg ðs; xðsÞÞi ds k¼1

0

Z t Z t 2 2 d1 ðxn ðsÞ; xðsÞÞds þ tE d1 ðfn ðs; xðsÞÞ; f ðs; xðsÞÞÞds 6 2ðm þ 1Þ LtE 0

þ4LmE

Z

t 0

0

# Z t m X

k  2 2  k d1 ðxn ðsÞ; xðsÞÞds þ 4 E d1 g n ðs; xðsÞÞ ; hg ðs; xðsÞÞi ds k¼1

" 6 2ðm þ 1Þ LðT þ 4mÞ

Z

t 0

Z m X þ4 E k¼1

T

0

0

Z

2

E sup d1 ðxn ðuÞ; xðuÞÞds þ TE u2½0;s

T

2

d1 ðfn ðs; xðsÞÞ; f ðs; xðsÞÞÞds

0

#

 2  d1 g kn ðs; xðsÞÞ ; hg k ðs; xðsÞÞi ds :

Again by the Gronwall inequality we infer that

" E sup u2½0;t

2 d1 ðxn ðuÞ; xðuÞÞ

6 2ðm þ 1Þ TE

Z 0

T

2 d1 ðfn ðs; xðsÞÞ; f ðs; xðsÞÞÞds

Z m X þ4 E k¼1

0

T

#

 2  d1 g kn ðs; xðsÞÞ ; hg k ðs; xðsÞÞi ds

2Lðmþ1ÞðTþ4mÞt

e

for t 2 I. Hence, by assumptions (3.13) and (3.14), the claim follows immediately. h For u 2 F ðRd Þ denote

lenðuÞ :¼ diamð½u0 Þ ¼ supfka  bk : a; b 2 ½u0 g: It is easy to see that len (u) = 0 iff there exists a 2 Rd such that u = hai (i.e. when u is precise). Also, if u = 1A for A 2 KðRd Þ then len (u) coincides with the classical notion of diameter of the set A. If the set A has interpretation that A is a set of possible values in the case of imperfect knowledge of the precise value a then len (u) could be a simple indicator of uncertainty contained in A. We can say that a fuzzy set u1 contains more uncertainty than a fuzzy set u2 if len (u1) > len (u2). Also u1 contains as much uncertainty as u2 if len (u1) = len (u2), and u1 does not contain less uncertainty than u2 if len (u1) P len (u2). The next result shows that solutions to stochastic fuzzy differential Eq. (3.1) possess trajectories with nondecreasing uncertainty in their values. Theorem 3.8. Assume that x : I  X ! F ðRd Þ is a strong solution to (3.1). Then with P.1 the function

I 3 t # lenðxðt; xÞÞ 2 F ðRd Þ does not decrease. Proof. Since x is solution to (3.1), from (3.3) we can infer that with P.1 for every s, t 2 I, s < t, it holds

½xðt; xÞ0 ¼ ½xðs; xÞ0 þ

Z s

Hence for any fixed a 2

hR

t

0 X m Z t k f ðs; x; xðs; xÞÞds þ g k ðs; xðsÞÞdB ðsÞ ðxÞ: k¼1

s

i0 P R  t k k m k¼1 s f ðs; x; xðs; xÞÞds þ s g ðs; xðsÞÞdB ðsÞ ðxÞ it holds t

½xðt; xÞ0  ½xðs; xÞ0 þ a: This fact implies that diam ([x(t, x)]0) P diam ([x(s, x)]0).

h

4. Solutions on [0, ‘) In this section we present some considerations concerning stochastic fuzzy differential equations with solutions defined on the half-line Rþ ¼ ½0; 1Þ. To this end, we establish a framework needed in this setting. Let ðX; A; fAt gtP0 ; PÞ be a complete, filtered probability space with a filtration fAt gtP0 satisfying the usual hypotheses. By N we denote (similarly as in previous material) the r-algebra of nonanticipating elements in the product Rþ  X, i.e. N :¼ fA 2 BðRþ Þ  A : At 2 At for every t 2 Rþ g, where BðRþ Þ denotes the Borel r-algebra of subsets of Rþ , and At = {x:(t, x) 2 A} for t 2 Rþ . Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

13

A fuzzy stochastic process x : Rþ  X ! F ðRd Þ, whose a-cuts ½xa : Rþ  X ! KðRd Þða 2 ½0; 1Þ are the N -measurable multifunctions, is called nonanticipating or N -measurable. By L2 ðRþ  X; N ; F ðRd ÞÞ we denote a set of nonanticipating and L2-integrably bounded fuzzy stochastic processes, i.e. RT x 2 L2 ðRþ  X; N ; F ðRd ÞÞ iff x is N -measurable and E 0 kj½xðtÞ0 jk2 dt < 1 for every T 2 Rþ . Let {B(t)}tP0 denote an m-dimensional fAt g-Brownian motion defined on ðX; A; fAt gtP0 ; PÞ, i.e. B = (B1, B2, . . . , Bm)0 , where 1 {B (t)}tP0,{B2(t)}tP0, . . . , {Bm(t)}tP0 are the independent, one-dimensional fAt gtP0 -Brownian motions. We are interested in the equations of the form Rþ P:1

P:1

dxðtÞ ¼ f ðt; xðtÞÞdt þ hgðt; xðtÞÞdBðtÞi;

xð0Þ ¼ x0 ;

ð4:1Þ

where

f : Rþ  X  F ðRd Þ ! F ðRd Þ; g : Rþ  X  F ðRd Þ ! Rd  Rm ;

and

x0 : X ! F ðR Þ is a fuzzy random variable: d

Since g = (g1, g2, . . . , gm), where g k : Rþ  X  F ðRd Þ ! Rd ðk ¼ 1; 2; . . . ; mÞ, we can rewrite (4.1) in the following form Rþ P:1

dxðtÞ ¼ f ðt; xðtÞÞdt þ

m X

k

hg k ðt; xðtÞÞdB ðtÞi;

P:1

xð0Þ¼ x0 :

k¼1

The definitions of the notions of solutions to (4.1) and their uniqueness are similar to those from preceding section. However, for clarity and completeness we state them here. Definition 4.1. By a strong solution to Eq. (4.1) we mean a fuzzy stochastic process x : Rþ  X ! F ðRd Þ such that (i) x 2 L2 ðRþ  X; N ; F ðRd ÞÞ, (ii) x is a d1-continuous fuzzy stochastic process, (iii) it holds Rþ P:1

xðtÞ ¼ x0 þ

Z

t

f ðs; xðsÞÞds þ

0

m Z X

t

 k g k ðs; xðsÞÞdB ðsÞ :

ð4:2Þ

0

k¼1

Definition 4.2. A strong solution x : Rþ  X ! F ðRd Þ to Eq. (4.1) is said to be strongly unique, if Rþ P:1

xðtÞ ¼ yðtÞ; where y : Rþ  X ! F ðRd Þ is any strong solution of (4.1). For T > 0, let fT ; g kT denote the truncations of f and gk to the interval [0, T], respectively. More precisely fT : ½0; T  X  F ðRd Þ ! F ðRd Þ; g kT : ½0; T  X  F ðRd Þ ! Rd ; k ¼ 1; 2; . . . ; m, are defined as follows

fT ðt; x; uÞ ¼ f ðt; x; uÞ; g kT ðt; x; uÞ ¼ g k ðt; x; uÞ;

ðt; x; uÞ 2 ½0; T  X  F ðRd Þ; ðt; x; uÞ 2 ½0; T  X  F ðRd Þ:

Theorem 4.3. Let x0 2 L2 ðX; A0 ; P; F ðRd ÞÞ. Assume that f : Rþ  X  F ðRd Þ ! F ðRd Þ; g k : Rþ  X  F ðRd Þ ! Rd ; k ¼ 1; 2; . . . ; m, are such that for every T > 0 their truncations fT : ½0; T  X  F ðRd Þ ! F ðRd Þ; g kT : ½0; T  X  F ðRd Þ ! Rd ; k ¼ 1; 2; . . . ; m, satisfy (c1)–(c3). Then Eq. (4.1) possesses a unique strong solution x : Rþ  X ! F ðRd Þ. Proof. Let us define a sequence xn : Rþ  X ! F ðRd Þ; n ¼ 0; 1; . . . as follows: Rþ P:1

x0 ðtÞ :¼ x0 ; and for n = 1, 2, . . . Rþ P:1

xn ðtÞ :¼ x0 þ

Z

t

f ðs; xn1 ðsÞÞds þ

0

m Z X k¼1

t

 k g k ðs; xn1 ðsÞÞdB ðsÞ :

0

Now let us fix T > 0. In the sequel we proceed similar as in the proof of Theorem 3.3, which concerns the case of finite interval [0, T], and obtain existence of a d1-continuous fuzzy stochastic process x : ½0; T  X ! F ðRd Þ for which P-a.e. it holds

sup d1 ðxn ðtÞ; xðtÞÞ ! 0;

as n ! 1:

t2½0;T

Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

14

M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

Since T is arbitrary, it follows the existence of a d1-continuous fuzzy stochastic process x : Rþ  X ! F ðRd Þ, which possesses a property that P-a.a. the trajectories xn(, x) converge to x(, x) uniformly on compact subsets of Rþ . Similarly as in the proof of Theorem 3.3 we obtain

sup d1 xðtÞ; x0 þ

Z

t2½0;T

t

f ðs; xðsÞÞds þ 0

m Z X

! P:1 g ðs; xðsÞÞdB ðsÞ ¼0

t

k

k

0

k¼1

for every T > 0. This leads us to the fact that

Z

P sup d1 ðxðtÞ; x0 þ t2½0;n

t

f ðs; xðsÞÞds þ

m Z X

0

t

!  g ðs; xðsÞÞdB ðsÞ Þ ¼ 0; 8n 2 N ¼ 1; k

k

0

k¼1

which implies that

sup d1 xðtÞ; x0 þ

Z

t2Rþ

t

f ðs; xðsÞÞds þ

m Z X

0

k¼1

t

k

g k ðs; xðsÞÞdB ðsÞ

!

P:1

¼ 0:



0

5. Some examples of modeling in random and fuzzy environment In the studies of stochastic differential equations, it is a main aim to ensure that a given equation possesses a unique solution. However, it is a very hard task to establish a closed and explicit form of solution to this equation (even in the crisp case). There exist explicit expressions of solutions for some linear crisp stochastic differential equations and in this section we will apply them to present explicit forms of solutions to some problems modeled with a help of stochastic fuzzy differential equations. 5.1. Population growth model In this part we extend a given in [24] example of application of stochastic fuzzy differential equations. The presented extension shows how the theoretical studies on stochastic fuzzy differential equations, driven by multidimensional Brownian motion and defined on unbounded interval, can be applied in modeling of a real-world phenomenon. We are interested in describing (in terms of a stochastic fuzzy differential equation) an evolution process of a population growth. Let us consider a population of some species, which lives on a given territory. If we denote by p(t) the precise number of individuals at the instant t, then one of the simplest crisp models of population evolution can be written as follows

p0 ðtÞ ¼ apðtÞ;

pð0Þ ¼ p0 ;

ð5:1Þ

where p0 denotes the initial number of individuals and to avoid a trivial case we consider a – 0. The associated, equivalent to (5.1), integral equation is

pðtÞ ¼ p0 þ

Z

t

apðsÞds;

ð5:2Þ

0

and the unique real-valued solution to these equations reads p(t) = p0eat. If we want to create a realistic model of a real world phenomena, we have to take into account some uncertainties in description of initial value (at least). Since the initial value may depend on random factors, probability space ðX; A; PÞ should be considered and initial value should be a random variable. Moreover, a description of initial population state may be expressed in linguistic terms, e.g. ‘‘very small’’, ‘‘small’’, ‘‘not big’’, ‘‘medium’’, ‘‘big’’, ‘‘large’’, etc., so modeling with a help of fuzzy sets is needed. Hence we arrive to the conclusion that initial value should be modeled by a fuzzy random variable being a natural model of random imperfect data. Also a more realistic model of population evolution could be expressed by a stochastic fuzzy differential equation in which epistemic and random uncertainties are combined. For further considerations let us assume that the territory has only m entrances in its boundary. Suppose, some individuals emigrate from the territory and some alien individuals immigrate to the population through each of m entrances, and this happens in a very chaotic manner in each entrance. Let the aggregated migration process, associated with kth entrance (k = 1, 2, . . . , m), be modeled by the process bkBk where bk is a positive constant and Bk is Brownian motion (Bk’s are independent). Then a model generalizing (5.2) and incorporating epistemic and random uncertainties could be proposed as follows Rþ P:1

xðt; xÞ ¼ x0 ðxÞ þ

Z 0

t

axðs; xÞds þ

m X

hbk Bk ðt; xÞi;

k¼1

where x symbolizes random factor, x0 is a fuzzy random variable, and the integral on the right hand side is the fuzzy stochastic Lebesgue–Aumann integral. This equation can be rewritten as (in the sequel we do not write the argument x)

Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

Rþ P:1

xðtÞ ¼ x0 þ

Z

t

axðsÞds þ

0

m Z X

t

 k bk dB ðsÞ

15

ð5:3Þ

0

k¼1

or in symbolic, differential form as m X k hbk dB ðtÞi;

Rþ P:1

dxðtÞ ¼ axðtÞdt þ

P:1

xð0Þ¼ x0 :

ð5:4Þ

k¼1

Hence we deal with Eq. (4.1), where f : Rþ  X  F ðRÞ ! F ðRÞ is defined by

f ðt; x; uÞ ¼ a  u for ðt; x; uÞ 2 Rþ  X  F ðRÞ; and g : Rþ  X  F ðRÞ ! R  Rm is given by

gðt; x; uÞ ¼ ðb1 ; b2 ; . . . ; bm Þ for ðt; x; uÞ 2 Rþ  X  F ðRÞ: It is easy to check that f, g satisfy assumptions of Theorem 4.3. Therefore, with appropriately chosen x0, there exists a unique solution to (5.4). To obtain an explicit form of solution x : Rþ  X ! F ðRÞ to (5.4), we introduce some notations for the a-level sets (a 2 [0,1]) of the solution x and a-level sets of initial value x0 : X ! F ðRÞ as follows

h i ½x0 a ¼ xa0;L ; xa0;U :

  ½xðtÞa ¼ xaL ðtÞ; xaU ðtÞ ;

It is known that xaL ; xaU : Rþ  X ! R are the stochastic processes, and xa0;L ; xa0;U : X ! R are the random variables. Since x satisfies (5.3), we are interested in solving the following systems of crisp stochastic integral equations: for a > 0

8 m X Rt Rt > Rþ P:1 k > > xaL ðtÞ ¼ xa0;L þ a 0 xaL ðsÞds þ b dB ðsÞ; > 0 k < k¼1

ð5:5Þ

m X > Rt Rt a > Rþ P:1 a k a > > b dB ðsÞ; : xU ðtÞ ¼ x0;U þ a 0 xU ðsÞds þ 0 k k¼1

and for a < 0

8 m X Rt Rt > Rþ P:1 k > > xa ðtÞ ¼ xa0;L þ a 0 xaU ðsÞds þ b dB ðsÞ; > 0 k < L k¼1

ð5:6Þ

m X > Rt Rt a Rþ P:1 a > k a > > b dB ðsÞ: : xU ðtÞ ¼ x0;U þ a 0 xL ðsÞds þ 0 k k¼1

Applying methods of solving the systems of linear crisp stochastic differential equations we obtain the unique solution to (5.5) as Rþ P:1

xaL ðtÞ ¼ eat xa0;L þ eat

m X

bk

k¼1

Z

t

k

eas dB ðsÞ

0

and Rþ P:1

xaU ðtÞ ¼ eat xa0;U þ eat

Z t m X bk eas dBðsÞ; k¼1

0

which implies that the solution x : Rþ  X ! F ðRÞ to (5.4) with a > 0 is of the form Rþ P:1

xðtÞ ¼ eat  x0 þ

*

eat

+ Z t m X k bk eas dB ðsÞ : k¼1

ð5:7Þ

0

For the unique solution to (5.6) we obtain Rþ P:1

xaL ðtÞ ¼ coshðatÞxa0;L þ sinhðatÞxa0;U þ eat

Z t m X k bk eas dB ðsÞ k¼1

0

and Rþ P:1

xaU ðtÞ ¼ sinhðatÞxa0;L þ coshðatÞxa0;U þ eat

Z t m X k bk eas dB ðsÞ: k¼1

0

Hence the solution x : Rþ  X ! F ðRÞ to (5.4) with a < 0 is of the form

Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

16

M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

*

Rþ P:1

xðtÞ ¼ coshðatÞ  x0 þ sinhðatÞ  x0 þ

eat

+ Z t m X k bk eas dB ðsÞ :

ð5:8Þ

0

k¼1

Let us notice that for a < 0 and t 2 (0, 1) the expressions cosh (at), sinh (at) are of the opposite sign, so one cannot rewrite the above solution in the form of solution which was established in the case a > 0. The values of trajectories of solutions (5.7) and (5.8) become fuzzier as time t grows. This is consistent with Theorem 3.8. To see this more explicitly in our model, let us consider the function Rþ 3 t # lenðxðt; xÞÞ 2 Rþ , where x 2 X and

lenðxðt; xÞÞ ¼ x0U ðt; xÞ  x0L ðt; xÞ: For solution (5.7) to Eq. (5.4) with a > 0 we obtain R P:1

lenðxðtÞÞ ¼ eat lenðx0 Þ and for solution (5.8) to Eq. (5.4) with a < 0 we have R P:1

lenðxðtÞÞ ¼ eat lenðx0 Þ: In each case P:1

lenðxðtÞÞ ! þ1 as t ! þ1: Considering the case a < 0 and taking into account the type of phenomenon which is modeled here, we can ask if it is possible to propose a model which is better fitted and better reflects evolution of the population. A negative value of a informs that the considered population dies out, i.e. the number of individuals decreases. Hence a precision in description of the states should grow. Thus one can expect that the number of individuals should be less fuzzy at instant t2 than that one at earlier instant t1. The studies contained in [15,16,19–21], where we considered set differential equations and random fuzzy differential equations without Itô integrals, direct us to examine the following model with a < 0

xðtÞ þ ð1Þ

Z

t

axðsÞds þ

Z t  m X Rþ P:1 k ð1Þ bk dB ðsÞ ¼ x0 :

0

ð5:9Þ

0

k¼1

This equation coincides with (5.3) in the crisp case. However, in the fuzzy case it is different from (5.3). If Eq. (5.9) has solution x : Rþ  X ! F ðRÞ then it should be satisfied

( P

a

x 2 X : ½xðt; xÞ þ ð1Þ

Z

t

m X ½axðs; xÞ ds þ ðbk ÞBk ðt; xÞ ¼ ½x0 a ; 8t 2 Rþ ; 8a 2 ½0; 1

a

0

)! ¼ 1:

k¼1

Hence we consider the system

8 m X Rt > Rþ P:1 > > xaL ðtÞ þ 0 ðaÞxaL ðsÞds þ ðbk ÞBk ðtÞ ¼ xa0;L ; > < k¼1

m X > Rt > Rþ P:1 a a > > ðbk ÞBk ðtÞ ¼ xa0;U : : xU ðtÞ þ 0 ðaÞxU ðsÞds þ

ð5:10Þ

k¼1

Its solution gives Rþ P:1

xðtÞ ¼ eat  x0 þ

* eat

+ Z t m X k bk eas dB ðsÞ : k¼1

0

But this time t ´ len (x(t, x)) is a decreasing function for almost all x 2 X. Hence the model (5.9) can be better in modeling population growth than model (5.3) with a < 0. 5.2. Financial model The following crisp stochastic differential equation is often used in financial modeling (see, e.g. [28])

dxðtÞ ¼ axðtÞdt þ bxðtÞdBðtÞ; xð0Þ ¼ x0 :

ð5:11Þ

It describes continuous-time evolution of a short-term interest rate in the Brennan–Schwartz model. Such an equation is also used in modeling dynamics of stock price in the Black–Scholes model. In real world, the data of a considered phenomenon may not be precise. For example, in financial engineering, the models with a constant interest rate may cause a dilemma because the interest rate may have different values in different banks. However, all of these values can be collected in a linguistic expression, for example, ‘‘around 5%’’. This leads to modeling with fuzzy sets in financial mathematics as well. Please cite this article in press as: M.T. Malinowski, Some properties of strong solutions to stochastic fuzzy differential equations, Inform. Sci. (2013), http://dx.doi.org/10.1016/j.ins.2013.02.053

M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

17

In what follows we examine a fuzzy counterpart of Eq. (5.11). Thus, let us consider the following stochastic fuzzy differential equation Rþ P:1

xðtÞ ¼ axðtÞdt þ



  b 1 xL ðtÞ þ x1U ðtÞ dBðtÞ ; 2

xð0Þ ¼ x0 ;

ð5:12Þ

where x : R þ  X ! F ðRÞ; a2 R; b > 0; B is a one-dimensional Brownian motion, x1L ; x1U : Rþ  X ! R are such that ½xðt; xÞ1 ¼ x1L ðt; xÞ; x1U ðt; xÞ , and x0 2 F ðRÞ. Note that equality x1L ¼ x1U does not have to hold in any sense for stochastic processes x1L ; x1U . Let us mention that this fuzzy equation coincides with Eq. (5.11) if the values of x were the real numbers. Note that the coefficients of Eq. (5.12) satisfy assumptions of Theorem 4.3. For the solution x of (5.12) we have Rþ P:1

xðtÞ ¼ x0 þ

Z

t

axðsÞds þ

Z

0

t

0

  b 1 xL ðsÞ þ x1U ðsÞ dBðsÞ : 2

Hence, utilizing notations introduced in Section 5.1, we can write the following system for the processes x1L ; x1U

8  R R  < x1 ðtÞRþ¼P:1x1 þ t ax1 ðsÞds þ t b x1 ðsÞ þ x1 ðsÞ dBðsÞ; L 0;L L L U 0 0 2 : x1 ðtÞRþ¼P:1x1 þ R t ax1 ðsÞds þ R t b x1 ðsÞ þ x1 ðsÞdBðsÞ: U

0;U

0

U

0 2

L

ð5:13Þ

U

Thus Rþ P:1

x1L ðtÞ þ x1U ðtÞ ¼ x10;L þ x10;U þ

Z 0

t

  a x1L ðsÞ þ x1U ðsÞ ds þ

and this equation has a unique solution

x1L ðtÞ

þ

Rþ P:1 x1U ðtÞ ¼

  x10;L þ x10;U exp

(

Z

t

0

  b x1L ðsÞ þ x1U ðsÞ dBðsÞ;

! ) 2 b t þ bBðtÞ : a 2

Now we can proceed to obtain a closed explicit form of solution to (5.12). Applying a similar procedure as in preceding subsection, for a P 0, we get the system

8  R R  < xa ðtÞRþ¼P:1xa þ t axa ðsÞds þ t b x1 ðsÞ þ x1 ðsÞ dBðsÞ; L 0;L L L U 0 0 2 : xa ðtÞRþ¼P:1xa þ R t axa ðsÞds þ R t b x1 ðsÞ þ x1 ðsÞdBðsÞ: U

0;U

0

U

0 2

L

U

which can be rewritten as

8  n  o R R  2 R P:1 > < xaL ðtÞ þ¼ xa0;L þ 0t axaL ðsÞds þ 0t 2b x10;L þ x10;U exp a  b2 s þ bBðsÞ dBðsÞ;   n  o > xa ðtÞRþ¼P:1xa þ R t axa ðsÞds þ R t b x1 þ x1 exp a  b2 s þ bBðsÞ dBðsÞ: : U 0;U U 0;L 0;U 0 0 2 2

ð5:14Þ

By Theorem 8.5.2 in [1] we obtain the unique solution to (5.14) in the matrix form as





xaL ðtÞ

Rþ P:1

¼

xaU ðtÞ 0 B ¼@



eat 0

2 ! Z  a t eas 6 x0;L þ 4 a x0;U 0 eat 0 0

0



eas

h i1  R b2 t eat xa0;L þ 2b x10;L þ x10;U 0 e 2 sþbBðsÞ dBðsÞ C h i A:  R t b2 sþbBðsÞ at a b 1 1 dBðsÞ e x0;U þ 2 x0;L þ x0;U 0 e 2

3 0  1   b2  b x10;L þ x10;U e a 2 sþbBðsÞ 2 7 B C @  AdBðsÞ5   b2  a 2 sþbBðsÞ b 1 1 x þ x e 0;L 0;U 2

Hence the solution x : Rþ  X ! F ðRÞ to (5.12) with a P 0 is of the form Rþ P:1

xðtÞ ¼ eat x0 þ

    Z t b2 b 1 e 2 sþbBðsÞ dBðsÞ : x0;L þ x10;U eat 2 0

To find the closed form of solution to (5.12) with a < 0 we have to solve the system

8  R R  < xa ðtÞRþ¼P:1xa þ t axa ðsÞds þ t b x1 ðsÞ þ x1 ðsÞ dBðsÞ; L U L 0;L U 0 0 2 : xa ðtÞRþ¼P:1xa þ R t axa ðsÞds þ R t b x1 ðsÞ þ x1 ðsÞdBðsÞ: U

Since

x1L ðtÞ

þ

0;U

Rþ P:1 x1U ðtÞ ¼

0

L

0 2

L

U

  n  o 2 x10;L þ x10;U exp a  b2 t þ bBðtÞ , we obtain the system

8  n  o R R  2 R P:1 > < xaL ðtÞ þ¼ xa0;L þ 0t axaU ðsÞds þ 0t 2b x10;L þ x10;U exp a  b2 s þ bBðsÞ dBðsÞ;  n  o R R  > : xaU ðtÞRþ¼P:1xa0;U þ t axaL ðsÞds þ t b x10;L þ x10;U exp a  b2 s þ bBðsÞ dBðsÞ: 0 0 2 2

ð5:15Þ

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M.T. Malinowski / Information Sciences xxx (2013) xxx–xxx

The unique solution to (5.15) reads

0  1 3   2 b x1 þ x1 e ab2 sþbBðsÞ 0;L 0;U sinhðatÞ coshðasÞ  sinhðasÞ B 2 Rþ P:1 coshðatÞ C 7 L þ ¼ @  AdBðsÞ5   b2  xaU ðtÞ sinhðatÞ coshðatÞ  sinhðasÞ coshðasÞ a 2 sþbBðsÞ b 1 1 0 x0;L þ x0;U e 2 0 1   R 2 b t xa coshðatÞ þ xa0;U sinhðatÞ þ 2b x10;L þ x10;U eat 0 e 2 sþbBðsÞ dBðsÞ B 0;L C ¼@   R A: b2 t xa0;L sinhðatÞ þ xa0;U coshðatÞ þ 2b x10;L þ x10;U eat 0 e 2 sþbBðsÞ dBðsÞ



xa ðtÞ







2

a 6 x0;L 4 a x0;U

!

Z t

This implies that the solution x : Rþ  X ! F ðRÞ to (5.12) with a < 0 is as follows Rþ P:1

xðtÞ ¼ coshðatÞ  x0 þ sinhðatÞ  x0 þ

    Z t b2 b 1 e 2 sþbBðsÞ dBðsÞ : x0;L þ x10;U eat 2 0

6. Concluding remarks In the paper we study stochastic fuzzy differential equations driven by multidimensional Brownian motion. Such equations are allways understood as integral equations. Therefore we recall the notions of fuzzy stochastic Lebesgue–Aumann integral and fuzzy stochastic Itô integral. Both the integrals are fuzzy random variables. This allows to obtain solutions of considered equations in forms of fuzzy stochastic processes. An existence and uniqueness theorem for solutions has been proven with a help of Picard’s iterations sequence. An estimation for the distance between exact solution and nth approximate solution has been presented. It has been shown that solutions are stable under small changes of equation parameters. The considered approach in formulating stochastic fuzzy differential equations has a potential in applications. We present some examples of modeling in random and fuzzy environment using fuzzy stochastic differential equations. Stochastic set-valued differential equations are particular cases of stochastic fuzzy differential equations. All the result presented in this paper apply automatically to stochastic set-valued differential equations. Following the approach presented in this paper, in a future research, one can think about equations with diffusion part driven by more general integrators than Brownian motion. These could be (for instance) fractional Brownian motion, Gaussian processes, Lévy processes, martingales, semimartingales. However such a change in the diffusion part only (this part of equation is necessarily single-valued) could constitute a small progress. To make the generalizations much stronger also the drift part of equations should be changed. It could be driven, for instance, by increasing processes or finite variation processes. In this paper we require the coefficients of the equation to satisfy Lipschitz condition. In a further research it is of interest to obtain some existence and uniqueness theorems for solutions under non-Lipschitzian conditions. Also the studies of stochastic fuzzy differential equations with solutions having a decreasing (or changing monotonicity) diameter of their values are needed. The problem of solutions having decreasing uncertainty in their values is studied in a submitted paper of the author. Acknowledgement The author would like to thank the Editor-in-Chief Professor Witold Pedrycz and the anonymous referees. References [1] L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley & Sons, New York, 1974. ´ az, D.A. Ralescu, A DE[0, 1] representation of random upper semicontinuous functions, Proc. Amer. ´ nguez-Menchero, M. López-Dı [2] A. Colubi, J.S. Domı Math. Soc. 130 (2002) 3237–3242. [3] W. Fei, Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients, Inform. Sci. 17 (2007) 4329–4337. [4] Y. Feng, Fuzzy stochastic differential systems, Fuzzy Sets Syst. 115 (2000) 351–363. [5] I.I. Gihman, A.V. Skorohod, Stochastic Differential Equations, Springer-Verlag, Berlin, 1972. [6] M.Á. Gil, M. López-Dı´az, D.A. Ralescu, Overview on the development of fuzzy random variables, Fuzzy Sets Syst. 157 (2006) 2546–2557. [7] F. Hiai, H. Umegaki, Integrals, conditional expectation, and martingales of multivalued functions, J. Multivar. Anal. 7 (1977) 149–182. [8] O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst. 24 (1987) 301–317. [9] J.H. Kim, On fuzzy stochastic differential equations, J. Korean Math. Soc. 42 (2005) 153–169. [10] Y.K. Kim, Measurability for fuzzy valued functions, Fuzzy Sets Syst. 129 (2002) 105–109. [11] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic Publishers., Dordrecht, 1991. [12] V. Lakshmikantham, R.N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, London, 2003. [13] M.T. Malinowski, On random fuzzy differential equations, Fuzzy Sets Syst. 160 (2009) 3152–3165. [14] M.T. Malinowski, Existence theorems for solutions to random fuzzy differential equations, Nonlinear Anal. TMA 73 (2010) 1515–1532. [15] M.T. Malinowski, Interval differential equations with a second type Hukuhara derivative, Appl. Math. Lett. 24 (2011) 2118–2123. [16] M.T. Malinowski, Random fuzzy differential equations under generalized Lipschitz condition, Nonlinear Anal. Real World Appl. 13 (2012) 860–881. [17] M.T. Malinowski, Strong solutions to stochastic fuzzy differential equations of Itô type, Math. Comput. Modell. 55 (2012) 918–928. [18] M.T. Malinowski, Itô type stochastic fuzzy differential equations with delay, Syst. Control Lett. 61 (2012) 692–701.

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