- Email: [email protected]
Hukuhara derivative of the fuzzy expected value
Fuzzy Sets and Systems 138 (2003) 593 – 600 www.elsevier.com/locate/fss
Hukuhara derivative of the fuzzy expected value Luis J. Rodr*+guez-Mu˜niza; b;∗ , Miguel L*opez-D*+aza; c a
b
Departamento de Estad stica e I.O. y D.M., Universidad de Oviedo, Spain Escuela Superior de la Marina Civil, Campus de Viesques, 33271 Gij on, Spain c Facultad de Ciencias, c/ Calvo Sotelo s/n, 33071 Oviedo, Spain
Received 21 May 2002; received in revised form 9 January 2003; accepted 16 January 2003
Abstract In this paper we establish su5cient conditions to guarantee the exchange of the fuzzy expected value for a fuzzy random variable depending on certain parameter and its Hukuhara derivative with respect to the parameter. Main results are obtained for two di6erent sets of general conditions, one of them requiring measurability conditions with respect to the product space the fuzzy random variable is de8ned from, and the second one just requiring conditions over the projections of the fuzzy random variables. c 2003 Elsevier B.V. All rights reserved. Keywords: Fuzzy expected value; Fuzzy random variable; Hukuhara derivative; Hukuhara di6erence
1. Introduction The study of the di6erentiability under the integral sign has been one of the classical topics in the theory of mathematical analysis, not only because of its theoretical interest but also for the wide variety of applications of these results. These mathematical results try to obtain su5cient conditions to exchange an integral of a mapping which depends on a certain parameter and a derivative/di6erential of the map with respect to the parameter. Of course, there have been di6erent studies about this kind of problem, since there exists a great variety of possibilities depending on the considered mapping, especially depending on the values it takes on, on the concept of integral used, and of course, on the di6erentiability criterion. The 8rst results that were obtained referred to real-valued random variables (see, for instance, [2,13]). Banks and Jacobs [1] were the 8rst ones who studied the problem of di6erentiability under the integral sign for set-valued mappings. Since, in some sense, fuzzy sets over a certain space ∗
Corresponding author. Fax: +34-985181902. E-mail address: [email protected] (L.J. Rodr*+guez-Mu˜niz).
c 2003 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter doi:10.1016/S0165-0114(03)00020-4
594
L.J. Rodr guez-Mu˜niz, M. L opez-D az / Fuzzy Sets and Systems 138 (2003) 593 – 600
are a generalization of the class of subsets of that space, a natural question arises: how to obtain general conditions for the di6erentiability under the integral sign when fuzzy-valued mappings are considered. The authors have obtained in Rodr*+guez-Mu˜niz et al. [16] some results regarding the di6erentiability under the integral sign for fuzzy random variables when four di6erent concepts of di6erentiability are considered, namely, the De Blasi di6erentiability [3,5], the -di6erentiability [1,11], the Fr*echet di6erentiability of the associated support function, and the s-di6erentiability [14,15] and the integral is the one established by Puri and Ralescu [12] for the concept of expected value of a fuzzy random variable. In the present paper, our motivation is to obtain new theoretical results about that topic, when we consider a wider class of fuzzy random variables than the one considered in Rodr*+guez-Mu˜niz et al. [16], as we will see, here we will obtain results just by requiring convexity but not continuity on the -level sets, as in that paper. On the other hand, the derivability criterion we will use is the Hukuhara’s one. In other words, we will enlarge the class of variables by restricting the derivability criterion. Main results in the paper are obtained under two general and di6erent frameworks, in the 8rst one product measurability conditions are required whereas in the second framework the product measurability condition is not required, which makes this result especially useful for those cases in which that measurability is not guaranteed. Our purpose is to use these theoretical results as a basis to obtain more results which can be applied in Statistical Decision Theory (for instance, in the equivalence between normal and extensive forms of Bayesian analysis). The paper is organized as follows: in Section 2 we provide some preliminaries. Section 3 is formed by several supporting results. Main results about the Hukuhara derivative of the fuzzy expected value are gathered in Section 4. Finally, some open problems are outlined in Section 5. 2. Preliminaries Let us denote by Kc the class of non-empty compact and convex subsets of R and by dH the Hausdor6 metric. Fc will denote the class of fuzzy sets A : R → [0; 1] such that A ∈Kc for all ∈[0; 1], being A ={x ∈R : A(x)¿} for ∈(0; 1] and A0 =supp A. This class can be endowed with a semilinear structure by means of Zadeh’s extension principle or, equivalently (see [10]), addition and product by a scalar can be levelwise calculated through (A+B) =A +B (Minkowski’s addition) and ( A) = A . Obviously, (Fc ; +; ·) is not a vector space. Hukuhara’s di=erence between two fuzzy sets A; B∈Fc can be de8ned as the set C ∈Fc (denoted by A −h B), whenever it exists, such that A=B + C (see [11]). Puri and Ralescu [10] de8ned the d∞ metric on Fc by the following expression: d∞ (A; B)= sup ∈ [0;1] dH (A ; B ) for every A; B∈Fc . For A∈Fc , A will denote d∞ (A; 1{0} ), being 1{0} the characteristic function of {0}. It is well known that (Fc ; d∞ ) is complete but non-separable [8,12]. Given ( ; A; P) a probability space, a mapping X : → Fc is said to be a fuzzy random variable [12] if X : → Kc , given by X (!)=(X (!)) , is a convex compact random set for all ∈[0; 1] (see [4]). A fuzzy random variable X is said to be integrably bounded with respect to P if X0 ∈L1 (P). For this kind of mappings it is possible to de8ne the expected value as the unique fuzzy set EX ∈Fc such that (EX ) =EX for every ∈[0; 1] (see [12]), where EX is the KudNo-Aumann
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integral of X . We will denote EX = X (!) dP(!) and when we work on R, we will denote b X (!) dP(!)= [a;b] X (!) dP(!). De8nitions hold when working with measures instead of proba abilities, as long as we handle integrable mappings. Throughout the paper, ([a; b]; M[a; b] ; m) will denote the Lebesgue measure space on the interval [a; b]. In [12] an extension to the Fc -valued case of the Hukuhara’s derivability criterion [6] for setvalued mappings can be found. Thus, a mapping F : T → Fc is said to be Hukuhara derivable at t0 if there exists F (t0 )∈Fc such that both limits lim +
Pt → 0
F(t0 + Pt) −h F(t0 ) Pt
and
lim +
Pt → 0
F(t0 ) −h F(t0 − Pt) Pt
exist and they are equal to F (t0 ), which is called the Hukuhara derivative of F at t0 .
3. Supporting results In this section, we recall some results and we prove some new ones that will be very useful for the proofs of the main results. Denition 3.1. Let F : [a; b] → Fc be a fuzzy valued mapping. Let P : [a; b] → Fc be a Hukuhara derivable mapping at every t ∈(a; b). P is said to be a primitive of F if the Hukuhara derivative of P equals F for every t ∈(a; b), that is, P (t) =F(t). The following result guarantees that two primitives of the same mapping di6er only in a constant when their Hukuhara di6erence exists. Proposition 3.1. Let F : [a; b] → Fc and G : [a; b] → Fc be two Hukuhara derivable mappings. If F and G are both primitives of the same mapping and there exists F(t) −h G(t) for every t ∈(a; b), then F(t)=G(t) + C, being C ∈Fc . Proof. Denote F(t)=G(t) + C(t). By taking the Hukuhara derivative at both sides, we have that F (t)=G (t) + C (t) and, whence, C (t)=1{0} for every t ∈(a; b), which implies (see, [5]) that C is constant. The next result is a weakened version of the First Fundamental Theorem of Calculus for fuzzy valued mappings. The version using continuity at every point can be found in Diamond and Kloeden [5]. The proof can be achieved by using a similar scheme to the continuous case, just by taking into account the continuity a.e. Proposition 3.2. If F : [a; b] → Fc is a continuous mapping at [a; b] a.e. [m], then G(t)= is Hukuhara derivable a:e: [m], with derivative G (t)=F(t) a:e: [m]. The following result can be found in Kaleva [7].
t a
F(s) dm(s)
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Proposition 3.3. Let F : [a; b] → Fc be Hukuhara derivable at[a; b]. If its derivative F is integrable t t on [a; b], then F(t2 )=F(t1 ) + t12 F (s) ds or, equivalently, t12 F (s) ds=F(t2 ) −h F(t1 ) for every t1 ; t2 ∈[a; b], being a6t1 6t2 6b. Given two fuzzy random variables their Hukuhara di6erence, in case of existence, is also a fuzzy random variable, as we prove in the following result. Lemma 3.4. Let X : → Fc and Y : → Fc be two integrably bounded fuzzy random variables such that there exists X (!) −h Y (!) for every !∈ . Denoting by X −h Y : → Fc the mapping given by (X −h Y )(!)=X (!) −h Y (!) then X −h Y is an integrably bounded fuzzy random variable and E(X −h Y )=E(X ) −h E(Y ). Proof. For every ∈[0; 1] we have that [X −h Y ] =X −h Y =[min X − min Y ; max X − max Y ]. Since X and Y are integrably bounded fuzzy random variables, then min X ; min Y ; max X and max Y are L1 (P) random variables. This implies that X −h Y is an integrably bounded fuzzy random variable. On the other hand, we have that X =(X −h Y ) + Y and, so, EX =E(X −h Y ) + E(Y ), which implies the existence of E(X ) −h E(Y ) and the fact that E(X −h Y )=E(X ) −h E(Y ).
4. Main results In this section we state the main results of the paper, i.e. we obtain mild conditions which guarantee the exchange of the expected value of a fuzzy random variable depending on a certain parameter and the Hukuhara derivative with respect to the parameter. These results are structured into two subsections because we have obtained them by using two di6erent methods. The 8rst one is based on the exchange of iterated expected values and so needs Fubini’s theorem for fuzzy random variables (see [9]) which requires the parameter space to be also a measurable space, whereas the second one is based on results about the continuity and derivability under the integral sign. 4.1. Method 1 We prove the 8rst result about the Hukuhara derivative of the fuzzy expected value within the class Fc . Theorem 4.1. Let ( ; A; P) be a probability space and let ([a; b]; M[a; b] ; m) be the Lebesgue measure space. We consider the product space ( × [a; b]; A ⊗ M[a; b] ; P ⊗ m). Let X : × [a; b] → Fc be a mapping such that (i) X is an integrably bounded fuzzy random variable with respect to P ⊗ m, (ii) for each !∈ , X! : [a; b] → Fc is continuous at [a; b] and Hukuhara derivable at (a; b), with a continuous Hukuhara derivative denoted by (@=@t)X (!; t),
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(iii) the mapping (@=@·)X (·; ·) : × (a; b) → Fc is an integrably bounded fuzzy random variable with respect to P ⊗ m, (iv) there exists h∈L1 (P) with (@=@t)X (!; t)6h(!) a:s: [P] for every t ∈(a; b), then @ @ X (!; t) dP(!) X (!; t) dP(!)= @t @t for all t ∈(a; b). Proof. Since (@=@t)X (!; t) is assumed to be continuous and (@=@t)X (!; t)6h(!) then, by using Theorem 4.3—Dominated Convergence Theorem for fuzzy random variables—in Puri and Ralescu [12], we obtain that the mapping t → (@=@t)X (!; t) dP(!) is continuous. Therefore, we can apply Proposition 3.2 to the last mapping in order to obtain that t @ @ @ X (!; t) dP(!)= X (!; s) dP(!) dm(s): @t a @t @s Now, by using a result from L*opez-D*+az and Gil [9] we can ensure the exchangeability of the preceeding two iterated integrals, so that t t @ @ @ @ X (!; s) dP(!) dm(s)= X (!; s) dm(s) dP(!): @t a @t a @s @s Because of the continuity of the Hukuhara derivative, we can apply Proposition 3.3 to obtain t @ @ @ (X (!; t) −h X (!; a)) dP(!): X (!; s) dm(s) dP(!)= @t @t a @s And, by using Lemma 3.4, we obtain that @ @ @ (X (!; t) −h X (!; a)) dP(!)= X (!; t) dP(!) −h X (!; a) dP(!): @t @t @t Since X (!; a) dP(!) is not dependent on t, its Hukuhara derivate vanishes, thus: @ @ @ X (!; t) dP(!) −h 1{0} = X (!; t) dP(!) X (!; t) dP(!)= @t @t @t for all t ∈(a; b), which concludes the proof. 4.2. Method 2 Assumptions for the second method are di6erent from those in Method 1, since the assumed measurability conditions in Method 2 does not necessarily imply the product measurability (which is a must in Method 1), as we will show with an example at the end of this section. On the other hand, it is easy to check that conditions in Method 1 does not imply those in Method 2.
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First we will obtain a continuity result under the integral sign which will be essential to obtain the second result about Hukuhara derivability under the integral sign. Proposition 4.2. Let ( ; A; P) be a probability space and let T ⊆ Rk be a non-empty convex closed subset with t0 ∈int(T ). If Y : × T → Fc veriCes that (i) Yt : → Fc is a fuzzy random variable such that there exists h∈L1 (P) with Yt 6h a:s: [P], for every t in a neighbourhood N ⊂ T of t0 , (ii) for almost every !∈ , Y! : T → Fc is continuous at t0 , then lim Y (!; t) dP(!) = Y (!; t0 ) dP(!): t − t0 → 0
Proof. Consider {tn }n ∈ N ⊂ T such that tn − t0 → 0 when n → ∞. Due to hypothesis (i), when n is large enough, we have that there exists h∈L1 (P) with Ytn 6h a:s: [P]. Denote by D ⊂ the set of points ! such that Y! is not continuous, and by N ∈A a set such that D ⊆ N and P(N )=0. So, by using Theorem 4.3 in Puri and Ralescu [12], we obtain that lim =
tn − t0 → 0
Y (!; tn ) dP(!) =
lim
tn − t0 → 0
lim
\ N tn − t0 → 0
Y (!; tn ) dP(!)=
Y (!; tn ) dP(!) Y (!; t0 ) dP(!)= Y (!; t0 ) dP(!)
\N
\N
and the proof is 8nished. The next theorem is the Hukuhara derivability result obtained without using any Fubini’s theorem, which allows us to work without measurability conditions on the product space. Theorem 4.3. Let ( ; A; P) be a probability space and let T ⊆ Rk be a non-empty convex closed subset with t0 ∈int(T ). If X : × T → Fc veriCes that: (i) for every t ∈T , Xt : → Fc is an integrably bounded fuzzy random variable, (ii) for almost every !∈ , X! : T → Fc is Hukuhara derivable at a neighbourhood N ⊂ T of t0 , (iii) there exists h∈L1 (P) such that (@=@t)X (!; t0 )6h(!) a:s: [P], where (@=@t)X (!; t0 ) denotes Hukuhara derivative of X! at t0 , then the mapping EX : T → Fc , given by EX (t)= X (!; t) dP(!), is Hukuhara derivable at t0 and its derivative is given by (@=@t)EX (t0 )=E(@=@t)X (!; t0 ). Proof. By hypothesis (iii), we assure the existence of E(@=@t)X (!; t0 ). Let t ∈T and {!n }n ∈ N ⊂ R+ be a sequence with null limit, due to hypothesis (ii), there exist the di6erences X (!; t0 + !n (t − t0 )) −h X (!; t0 ) a.s. [P] and, so, by using Lemma 3.4, we deduce that EX (!; t0 + !n (t − t0 )) −h EX (!; t0 ) X (!; t0 + !n (t − t0 )) −h X (!; t0 ) : =E !n !n
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By applying Dominated Convergence Theorem (more precisely, its version in the case of fuzzy random variables, see [12]) to the sequence of mappings X (!; t0 + !n (t − t0 )) −h X (!; t0 ) ; !n we obtain
EX (!; t0 + !n (t − t0 )) −h EX (!; t0 ) X (!; t0 + !n (t − t0 )) −h X (!; t0 ) = lim + E lim !n →0+ !n → 0 !n !n @ X (!; t0 + !n (t − t0 )) −h X (!; t0 ) = E lim + =E X (!; t0 ): !n → 0 !n @t
Similarly, we deduce that: lim +
!n → 0
EX (!; t0 ) −h EX (!; t0 − !n (t − t0 )) @ =E X (!; t0 ); !n @t
and therefore the proof is 8nished. We can easily extend the previous result to a more general case. Theorem 4.4. Let ( ; A; P) be a probability space and let T ⊆ Rk be a non-empty closed and convex subset. If X : × T → Fc veriCes that (i) for every t ∈T , Xt : → Fc is an integrably bounded fuzzy random variable, (ii) for almost all !∈ , X! : T → Fc is Hukuhara derivable at int(T ), (iii) there exists h∈L1 (P) such that (@=@t)X (!; t)6h(!) a.s. [P], then the mapping EX : I → Fc , given by EX (t)= X (!; t) dP(!), is Hukuhara derivable at int(T ) and its derivative is @ @ EX (t)= X (!; t) dP(!); @t @t for every t ∈ int(T ). Finally, we want to underline that measurability conditions in Method 2 does not imply those in Method 1, as we can see from the following example. Consider ([0; 1]; B[0;1] ; m) and C the Cantor subset and let f : R → R2 be de8ned as f(x)=(x; x). Assume that A=f(C) and g : [0; 1] × [0; 1] → R given by g(!; x)=1A (!; x). De8ne now the mapping G : [0; 1]2 → Fc given by G(!; x)=1{g(!; x)} . Clearly, G satis8es the conditions in Theorem 4.3 but it is obviously not product measurable (since that would imply the Borel measurability of g). 5. Concluding remarks In this paper we have stated two main results about the Hukuhara derivability of the fuzzy expected value of a fuzzy random variable depending on a parameter. The main contribution of
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the paper is to obtain an exchange result within two di6erent frameworks. In the 8rst one, we have supposed the applicability of a version of Fubini’s theorem for fuzzy random variables, and we have deduced the exchanging result on the basis of this theorem. In the second framework, we have obtained the exchanging result without supposing the applicability of Fubini’s theorem. Regarding future directions, this results can be very useful in Statistical Decision Theory with fuzzy-valued utilities, which we are developing and it will appear in a forthcoming paper. Acknowledgements The authors wish sincerely to acknowledge their colleague Prof. M.A. Gil for her kindness and helpful partnership. We also want to express our gratitude to the Area Editor and the referees for their suggestions to improve the paper. We would also like to thank the Spanish Ministry of Science and Technology for their 8nancial support (Grant BFM2002-03262). References [1] H.T. Banks, M.Q. Jacobs, A di6erential calculus for multifunctions, J. Math. Anal. Appl. 29 (1970) 246,272. [2] C.W. Burrill, Measure, Integration, and Probability, McGraw-Hill Book Co., New York-DWusseldorf-Johannesburg, 1972. [3] F.S. De Blasi, On the di6erentiability of multifunctions, Paci8c J. Math. 66 (1976) 67–81. [4] G. Debreu, Integration of correspondences, in: Proceedings of the Fifth Berkeley Symposium on Mathematics Statistical and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1, Univ. California Press, Berkeley, CA, 1967, pp. 351–372. [5] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets, World Scienti8c, New Jersey, 1994. [6] M. Hukuhara, Int*egration des applications measurables dont la valeur est un compact convexe, Funkcial. Ekvac. 10 (1967) 205–223. [7] O. Kaleva, Fuzzy di6erential equations, Fuzzy Sets and Systems 24 (1987) 301,317. [8] E.P. Klement, M.L. Puri, D.A. Ralescu, Limit theorems for fuzzy random variables, Proc. R. Soc. Lond. A 407 (1986) 171–182. [9] M. L*opez-D*+az, M.A. Gil, An extension of Fubini’s theorem for fuzzy random variables, Inform. Sci. 115 (1999) 29,41. [10] M.L. Puri, D.A. Ralescu, Di6*erentielle d’une fonction Youe, C.R. Acad. Sci. Paris, S*er. A 293 (1981) 237,239. [11] M.L. Puri, D.A. Ralescu, Di6erentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552–558. [12] M.L. Puri, D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409,422. [13] C. Ramamohana Rao, On di6erentiation under the integral sign, Math. Student 25 (1957) 143–146. [14] L.J. Rodr*+guez-Mu˜niz, Several notions of di6erentiability for fuzzy set-valued mappings, in: C. Bertoluzza, M.A. Gil, D.A. Ralescu (Eds.), Statistical Modelling, Analysis and Management of Fuzzy Data, Studies in Fuzzines and Soft Computing, No. 87, Physica-Verlag, Heidelberg, 2002, pp. 104,116. [15] L.J. Rodr*+guez-Mu˜niz, M. L*opez-D*+az, M.A. Gil, D.A. Ralescu, The s-di6erentiability of a fuzzy-valued mapping, Inform. Sci., 2003a, to appear. [16] L.J. Rodr*+guez-Mu˜niz, M. L*opez-D*+az, M.A. Gil, Di6erentiating random upper semicontinuous functions under the integral sign, TEST, 2003b, to appear.
Hukuhara derivative of the fuzzy expected value Luis J. Rodr*+guez-Mu˜niza; b;∗ , Miguel L*opez-D*+aza; c a
b
Departamento de Estad stica e I.O. y D.M., Universidad de Oviedo, Spain Escuela Superior de la Marina Civil, Campus de Viesques, 33271 Gij on, Spain c Facultad de Ciencias, c/ Calvo Sotelo s/n, 33071 Oviedo, Spain
Received 21 May 2002; received in revised form 9 January 2003; accepted 16 January 2003
Abstract In this paper we establish su5cient conditions to guarantee the exchange of the fuzzy expected value for a fuzzy random variable depending on certain parameter and its Hukuhara derivative with respect to the parameter. Main results are obtained for two di6erent sets of general conditions, one of them requiring measurability conditions with respect to the product space the fuzzy random variable is de8ned from, and the second one just requiring conditions over the projections of the fuzzy random variables. c 2003 Elsevier B.V. All rights reserved. Keywords: Fuzzy expected value; Fuzzy random variable; Hukuhara derivative; Hukuhara di6erence
1. Introduction The study of the di6erentiability under the integral sign has been one of the classical topics in the theory of mathematical analysis, not only because of its theoretical interest but also for the wide variety of applications of these results. These mathematical results try to obtain su5cient conditions to exchange an integral of a mapping which depends on a certain parameter and a derivative/di6erential of the map with respect to the parameter. Of course, there have been di6erent studies about this kind of problem, since there exists a great variety of possibilities depending on the considered mapping, especially depending on the values it takes on, on the concept of integral used, and of course, on the di6erentiability criterion. The 8rst results that were obtained referred to real-valued random variables (see, for instance, [2,13]). Banks and Jacobs [1] were the 8rst ones who studied the problem of di6erentiability under the integral sign for set-valued mappings. Since, in some sense, fuzzy sets over a certain space ∗
Corresponding author. Fax: +34-985181902. E-mail address: [email protected] (L.J. Rodr*+guez-Mu˜niz).
c 2003 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter doi:10.1016/S0165-0114(03)00020-4
594
L.J. Rodr guez-Mu˜niz, M. L opez-D az / Fuzzy Sets and Systems 138 (2003) 593 – 600
are a generalization of the class of subsets of that space, a natural question arises: how to obtain general conditions for the di6erentiability under the integral sign when fuzzy-valued mappings are considered. The authors have obtained in Rodr*+guez-Mu˜niz et al. [16] some results regarding the di6erentiability under the integral sign for fuzzy random variables when four di6erent concepts of di6erentiability are considered, namely, the De Blasi di6erentiability [3,5], the -di6erentiability [1,11], the Fr*echet di6erentiability of the associated support function, and the s-di6erentiability [14,15] and the integral is the one established by Puri and Ralescu [12] for the concept of expected value of a fuzzy random variable. In the present paper, our motivation is to obtain new theoretical results about that topic, when we consider a wider class of fuzzy random variables than the one considered in Rodr*+guez-Mu˜niz et al. [16], as we will see, here we will obtain results just by requiring convexity but not continuity on the -level sets, as in that paper. On the other hand, the derivability criterion we will use is the Hukuhara’s one. In other words, we will enlarge the class of variables by restricting the derivability criterion. Main results in the paper are obtained under two general and di6erent frameworks, in the 8rst one product measurability conditions are required whereas in the second framework the product measurability condition is not required, which makes this result especially useful for those cases in which that measurability is not guaranteed. Our purpose is to use these theoretical results as a basis to obtain more results which can be applied in Statistical Decision Theory (for instance, in the equivalence between normal and extensive forms of Bayesian analysis). The paper is organized as follows: in Section 2 we provide some preliminaries. Section 3 is formed by several supporting results. Main results about the Hukuhara derivative of the fuzzy expected value are gathered in Section 4. Finally, some open problems are outlined in Section 5. 2. Preliminaries Let us denote by Kc the class of non-empty compact and convex subsets of R and by dH the Hausdor6 metric. Fc will denote the class of fuzzy sets A : R → [0; 1] such that A ∈Kc for all ∈[0; 1], being A ={x ∈R : A(x)¿} for ∈(0; 1] and A0 =supp A. This class can be endowed with a semilinear structure by means of Zadeh’s extension principle or, equivalently (see [10]), addition and product by a scalar can be levelwise calculated through (A+B) =A +B (Minkowski’s addition) and ( A) = A . Obviously, (Fc ; +; ·) is not a vector space. Hukuhara’s di=erence between two fuzzy sets A; B∈Fc can be de8ned as the set C ∈Fc (denoted by A −h B), whenever it exists, such that A=B + C (see [11]). Puri and Ralescu [10] de8ned the d∞ metric on Fc by the following expression: d∞ (A; B)= sup ∈ [0;1] dH (A ; B ) for every A; B∈Fc . For A∈Fc , A will denote d∞ (A; 1{0} ), being 1{0} the characteristic function of {0}. It is well known that (Fc ; d∞ ) is complete but non-separable [8,12]. Given ( ; A; P) a probability space, a mapping X : → Fc is said to be a fuzzy random variable [12] if X : → Kc , given by X (!)=(X (!)) , is a convex compact random set for all ∈[0; 1] (see [4]). A fuzzy random variable X is said to be integrably bounded with respect to P if X0 ∈L1 (P). For this kind of mappings it is possible to de8ne the expected value as the unique fuzzy set EX ∈Fc such that (EX ) =EX for every ∈[0; 1] (see [12]), where EX is the KudNo-Aumann
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integral of X . We will denote EX = X (!) dP(!) and when we work on R, we will denote b X (!) dP(!)= [a;b] X (!) dP(!). De8nitions hold when working with measures instead of proba abilities, as long as we handle integrable mappings. Throughout the paper, ([a; b]; M[a; b] ; m) will denote the Lebesgue measure space on the interval [a; b]. In [12] an extension to the Fc -valued case of the Hukuhara’s derivability criterion [6] for setvalued mappings can be found. Thus, a mapping F : T → Fc is said to be Hukuhara derivable at t0 if there exists F (t0 )∈Fc such that both limits lim +
Pt → 0
F(t0 + Pt) −h F(t0 ) Pt
and
lim +
Pt → 0
F(t0 ) −h F(t0 − Pt) Pt
exist and they are equal to F (t0 ), which is called the Hukuhara derivative of F at t0 .
3. Supporting results In this section, we recall some results and we prove some new ones that will be very useful for the proofs of the main results. Denition 3.1. Let F : [a; b] → Fc be a fuzzy valued mapping. Let P : [a; b] → Fc be a Hukuhara derivable mapping at every t ∈(a; b). P is said to be a primitive of F if the Hukuhara derivative of P equals F for every t ∈(a; b), that is, P (t) =F(t). The following result guarantees that two primitives of the same mapping di6er only in a constant when their Hukuhara di6erence exists. Proposition 3.1. Let F : [a; b] → Fc and G : [a; b] → Fc be two Hukuhara derivable mappings. If F and G are both primitives of the same mapping and there exists F(t) −h G(t) for every t ∈(a; b), then F(t)=G(t) + C, being C ∈Fc . Proof. Denote F(t)=G(t) + C(t). By taking the Hukuhara derivative at both sides, we have that F (t)=G (t) + C (t) and, whence, C (t)=1{0} for every t ∈(a; b), which implies (see, [5]) that C is constant. The next result is a weakened version of the First Fundamental Theorem of Calculus for fuzzy valued mappings. The version using continuity at every point can be found in Diamond and Kloeden [5]. The proof can be achieved by using a similar scheme to the continuous case, just by taking into account the continuity a.e. Proposition 3.2. If F : [a; b] → Fc is a continuous mapping at [a; b] a.e. [m], then G(t)= is Hukuhara derivable a:e: [m], with derivative G (t)=F(t) a:e: [m]. The following result can be found in Kaleva [7].
t a
F(s) dm(s)
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Proposition 3.3. Let F : [a; b] → Fc be Hukuhara derivable at[a; b]. If its derivative F is integrable t t on [a; b], then F(t2 )=F(t1 ) + t12 F (s) ds or, equivalently, t12 F (s) ds=F(t2 ) −h F(t1 ) for every t1 ; t2 ∈[a; b], being a6t1 6t2 6b. Given two fuzzy random variables their Hukuhara di6erence, in case of existence, is also a fuzzy random variable, as we prove in the following result. Lemma 3.4. Let X : → Fc and Y : → Fc be two integrably bounded fuzzy random variables such that there exists X (!) −h Y (!) for every !∈ . Denoting by X −h Y : → Fc the mapping given by (X −h Y )(!)=X (!) −h Y (!) then X −h Y is an integrably bounded fuzzy random variable and E(X −h Y )=E(X ) −h E(Y ). Proof. For every ∈[0; 1] we have that [X −h Y ] =X −h Y =[min X − min Y ; max X − max Y ]. Since X and Y are integrably bounded fuzzy random variables, then min X ; min Y ; max X and max Y are L1 (P) random variables. This implies that X −h Y is an integrably bounded fuzzy random variable. On the other hand, we have that X =(X −h Y ) + Y and, so, EX =E(X −h Y ) + E(Y ), which implies the existence of E(X ) −h E(Y ) and the fact that E(X −h Y )=E(X ) −h E(Y ).
4. Main results In this section we state the main results of the paper, i.e. we obtain mild conditions which guarantee the exchange of the expected value of a fuzzy random variable depending on a certain parameter and the Hukuhara derivative with respect to the parameter. These results are structured into two subsections because we have obtained them by using two di6erent methods. The 8rst one is based on the exchange of iterated expected values and so needs Fubini’s theorem for fuzzy random variables (see [9]) which requires the parameter space to be also a measurable space, whereas the second one is based on results about the continuity and derivability under the integral sign. 4.1. Method 1 We prove the 8rst result about the Hukuhara derivative of the fuzzy expected value within the class Fc . Theorem 4.1. Let ( ; A; P) be a probability space and let ([a; b]; M[a; b] ; m) be the Lebesgue measure space. We consider the product space ( × [a; b]; A ⊗ M[a; b] ; P ⊗ m). Let X : × [a; b] → Fc be a mapping such that (i) X is an integrably bounded fuzzy random variable with respect to P ⊗ m, (ii) for each !∈ , X! : [a; b] → Fc is continuous at [a; b] and Hukuhara derivable at (a; b), with a continuous Hukuhara derivative denoted by (@=@t)X (!; t),
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(iii) the mapping (@=@·)X (·; ·) : × (a; b) → Fc is an integrably bounded fuzzy random variable with respect to P ⊗ m, (iv) there exists h∈L1 (P) with (@=@t)X (!; t)6h(!) a:s: [P] for every t ∈(a; b), then @ @ X (!; t) dP(!) X (!; t) dP(!)= @t @t for all t ∈(a; b). Proof. Since (@=@t)X (!; t) is assumed to be continuous and (@=@t)X (!; t)6h(!) then, by using Theorem 4.3—Dominated Convergence Theorem for fuzzy random variables—in Puri and Ralescu [12], we obtain that the mapping t → (@=@t)X (!; t) dP(!) is continuous. Therefore, we can apply Proposition 3.2 to the last mapping in order to obtain that t @ @ @ X (!; t) dP(!)= X (!; s) dP(!) dm(s): @t a @t @s Now, by using a result from L*opez-D*+az and Gil [9] we can ensure the exchangeability of the preceeding two iterated integrals, so that t t @ @ @ @ X (!; s) dP(!) dm(s)= X (!; s) dm(s) dP(!): @t a @t a @s @s Because of the continuity of the Hukuhara derivative, we can apply Proposition 3.3 to obtain t @ @ @ (X (!; t) −h X (!; a)) dP(!): X (!; s) dm(s) dP(!)= @t @t a @s And, by using Lemma 3.4, we obtain that @ @ @ (X (!; t) −h X (!; a)) dP(!)= X (!; t) dP(!) −h X (!; a) dP(!): @t @t @t Since X (!; a) dP(!) is not dependent on t, its Hukuhara derivate vanishes, thus: @ @ @ X (!; t) dP(!) −h 1{0} = X (!; t) dP(!) X (!; t) dP(!)= @t @t @t for all t ∈(a; b), which concludes the proof. 4.2. Method 2 Assumptions for the second method are di6erent from those in Method 1, since the assumed measurability conditions in Method 2 does not necessarily imply the product measurability (which is a must in Method 1), as we will show with an example at the end of this section. On the other hand, it is easy to check that conditions in Method 1 does not imply those in Method 2.
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First we will obtain a continuity result under the integral sign which will be essential to obtain the second result about Hukuhara derivability under the integral sign. Proposition 4.2. Let ( ; A; P) be a probability space and let T ⊆ Rk be a non-empty convex closed subset with t0 ∈int(T ). If Y : × T → Fc veriCes that (i) Yt : → Fc is a fuzzy random variable such that there exists h∈L1 (P) with Yt 6h a:s: [P], for every t in a neighbourhood N ⊂ T of t0 , (ii) for almost every !∈ , Y! : T → Fc is continuous at t0 , then lim Y (!; t) dP(!) = Y (!; t0 ) dP(!): t − t0 → 0
Proof. Consider {tn }n ∈ N ⊂ T such that tn − t0 → 0 when n → ∞. Due to hypothesis (i), when n is large enough, we have that there exists h∈L1 (P) with Ytn 6h a:s: [P]. Denote by D ⊂ the set of points ! such that Y! is not continuous, and by N ∈A a set such that D ⊆ N and P(N )=0. So, by using Theorem 4.3 in Puri and Ralescu [12], we obtain that lim =
tn − t0 → 0
Y (!; tn ) dP(!) =
lim
tn − t0 → 0
lim
\ N tn − t0 → 0
Y (!; tn ) dP(!)=
Y (!; tn ) dP(!) Y (!; t0 ) dP(!)= Y (!; t0 ) dP(!)
\N
\N
and the proof is 8nished. The next theorem is the Hukuhara derivability result obtained without using any Fubini’s theorem, which allows us to work without measurability conditions on the product space. Theorem 4.3. Let ( ; A; P) be a probability space and let T ⊆ Rk be a non-empty convex closed subset with t0 ∈int(T ). If X : × T → Fc veriCes that: (i) for every t ∈T , Xt : → Fc is an integrably bounded fuzzy random variable, (ii) for almost every !∈ , X! : T → Fc is Hukuhara derivable at a neighbourhood N ⊂ T of t0 , (iii) there exists h∈L1 (P) such that (@=@t)X (!; t0 )6h(!) a:s: [P], where (@=@t)X (!; t0 ) denotes Hukuhara derivative of X! at t0 , then the mapping EX : T → Fc , given by EX (t)= X (!; t) dP(!), is Hukuhara derivable at t0 and its derivative is given by (@=@t)EX (t0 )=E(@=@t)X (!; t0 ). Proof. By hypothesis (iii), we assure the existence of E(@=@t)X (!; t0 ). Let t ∈T and {!n }n ∈ N ⊂ R+ be a sequence with null limit, due to hypothesis (ii), there exist the di6erences X (!; t0 + !n (t − t0 )) −h X (!; t0 ) a.s. [P] and, so, by using Lemma 3.4, we deduce that EX (!; t0 + !n (t − t0 )) −h EX (!; t0 ) X (!; t0 + !n (t − t0 )) −h X (!; t0 ) : =E !n !n
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By applying Dominated Convergence Theorem (more precisely, its version in the case of fuzzy random variables, see [12]) to the sequence of mappings X (!; t0 + !n (t − t0 )) −h X (!; t0 ) ; !n we obtain
EX (!; t0 + !n (t − t0 )) −h EX (!; t0 ) X (!; t0 + !n (t − t0 )) −h X (!; t0 ) = lim + E lim !n →0+ !n → 0 !n !n @ X (!; t0 + !n (t − t0 )) −h X (!; t0 ) = E lim + =E X (!; t0 ): !n → 0 !n @t
Similarly, we deduce that: lim +
!n → 0
EX (!; t0 ) −h EX (!; t0 − !n (t − t0 )) @ =E X (!; t0 ); !n @t
and therefore the proof is 8nished. We can easily extend the previous result to a more general case. Theorem 4.4. Let ( ; A; P) be a probability space and let T ⊆ Rk be a non-empty closed and convex subset. If X : × T → Fc veriCes that (i) for every t ∈T , Xt : → Fc is an integrably bounded fuzzy random variable, (ii) for almost all !∈ , X! : T → Fc is Hukuhara derivable at int(T ), (iii) there exists h∈L1 (P) such that (@=@t)X (!; t)6h(!) a.s. [P], then the mapping EX : I → Fc , given by EX (t)= X (!; t) dP(!), is Hukuhara derivable at int(T ) and its derivative is @ @ EX (t)= X (!; t) dP(!); @t @t for every t ∈ int(T ). Finally, we want to underline that measurability conditions in Method 2 does not imply those in Method 1, as we can see from the following example. Consider ([0; 1]; B[0;1] ; m) and C the Cantor subset and let f : R → R2 be de8ned as f(x)=(x; x). Assume that A=f(C) and g : [0; 1] × [0; 1] → R given by g(!; x)=1A (!; x). De8ne now the mapping G : [0; 1]2 → Fc given by G(!; x)=1{g(!; x)} . Clearly, G satis8es the conditions in Theorem 4.3 but it is obviously not product measurable (since that would imply the Borel measurability of g). 5. Concluding remarks In this paper we have stated two main results about the Hukuhara derivability of the fuzzy expected value of a fuzzy random variable depending on a parameter. The main contribution of
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the paper is to obtain an exchange result within two di6erent frameworks. In the 8rst one, we have supposed the applicability of a version of Fubini’s theorem for fuzzy random variables, and we have deduced the exchanging result on the basis of this theorem. In the second framework, we have obtained the exchanging result without supposing the applicability of Fubini’s theorem. Regarding future directions, this results can be very useful in Statistical Decision Theory with fuzzy-valued utilities, which we are developing and it will appear in a forthcoming paper. Acknowledgements The authors wish sincerely to acknowledge their colleague Prof. M.A. Gil for her kindness and helpful partnership. We also want to express our gratitude to the Area Editor and the referees for their suggestions to improve the paper. We would also like to thank the Spanish Ministry of Science and Technology for their 8nancial support (Grant BFM2002-03262). References [1] H.T. Banks, M.Q. Jacobs, A di6erential calculus for multifunctions, J. Math. Anal. Appl. 29 (1970) 246,272. [2] C.W. Burrill, Measure, Integration, and Probability, McGraw-Hill Book Co., New York-DWusseldorf-Johannesburg, 1972. [3] F.S. De Blasi, On the di6erentiability of multifunctions, Paci8c J. Math. 66 (1976) 67–81. [4] G. Debreu, Integration of correspondences, in: Proceedings of the Fifth Berkeley Symposium on Mathematics Statistical and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1, Univ. California Press, Berkeley, CA, 1967, pp. 351–372. [5] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets, World Scienti8c, New Jersey, 1994. [6] M. Hukuhara, Int*egration des applications measurables dont la valeur est un compact convexe, Funkcial. Ekvac. 10 (1967) 205–223. [7] O. Kaleva, Fuzzy di6erential equations, Fuzzy Sets and Systems 24 (1987) 301,317. [8] E.P. Klement, M.L. Puri, D.A. Ralescu, Limit theorems for fuzzy random variables, Proc. R. Soc. Lond. A 407 (1986) 171–182. [9] M. L*opez-D*+az, M.A. Gil, An extension of Fubini’s theorem for fuzzy random variables, Inform. Sci. 115 (1999) 29,41. [10] M.L. Puri, D.A. Ralescu, Di6*erentielle d’une fonction Youe, C.R. Acad. Sci. Paris, S*er. A 293 (1981) 237,239. [11] M.L. Puri, D.A. Ralescu, Di6erentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552–558. [12] M.L. Puri, D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409,422. [13] C. Ramamohana Rao, On di6erentiation under the integral sign, Math. Student 25 (1957) 143–146. [14] L.J. Rodr*+guez-Mu˜niz, Several notions of di6erentiability for fuzzy set-valued mappings, in: C. Bertoluzza, M.A. Gil, D.A. Ralescu (Eds.), Statistical Modelling, Analysis and Management of Fuzzy Data, Studies in Fuzzines and Soft Computing, No. 87, Physica-Verlag, Heidelberg, 2002, pp. 104,116. [15] L.J. Rodr*+guez-Mu˜niz, M. L*opez-D*+az, M.A. Gil, D.A. Ralescu, The s-di6erentiability of a fuzzy-valued mapping, Inform. Sci., 2003a, to appear. [16] L.J. Rodr*+guez-Mu˜niz, M. L*opez-D*+az, M.A. Gil, Di6erentiating random upper semicontinuous functions under the integral sign, TEST, 2003b, to appear.