STOCHASTIC COMPARISONS OF FUZZY

Transcription

1 CHAPTER FIVE STOCHASTIC COPARISONS OF FUZZY STOCHASTIC PROCESSES Abstract Chapter 5 investigates the stochastic comparison of fuzzy stochastic processes. This chapter introduce the concept of stochastic comparison of fuzzy stochastic processes. The condition that manifests the stochastic inequality is realized in terms of an increasing functional f. Chapter 5 ends with the concluding section. This section of conclusion includes the summary of the results of this thesis. The contents of this chapter form the substance of the paper entitled Stochastic comparisons of fuzzy stochastic processes, accepted for publication in the International Journal Reflection des ERA-Journal of athematical Sciences, India. 111

2 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes Introduction The theory of fuzzy random variables is a natural extension of classical real valued random variables or random vectors. Fuzzy random variables have many special properties. This allows new meanings for the classical probability theory. As a result of advancement in this area in the past three decades the theory of fuzzy random variables with diverse applications has become one of new and active branches in probability theory. In reality we often come across with random experiments whose outcomes are not numbers but are expressed in inexact linguistic terms, which varies with time t. Such linguistic terms will be represented by a dynamic fuzzy set [49]. This is a typical fuzzy stochastic phenomenon with prolonged time. Fuzzy random variables [33, 34, 44, 67] are mathematical characterizations for fuzzy stochastic phenomena, but only one point of time description. For the formulation of a fuzzy stochastic process,

3 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 113 fuzzy random variables should be considered repeatedly and even continuously to describe and investigate the structure of their family. As a desideratum, the study of fuzzy stochastic processes is essential. Kwakernaak [33, 34] introduced the notion of a fuzzy random variable as a measurable functions F : Ω FR, where Ω, A, P is a probability space and FR denotes all piecewise continuous functions u : R [0, 1]. Puri and Ralescu [44] defined the concept of a fuzzy random variable as a function F : Ω FR n where Ω, A, P is a probability space and FR n denotes all functions u : R n [0, 1] such that {x R n ; ux } is a non-empty and compact for each 0, 1]. In this chapter, a concept of fuzzy random variable, slightly different than that of Kwakernaak [33, 34] and Puri [44] is introduced. It is defined as a measurable fuzzy set valued function X : Ω F 0 R, where R is the real line, Ω, A, P is a probability space,

4 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 114 F 0 R = {A : R [0, 1]} and {x R; Ax } is a bounded closed interval for each 0, 1]. Guangyuan Wang et.al., [18] have introduced the general theory of fuzzy stochastic processes, which include the definitions of fuzzy random function, fuzzy stochastic processes. Earnest Lazarus Piriyakuar et.al., [12] have studied various stochastic comparison of fuzzy random variables. In this chapter the concept of stochastic comparison is extended to fuzzy stochastic processes. Congruous to stochastic comparisons of classical random variables, stochastic comparisons for functionals of fuzzy stochastic processes, which are of practical importance are derived. The stochastic comparison of two fuzzy random variables whose end points of each -cut is univariate in nature can be generalized and the resulting stochastic comparison is nothing but the stochastic comparison of two fuzzy stochastic processes. In many applied problems the exact calculation of quantities of

5 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 115 interest which are obscured by perceptional deficiencies the resulting fuzzy stochastic process pose a variety of complexities. In such cases the only remedy is to compute bounds on these parameters by comparing the given fuzzy stochastic process with a simpler fuzzy stochastic process. This kind of stochastic comparison has great relevance in reliability problems. In this chapter, stronger type of comparison of two fuzzy stochastic processes is introduced. Gordon Pledger et. al [15] have discussed stochastic comparison of random processes with applications in reliability. In Section 5.2, some results related to dynamic fuzzy sets, fuzzy random variables, fuzzy random vectors, fuzzy random function and fuzzy stochastic processes are introduced. In Section 5.3, the concept of stochastic comparison of fuzzy stochastic processes is introduced. Conditions are obtained under which the fuzzy stochastic process {Xt; t 0} stochas-

6 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 116 tically larger than its counter part {Yt; t 0} implies that f {Xt; t 0} st f {Yt; t 0} for increasing functional f, and other properties of stochastic comparison are introduced. 5.2 Preliminaries Let R be the real line and R, B be the Borel measurable space. Let F 0 R denote the set of fuzzy subsets A : R [0, 1] with the following properties: 1. {x R; Ax = 1} φ. 2. A = {x R; Ax } is a bounded closed interval in R for each 0, 1]. i.e., A = [ A L, A U] where A L = inf A and A U = sup A A L, A U A, < A L and A U < for each 0, 1]. A F 0 R is called a bounded closed fuzzy number.

7 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 117 Definition [49]. {At, t T R} is known as a dynamic fuzzy set in U where U is a non empty set with respect to T if At FU, the set of all fuzzy subsets of U, for each t T. In particular {At; t T} is called a normal dynamic fuzzy set if At F 0 R for each t T. Definition Let At be a normal dynamic fuzzy set with respect to T and IR = {[x, y]; x, y R, x y}. Let A : T IR defined as t A t = At = [ A L t, A U t ]. Then A t is known as the level function of At. A is an interval valued mapping on T. Definition [18]. Let Ω, A, P be a probability space. A fuzzy set valued mapping X : Ω F 0 R is called a fuzzy random variable if for each B B and every 0, 1], X 1 B = {ω Ω; X ω B φ} A. A fuzzy set valued mapping X : Ω F m 0 R = F 0R F 0 R

8 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 118 represented by Xω = X1, ω, X2, ω,..., Xm, ω is known as a fuzzy random vector if for each k, 1 k m, Xk, ω is a fuzzy random variable. Definition [18]. Let Ω, A, P be a probability space and X a set valued mapping X : Ω IR defined as ω Xω = [X L ω, X U ω] Then Xω = [X L ω, X U ω] is called a random interval if X L ω and X U ω are both random variables on Ω, A, P. Theorem [18]. Xω is a fuzzy random variable if and only if X ω = [ X L ω, X U ω ] is a random interval for each 0, 1] and Xω = X ω = [ X L ω, X U ω ] ,1] 0,1] This theorem is useful in the construction of various fuzzy sets made in terms of their corresponding -cuts. ost of the results of this chapter are proved for the corresponding -cuts.

9 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 119 With the help of the above theorem we can establish the results for the corresponding fuzzy sets. A family of fuzzy random variables Xt = {Xt, ω; t T} is known as a fuzzy random function. The parameter set T can be viewed as any one of the following: R, R + = [0,, [a, b] R, Z = {0, ±1, ±2,...}, Z + = {0, 1, 2,...}, {1, 2,..., m} and so on. In all these cases the parameter t T can be viewed as time. If T = Z or Z + then a fuzzy random sequence can be realized. If T = R or R + or [a, b], Xt is known as a fuzzy stochastic process. Definition [18]. A fuzzy random function Xt = {Xt, ω, t T} is a fuzzy set valued function from the space T Ω to F 0 R. Xt, is a fuzzy random variable on Ω, A, P for each fixed t T and X, ω is a normal Dynamic fuzzy set with respect to the parameter set T, for each fixed ω Ω. X, ω is called a fuzzy sample function or a fuzzy trajectory.

10 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 120 Definition [18]. Let Ω, A, P be a probability space, T R and X a set valued mapping. X : T Ω IR defined as t, ω Xt, ω = [X L t, ω, X U t, ω] is known as an interval valued random function if X L t,ω and X U t,ω are both random functions. The following theorem is important for the construction of fuzzy stochastic processes using their corresponding -cuts. In this chapter the stochastic comparison of fuzzy stochastic processes are proved in terms of their corresponding -cuts. With the aid of the following theorem stochastic comparison of fuzzy stochastic processes can be realized. Theorem [18]. Xt = {Xt, ω; t T} is a fuzzy random function if and only if for each 0, 1]. X t = {X t, ω, t T} = {[ X L t, ω, X U t, ω ], t T } is

11 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 121 an interval random function for each t T. and every ω Ω and Xt, ω = X t, ω = [ X L t, ω, X U t, ω ] 0,1] 0,1] Definition The fuzzy random variable X is stochastically larger than the fuzzy random variable Y if { P X L > a P X U > a } { P Y L > a P Y U > a } symbolically it is denoted as X st Y. 5.3 Stochastic comparison of fuzzy stochastic processes Theorem Let X and Y be fuzzy random variables. If X st Y then E[X] E[Y]. Proof. Assume first that X and Y are non-negative fuzzy random variables. Then for 0, 1], E [ X L] E [ X U] = P X L > a da P X U > a da 0 0

12 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 122 P Y L > a da P Y U > a da 0 0 = E [ Y L] E [ Y U] Generally one can express any fuzzy random variable Z as the difference of two non-negative fuzzy random variables. Let Z = Z + Z. i.e., For each x R, Zωx; Zωx 0 Z + ωx = 0; Zωx < 0 0; Zωx 0 Z ωx = Zωx; Zωx < 0 X st Y implies P { X L > a } P { X U > a } P { Y L > a } P { Y U > a } Let Xωx 0 and Yωx 0. Then P X + L > a P X + U > a P Y + L > a P Y + U > a

13 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 123 This shows that X + st Y +. Similarly one can prove X st Y. The proof of other cases are similar. E [ X L] E [ X U] = E [ + ] L X E [ + ] U X {[ E X ] L [ E X ]} U E [ + ] L Y E [ + ] U Y {[ E Y ] L [ E Y ]} U = E Y L E Y U The above inequality is true for each 0, 1]. E X L 0,1] E X U 0,1] E Y L E 0,1] Y U 0,1] This shows that E [ [ ] X L, X U] 0,1] E [ [ ] Y L, Y U] [0,1] i.e., E[X] E[Y].

14 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 124 Theorem If X and Y are fuzzy random variables then X st Y if and only if E[ f X] E[ f Y] for all increasing functions f. Proof. Let X st Y, and f be an increasing function. Let f 1 a = inf{x; f x a}. Then P { f X L > a } P { f X U > a } = P { X L > f 1 a } P { X U > f 1 a } P { Y L > f 1 a } P { Y U > f 1 a } = P { f Y L > a } P { f Y U > a } f X L f X U st f Y L f Y U. The above inequality is true for each 0, 1]. f X L f X U st 0,1] f Y L f Y U 0,1] This shows that [ f X L, f X U] st 0,1] [ f Y L, f Y U] 0,1]

15 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 125 This shows that f X st f Y. By Theorem 5.3.1, E[ f X] E[ f Y]. Definition Fuzzy stochastic process {Xt; t 0} is said to be stochastically larger than the fuzzy stochastic process {Yt; t 0} denoted as {Xt; t 0} st {Yt; t 0} if for 0, 1] and for each choice of 0 t 1 < t 1 < t 2 < < t n ; n = 1, 2,.... X L t 1 X L t n st Y L t 1 Y L t n and X U t 1 X U t n st Y U t 1 Y U t n If we consider continuous increasing funtionals f, then the stochastic comparison of f Xt; t 0 with f Yt; t 0 is a particular case of For > 0, let D[0, ] denote the space of all real interval valued functions on [0, ] whose end points are right

16 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 126 continuous and have left hand limits. Let S denote the class of fuzzy stochastic processes {zt; t 0} such that P [ {zt = [z L t, w, z U t, w]; 0 t } D0, IR ] = 1 Let z denote the fuzzy stochastic processes {zt; 0 t }. For n = 1, 2,..., we define z n, the n th approximation of z by z L in 1 for in 1 t i + 1n 1 ; Z n L t = 0 i n 1 z L for t = and z U in 1 for in 1 t i + 1n 1 ; Z n U t = 0 i n 1 z U for t = Theorem Let X and Y be n-dimensional fuzzy random vectors and X and Y be n -dimensional random vectors such

17 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 127 that X st Y, X st Y with X and X independent and Y and Y independent then X, X st Y, Y where X, X denotes the n + n dimensional fuzzy random vector X 1,..., X n, X 1,..., X n. Proof. Let f x, x be a real valued increasing function of n + n 1 arguments such that E f X, X and E f Y, Y exist. Let X be an n -dimensional fuzzy random vector which is independent of X and of Y and follows the same distribution as X. Then E f { X L, X L X L = x L } = E f X L, X L X L = x L E f Y L, X L X L = x L and E f X U, X U X U = x U = E f X U, X U X U = x U E f Y U, X U X U = x U

18 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 128 where x is a fuzzy number. Since f is increasing in its first n arguments, it follows that E f X L, X L E f Y L, X L E f X U, X U E f Y U, X U and Invoking the same argument, it follows that E f Y L, X L E f Y L, Y L E f Y U, X U E f Y U, Y U and Combining the above set of inequalities, E f X L, X L E f Y L, Y L E f X U, X U E f Y U, Y U and The above inequalities are true for each 0, 1]. E f X L, X L 0,1] 0,1] E f Y L, Y L 0,1] 0,1]

19 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 129 and E f X U, X U 0,1] 0,1] E f Y U, Y U 0,1] 0,1] These two inequalities together imply E f [ X L, X L, X U, X U ] 0,1] E f [ Y L, Y L, Y U, Y U ] 0,1] This shows that E f X, X E f Y, Y. Applying Theorem it follows that X, X st Y, Y Theorem Let > 0, {Xt, t 0} S, {Yt; t 0} S and Γ D[0, ] such that for 0, 1] and for n = 1, 2,... P[X Γ, Y Γ, X n, Γ, Y n, Γ] = 1. Let f : Γ, be continuous and increasing. Then {Xt, t 0} st {Yt, t 0} implies f X st f Y.

20 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 130 Proof. By stipulation for each 0, 1], P { X Γ, Y Γ, X n, Γ, Y n, Γ, n = 1, 2,... } = 1 Then as n and for each 0, 1], and [ X L L L L ] P n, X, Y n, Y = 1 [ X U U U U ] P n, X, Y n, Y = 1 Since f is continuous on Γ as n it follows that and [ P f L L L L ] Xn, f X, f Y n, f Y = 1 [ P f U U U U ] Xn, f X, f Y n, f Y = 1 Since {Xt; t 0} st {Yt; t 0} implies for each 0, 1], and X L L 0, X,..., L X n,..., L Y Y L L st 0, Y n X U U 0, X, U X n n 1 st Y U 0, Y U n,..., Y,..., U X U Since f is an increasing function over the n + 1 values X L 0,..., X L and X U 0... X U.

21 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 131 Then by the stochastic comparison of two vectors, f X n, L st f Y n, L and f X n, U st f Y n, U Then as n L st L and f X U st U. f X f Y f Y Then and f f 0,1] X 0,1] X L U st f st f 0,1] Y 0,1] Y L U These two inequalities together imply that f X st f Y Theorem For the fuzzy stochastic process Xt, Yt, X t and Y t let {Xt; t 0} st {Yt; t 0} and {X t; t 0} st {Y t; t 0} with {Xt; t 0} independent of {X t, t 0} and {Yt, t 0}

22 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 132 independent of {Y t, t 0}. Then {Xt + X t, t 0} st {Yt + Y t; t 0} Proof. Let f be an increasing function on 0 t 1 t k. Then by Theorem f [ X L t 1 + X L t 1,..., X L t k + X L t k ] st f [ Y L t 1 + Y L t 1,..., Y L t k + Y L t k ] f [ X U t 1 + X U t 1,..., X U t k + X U t k ] st f [ Y U t 1 + Y U t 1,..., Y U t k + Y U t k ] It follows from the definition that {Xt + X t; t 0} st {Yt + Y t; t 0}.

23 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes Conclusions The embodiment of this thesis establish the following important results. 1. Let { fn } and { gn } be sequence of canonical positive fuzzy numbers. Let {X n } and {Y n } be sequence of fuzzy random variables, X n = o p fn denotes {Xn } is of smaller order in probability than { fn } and Xn = O p gn denotes {Xn } is at most of order { gn } in probability then i X n Y n = o p fn g n, Xn Y n = O p fn g n ii X n S = o p f s n, Xn s = O p f s n iii X n +Y n =o p max fn, g n, Xn + Y n = O p max fn, g n. 2. Helly s theorem If i non-decreasing sequence of fuzzy probability distribution function {F n x} converges to the fuzzy probability distribution function Fx, ii the fuzzy

24 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 134 valued function gx is everywhere continuous and iii a, b are continuity points of Fx then for 0, 1] b lim n a = b g L xdf min n 3. Helly Bray Theorem a g L xdf min x β x β g U xdf max n x β g U xdf max x β i The fuzzy valued function gx is continuous. ii The fuzzy probability distribution function F n x Fx in each continuity point of Fx and iii For any ε > 0 we can find A such that A g L x df min n + A g x β U x df max n g L x df min n x β g x β U x df max n for all n = 1, 2, 3,..., then x β < ε

25 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 135 lim n = g L xdf min n g L xdf min x β x β g U xdf max n x β g U xdf max x 4. If Y is a fuzzy random variable on Ω, σ, m, G a fuzzy sub σ-algebra of σ, then there is always a regular conditional distribution function for Y given G. β 5. If Y be a fuzzy random variable on Ω, σ, m, G a fuzzy sub σ-algebra of σ, then there exists a regular conditional probability for Y given G. 6. Baye s theorem is valid in the Krzysztof Piasecki s probability space defined in terms of fuzzy relation less than.

26 Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes Let X and Y be n-dimensional fuzzy random vectors X and Y be n -dimensional random vectors such that X st Y, X st Y with X and X are independent and Y and Y are independent then X, X st Y, Y where X, X denotes the n+n dimensional fuzzy random vector X 1,..., X n, X 1,..., X n.