Journal of Fuzzy Set Valued Analysis 2015 No.3 (2015) 232-244 Available online at www.ispacs.com/jfsva Volume 2015, Issue 3, Year 2015 Article ID jfsva-00256, 13 Pages doi:10.5899/2015/jfsva-00256 Research Article
Fuzzy Itˆo Integral Driven by a Fuzzy Brownian Motion Didier Kumwimba Seya1∗ , Rostin Mabela Makengo2 , Marcel R´emon3 , Walo Omana Rebecca2 (1) University of Lubumbashi, Faculty of Science, Department of Mathematic (2) University of Kinshasa, Faculty of Science, Department of Mathematic and computer Science (3) University of Namur, Faculty of Science, Department of Mathematic c Didier Kumwimba Seya, Rostin Mabela Makengo, Marcel R´emon and Walo Omana Rebecca. This is an open access article disCopyright 2015 ⃝ tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract In this paper we take into account the fuzzy stochastic integral driven by fuzzy Brownian motion. To define the metric between two fuzzy numbers and to take into account the limit of a sequence of fuzzy numbers, we invoke the Hausdorff metric. First this fuzzy stochastic integral is constructed for fuzzy simple stochastic functions, then the construction is done for fuzzy stochastic integrable functions. Keywords: Fuzzy random variable, Fuzzy stochastic process, Fuzzy Brownian Motion, Fuzzy function, Fuzzy random function,
Fuzzy Itˆo Integral.
1 Introduction The notions of Fuzzy Stochastic Integral with respect to a crisp Brownian Motion have been introduced by Kim and Kim in [7] and since then they were successfully used in the topic of set valued fuzzy differential equations [8, 11, 15, 4]. In [12] Malinowski and Michta exploit the properties of set-valued stochastic trajectory integrals to define a notion of fuzzy stochastic Lebesgue-Stieltjes trajectory integral. Meanwhile they consider a notion of fuzzy stochastic trajectory integral with respect to martingales ( crisp martingales). The notion of Fuzzy Stochastic Itˆo integral given in[12] allowed the authors of the paper to define stochastic fuzzy differential equations driven by Brownian Motion. Some other results containing stochastic fuzzy Integral have been published [13, 14, 5]. In this paper, we define the stochastic Integral of a fuzzy process [20, 17, 21] with respect to a fuzzy Brownian Motion [6, 10] and give its properties. The fuzzy Itˆo Integral with respect to a fuzzy Brownian motion is a natural tool in the study of the theory of fuzzy stochastic differential equations driven by a fuzzy Brownian Motion. The paper is organized as follows. In section 2, we give some preliminaries on fuzzy random variables and fuzzy Brownian Motions. In section 3, we introduce the new version of the notion of fuzzy stochastic integral with respect to a fuzzy Brownian Motion, for the first time, by using simple fuzzy functions. The more general case of measurable fuzzy functions is also considered. 2 Preliminaries and notations We give different definitions and elementary concepts of fuzzy arithmetic, fuzzy random variables and fuzzy Brownian motions that will be used in the next section. The reader is referred to [19, 9, 18, 1, 22, 23] for more details. ∗ Corresponding
author. Email address:
[email protected], Tel:+243995557217
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Definition 2.1. Let X = R be a Universal set, we define F (R) = {u˜ : R 7−→ [0, 1] : u˜ satis f ies (i) to (iii)} , where (i) u˜ is normal,(ii) u˜ is convex and (iii) u˜ is upper semi continuous. We call a fuzzy number, any u˜ ∈ F (R). Definition 2.2. (α -level set) ˜ for every α ∈ (0, 1], the set Let u˜ ∈ F (R) be a fuzzy number, we call a α -level set of u, ] [ ˜ ≥ α , α ∈ (0, 1]} = u˜Lα , u˜Uα , [u] ˜ α = {x ∈ R : u(x)
(2.1)
where [u] ˜ Lα = infx∈R {x ∈ [u] ˜ α } and [u] ˜ Uα = supx∈R {x ∈ [u] ˜ α }. The support set of u˜ is given by [u] ˜ 0+ = Supp u˜ = Cl {x ∈ R : u(x) ˜ > 0}. Definition 2.3. (Metric of Hausdorff) Let A ⊆ R and B ⊆ R be two intervals, the Hausdorff’s metric between them is defined as follows: { } dH (A, B) = max sup inf |a − b| , sup inf |a − b| . a∈A b∈B
(2.2)
b∈B a∈A
According to Puri and Ralescu [17] one can define a metric on F (R) as follows: Definition 2.4. (Metric on F (R)) Let a˜ and b˜ be two fuzzy numbers, a metric between a˜ and b˜ is given by dF = sup dH (a˜α , b˜ α ).
(2.3)
α ∈(0,1]
e be two fuzzy numbers. Then ∀α ∈ [0, 1] we have Theorem 2.1. [2] Let Fe and G } ( ) { eα = max FeαL − G eLα , FeαU − G eUα . dH Feα , G
(2.4)
A fuzzy number a˜ of R can be also defined by its memberships function µa˜ : R 7−→ [0, 1]. Between two fuzzy numbers a˜ and b˜ we define by ”‘⊙”’any binary operation ⊕,⊖ or ⊗ such that[3] } { (2.5) µa⊙ ˜ b˜ (z) = sup min µa˜ (x), µb˜ (y) , x◦y=z
for ⊙ = ⊕,⊖ or ⊗ and ◦ = +, − or ×. Between two closed intervals a˜α = [a˜Lα , a˜Uα ] and e bα = [b˜ Lα , b˜ Uα ] we define by ”‘⊙int ”’any binary operation ⊕int , ⊖int or ⊗int such that[3] } { (2.6) aeα ⊙int b˜ α = z ∈ R|z = x ◦ y, ∃x ∈ a˜α , ∃y ∈ b˜ α . Definition 2.5. (Hukuhara difference of fuzzy numbers, [19]) Let a˜ and b˜ be two fuzzy numbers, the Hukuhara is defined by a˜ ⊖H b˜ = c˜ if and only if a˜ = b˜ ⊕H c˜ exists, it is unique and its α -levels are [difference ] or [H-difference ] L L U U ˜ ˜ ˜ a˜ ⊖H b α = a˜α − bα , a˜α − bα . Clearly, a˜ ⊖H a˜ = {0}. Definition 2.6. (Interval-valued stochastic process, GUO [6]) ¯ A random interval-valued stochastic process {X(t),t ∈ [0, T ] , 0 < T < ∞} on probability space (Ω, A , P) is the mapping from Ω to L (R) ⊂ K (R)
for all t ∈ [0, T ] uniformly, where
X¯t : Ω −→ K (R) ω −→ X¯t = [X¯tL , X¯tU ],
(2.7)
{ } L (R) = [a¯L , a¯U ], a¯L ≤ a¯U , a¯L , a¯U ∈ R .
(2.8)
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Definition 2.7. ( Interval-valued Gaussian process, GUO [6]) ¯ A family {X(t),t ∈ [0, T ] , 0 < T < ∞} is an interval-valued Gaussian process on probability space (Ω, A , P) if and only if ¯ ω ) = [X¯ L (t, ω ), X¯ U (t, ω )], X(t, (2.9) where both X¯ L (t, ω ) and X¯ U (t, ω ) are Gaussian processes such that P[X¯ L (t, ω ) ≤ X¯ U (t, ω )] = 1.
(2.10)
Definition 2.8. ( Interval-valued Brownian motion, GUO [6]) ¯ A family {B(t),t ∈ [0, T ] , 0 < T < ∞} is an interval-valued Brownian motion on probability space (Ω, A , P) if and only if ¯ ω ) = [B¯ L (t, ω ), B¯U (t, ω )], B(t, (2.11) where both B¯ L (t, ω ) and B¯U (t, ω ) are Brownian motions such that P[B¯ L (t, ω ) ≤ B¯U (t, ω )] = 1.
(2.12)
Definition 2.9.{ ( Fuzzy stochastic process,}GUO [6]) ˜ ∈ [0, T ] , 0 < T < ∞ is called a fuzzy stochastic process on probability space (Ω, A , P) if and A fuzzy family X(t),t only if for all α ∈ [0, 1], the process X˜α (t, ω ) = [X˜αL (t, ω ), X˜αU (t, ω )], (2.13) is an interval-stochastic process on (Ω, A , P) and ˜ ω) = X(t,
∪
X˜α (t, ω ).
(2.14)
α ∈[0,1]
Definition 2.10. ( Fuzzy Gaussian process, GUO [6])} { ˜ A fuzzy stochastic process X(t),t ∈ [0,{T ] , 0 < T < ∞ is a fuzzy Gaussian process on probability space (Ω, A , P) if } and only if for all α ∈ [0, 1], the family X˜α (t),t ∈ [0, T ] , 0 < T < ∞ is an interval-valued Gaussian process. Definition 2.11. ( Fuzzy Brownian Motion, GUO [6])} { ˜ A fuzzy stochastic process B(t),t ∈ [0, T ] , 0 < T < ∞ is a fuzzy Brownian Motion on probability space (Ω, A , P) if and only if for all α ∈ [0, 1], the process B˜ α (t, ω ) = [B˜ Lα (t, ω ), B˜Uα (t, ω )],
(2.15)
is an interval-Brownian Motion on (Ω, A , P) and ˜ ω) = B(t,
∪
B˜ α (t, ω ).
(2.16)
α ∈[0,1]
˜ ω ) is a fuzzy number. Remark 2.1. (i) For all t ∈ [0, T ] and ω ∈ Ω, B(t, { } ˜ (ii) A {nonnegative fuzzy stochastic process B(t),t ∈ [0, T ] is a nonnegative fuzzy Brownian motion if and only if } B˜ αi (t, ω ),t ∈ [0, T ] is a non negative Brownian Motion where i ∈ {L,U} and α ∈ [0, 1]. Theorem 2.2. (Shoumei [10]) { } { } ˜ Then B(t, ˜ ˜ = 0. ˜ ω ),t ≥ 0 is a fuzzy Brownian motion Let B(t),t ≥ 0 be a fuzzy stochastic process such that B(t) if and only if it is Gaussian and for all α ∈ [0, 1] [ ] (i) E B˜ iα (t, ω ) = 0 for all t ≥ 0, i ∈ {L,U}, [ ] (ii) E B˜ iα (t, ω ).B˜ iα (s, ω ) = t ∧ s for all s,t ≥ 0, i ∈ {L,U}, [ ] (iii) E B˜ iα (t, ω ).B˜ αj (s, ω ) = 0 for all s,t ≥ 0, i, j ∈ {L,U} and i ̸= j.
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Theorem 2.3.{ (Shoumei [10]) } ˜ B(t),t ˜ = 0, ˜ Let B(t) ≥ 0 be a fuzzy Brownian motion. Then we have { } ˜ + t0 ),t ≥ 0 is a fuzzy Brownian motion for all t0 ≥ 0, (i) B(t { } ˜ (ii) v˜ ⊕ B(t),t ≥ 0 is a fuzzy Brownian motion for all v˜ ∈ Fc (R), { } ˜ λ t),t ≥ 0 is a fuzzy Brownian motion for all λ > 0, (iii) √1 ⊗ B( λ
} { ˜ 1 ),t ≥ 0 is a fuzzy Brownian motion . (iv) t ⊗ B( t Definition 2.12. { ( Fuzzy } Martingale, PURI [17]) The sequence X˜n , Fn n of fuzzy random variables and σ -algebra is a fuzzy martingale if for each n ≥ 1:
(i) X˜n is F˜n -measurable and E SuppX˜n < ∞, (ii) E(X˜n+1 | Fn ) = X˜n . If property (ii) is replaced by (ii’) E(X˜n+1 | Fn ) ≥ X˜n (E(X˜n+1 | Fn ) ≤ X˜n ). { } Then X˜n , Fn n is called a fuzzy sub-martingale ( surmartingale), respectively. Theorem 2.4. (Shoumei [10]) { } { } ˜ : s ≤ t) and B(t) ˜ : t ≥ 0 is a fuzzy Brownian motion. Then B(t), ˜ let Ft = σ (B(s) Ft : t ≥ 0 is a fuzzy martingale. 3 Fuzzy Itˆo Integral In this section we define a fuzzy counterpart of stochastic Integral or Itˆo Integral. The fuzzy stochastic Integral considered here is driven by a fuzzy Brownian motion. Definition 3.1. ( Fuzzy Simple function) A fuzzy simple function is a fuzzy function of the form ˜ ω) = Φ(t,
⊕
E˜ j (ω ) ⊗ 1[ j2−n ,( j+1)2−n ] (t)
(3.17)
j≥0
where 1A denotes the indicator function , n is a natural number and E˜ j (ω ) a fuzzy number. We define the fuzzy Itˆo integral of a fuzzy simple function driven by a fuzzy Brownian motion as follows. ˜ ω ) with respect to a fuzzy Brownian Definition 3.2. The fuzzy Itˆo integral of a fuzzy simple function defined by Φ(t, motion is defined on [S, T ] by ∫ T S
where
(n) tk = tk
−n k2 S = T
˜ ω )d B(t, ˜ ω) = Φ(t,
⊕
˜ j+1 , ω ) ⊖H B(t ˜ j , ω )], E˜ j (ω ) ⊗ [B(t
(3.18)
j≥0
if S ≤ k2−n ≤ T if k2−n < S if k2−n > T
˜ j ) and E˜ j are fuzzy numbers for all j ≥ 0. By stability of ⊕, ⊗ and ⊖H in Fc (R), the fuzzy Remark 3.1. In (3.18), B(t Itˆo Integral of fuzzy simple function with respect to a fuzzy Brownian Motion is a fuzzy number.
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By the definition of α -level set of fuzzy number (3.17) we can have: ˜ Lα (t, ω ) = Φ
∑ (E˜ j (ω ))Lα 1[ j2−n ,( j+1)2−n ] (t)
(3.19)
∑ (E˜ j (ω ))Uα 1[ j2−n ,( j+1)2−n ] (t).
(3.20)
j≥0
and ˜ Uα (t, ω ) = Φ
j≥0
Theorem 3.1. Let B˜t be a fuzzy Brownian Motion, the following assertions are trues: ˜ ω ) is a non negative fuzzy simple function, then (i) If Φ(t, [∫ T ∫ ∫ T L L ˜ ˜ ˜ ˜ ( Φ(t, ω )d B(t, ω ))α = Φα (t, ω )d Bα (t, ω ), S
S
T
S
˜ ω ) is a non positive fuzzy simple function, then (ii) If Φ(t, [∫ T ∫ T ∫ ˜ ω )d B(t, ˜ Lα (t, ω )d B˜Uα (t, ω ), ˜ ω ))α = ( Φ(t, Φ S
S
T
S
] U U ˜ ˜ Φα (t, ω )d Bα (t, ω ) ,
(3.21)
] ˜ Uα (t, ω )d B˜ Lα (t, ω ) . Φ
(3.22)
Proof. The proof follows from Theorem 4.1 in Wu [2]. Indeed, let’s show (ii). )U (∫ T )U ( ⊕ ˜ ˜ ˜ ˜ ˜ = E j (ω ) ⊗ [B(t j+1 , ω ) ⊖H B(t j , ω )] Φ(t, ω )d B(t, ω ) α
S
j≥0
( ) ˜ j+1 , ω ) ⊖H B(t ˜ j , ω )] U = ∑ E˜ j (ω ) ⊗ [B(t α j≥0
=
∑
( )U ˜ j+1 , ω ) ⊖H B(t ˜ j , ω )]Lα E˜ j (ω ) α [B(t
∑
( )U E˜ j (ω ) α [B˜ Lα (t j+1 , ω ) − B˜ Lα (t j , ω )]
j≥0
=
j≥0
∫ T
= S
˜ Uα (t, ω )d B˜ Lα (t, ω ). Φ
Similarly, for the lower bound case we have: (∫ T )L ∫ ˜ ˜ Φ(t, ω )d B(t, ω ) = S
α
α
S
T
˜ Lα (t, ω )d B˜Uα (t, ω ). Φ
˜ ω ) is a simple positive or negative fuzzy function integrable with respect to a fuzzy Corollary 3.1. Suppose that Φ(t, Brownian motion. Then the followings assertions are true. ˜ ω ) is positive then (i) If Φ(t,
∫T
˜ ω ) is negative then (ii) If Φ(t,
S
˜ ω )d B(t, ˜ ω ) is a positive fuzzy number. Φ(t,
∫T S
˜ ω )d B(t, ˜ ω ) is a negative fuzzy number. Φ(t,
Proof. The result follows from Definition 3.2 and Remark 3.1 immediately. Definition 3.3. Let VF = VF (S, T ) be the class of all fuzzy functions ˜ ω ) : [0, ∞] × Ω 7−→ Fc (R), F(t, such that:
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˜ ω )is B × Ft - measurable, where B denotes the Borel σ -algebra on [0, ∞), (i) (t, ω ) 7−→ F(t, ˜ ω ) is Ft -adapted, (ii) F(t, [∫
] ˜ ω 2 dt < ∞. (iii) E ST F(t,
( ) ˜ ω ) = supα ∈(0,1] dH F˜α (t, ω ), 0˜ α Where F(t, Theorem 3.2. Let F˜ ∈ VF (S, T ), its fuzzy Itˆo integral is defined by ˜ ω) = I[F](
∫ T S
˜ ω )d B(t, ˜ ω ). F(t,
(3.23)
Proof. To prove this identity, we first consider the fuzzy Itˆo integral of a fuzzy simple function as in definition 3.2 and ˜ n )n≥1 of such fuzzy simple function as follows show that F˜ can be approximated by a sequence (Φ ∫ T
˜ ω )d B(t, ˜ ω ) = lim F(t,
∫ T
n→∞ S
S
˜ n (t, ω )d B(t, ˜ ω ). Φ
(3.24)
In the following Lemma, we give the property of isometry for the fuzzy Itˆo integral. Lemma 3.1. (Itˆo isometry) ˜ ω ) be a bounded fuzzy simple function, then Let Φ(t, ∫ T
E[( S
∫ T
˜ ω )d B(t, ˜ ω ))2 ] = E[ Φ(t,
S
˜ ω )2 dt]. Φ(t,
(3.25)
Proof. We must show that for each α ∈ (0, 1], ∫ T
[E[( S
∫ T
˜ ω )d B(t, ˜ ω ))2 ]]α = [E[ Φ(t,
S
˜ ω )2 dt]]α . Φ(t,
˜ ω ) ≽ 0˜ for each t ∈ [0, ∞[ and ω ∈ Ω . We first consider the case Φ(t, By the definition of the expectation, the Theorem 3.1 and the classical Itˆo isometry we have that: ∫ ˜ ω )d B(t, ˜ ω ))2 ]]α [E[( ST Φ(t, ∫ T
= [E[( S
∫ T
= [E[( S
∫ T
= [E[( ∫
= [E[ S
S T
∫ T
˜ Lα (t, ω )d B˜ Lα (t, ω ))2 ], ( Φ
S
˜ Uα (t, ω )d B˜Uα (t, ω ))2 ] Φ
∫ T
˜ Lα (t, ω )d B˜ Lα (t, ω ))2 ], E[ Φ ∫ T
˜ Lα (t, ω )2 dt], E[ Φ
S
S
˜ Lα (t, ω )d B˜Uα (t, ω ))2 ]] Φ
˜ Uα (t, ω )2 dt]] Φ
˜ ω )2 dt]]α . Φ(t,
Similarly we can show the result for a non positive simple fuzzy function. In the sequel, we shall use this Itˆo isometry property to define the fuzzy Itˆo integral of VF (S, T )-fuzzy functions. We first prove some useful results. Definition 3.4. A fuzzy function G˜ : R 7−→ Fc (R) is bounded if and only if for every α ∈ (0, 1] and i ∈ {L,U}, G˜ iα is a real bounded fuzzy function.
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˜ ω ) be continuous for each ω . There exists a Theorem 3.3. Let G˜ ∈ VF (S, T ) be a bounded fuzzy function and let G(·, ˜ sequence of fuzzy simple functions Φn such that lim E
∫ T
n→∞
G(t, ˜ ω ) ⊖H Φ ˜ n (t, ω ) 2 dt = 0.
S
Proof. We define ˜ n (t, ω ) = Φ
⊕
(3.26)
˜ j , ω ) ⊗ 1[t ,t ) (t). G(t j j+1
j≥0
˜ n (t, ω ) is a fuzzy simple function for all n ≥ 1. Hence the definition we have Clearly Φ ( )i ˜ n (t, ω ) = Φ α
∑ G˜ iα (t j , ω )1[t j ,t j+1 ) (t),
j≥0
for each α ∈ [0, 1] and i ∈ {L,U}, where the function G˜ αi (·, ω ) are continuous. The Itˆo isometry property for bounded function G˜ iα leads to ( see [16]) ∫ T( ( ) i )2 ˜n dt = 0 lim G˜ iα − Φ α n→∞ S
∀ω ∈ Ω, α ∈ [0, 1] and i ∈ {L,U}. That is [∫
T
lim E
n→∞
S
(
] ( ) i )2 ˜n G˜ iα − Φ dt = 0, α
∀ω ∈ Ω, α ∈ [0, 1] and i ∈ {L,U}. Thus
∫ ˜ n 2 dt limn→∞ E ST G˜ ⊖H Φ ∫ T
= lim E n→∞
( ) ˜ L ( ˜ )L 2 ˜ U ( ˜ )U 2 sup max Gα − Φn α , Gα − Φn α dt
S α ∈[0,1]
( ∫ ≤ sup max lim E α ∈[0,1]
n→∞
T
S
∫ L G˜ α − (Φ ˜ n )Lα 2 dt, lim E n→∞
T
S
U G˜ α − (Φ ˜ n )Uα 2 dt
)
= sup max (0, 0) α ∈[0,1]
= 0.
Corollary 3.2. Let H˜ ∈ VF (S, T ) be a bounded fuzzy function, then there exists a sequence of fuzzy bounded functions G˜ n ∈ VF (S, T ) such that G˜ n (·, ω ) is continuous for every ω ∈ Ω and n ≥ 1, and we have lim E
n→∞
∫ T S
H˜ ⊖H G˜ n 2 dt = 0.
(3.27)
˜ ω ) ≤ M for every t ∈ [S, T ] and ω ∈ Ω. For every n, let’s consider Ψn a real positive Proof. Suppose that H(t, continuous function such that (i) Ψn (x) = 0 for every x ≤ (ii)
∫ −∞ ∞
−1 n
and x ≥ 0
Ψn (x)dx = 1.
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We define G˜ n (t, ω ) = such that
∫ t 0
˜ ω )ds Ψn (s − t) ⊗ H(s,
∫ t ( )i ˜ Gn α (t, ω ) = Ψn (s − t) ⊗ H˜ αi (s, ω )ds 0
( ) ( )i i ∀ω ∈ Ω, α ∈ [0, 1] , t ∈ [S, T ] and i ∈ {L,U}. Since G˜ n α (·, ω ) is continuous for every ω and G˜ n α (t, ω ) ≤ M. ( )i By measurability of H˜ αi (s, ω ), G˜ n α is measurable for every α ∈ [0, 1] , t ∈ [S, T ] and i ∈ {L,U}. Thus ∫ T H˜ αi (t, ω ) − (G˜ n )iα (t, ω ) 2 dt = 0. lim n→∞ S
By the classical bounded convergence Theorem, we have lim E
∫ T
n→∞
H˜ αi (t, ω ) − (G˜ n )iα (t, ω ) 2 dt = 0.
S
∀ω ∈ Ω, α ∈ [0, 1] , t ∈ [S, T ] and i ∈ {L,U} . The result follows by using similar arguments as in the proof of Theorem 3.3. Corollary 3.3. Let F˜ ∈ VF (S, T ) , then there exists a sequence of fuzzy functions H˜ n ∈ VF (S, T ) such that H˜ n is bounded for every n ≥ 1 and we have lim E
n→∞
Proof. Let H˜ n suchthat −n if F˜αi ≤ −n ( )i F˜ i if −n ≤ F˜αi ≤ n H˜ n (t, ω ) α = α n if F˜αi > n convergence Theorem, we have
S
F˜ ⊖H H˜ n 2 dt = 0.
(3.28)
∀ω ∈ Ω, α ∈ [0, 1] , t ∈ [S, T ] and i ∈ {L,U}. By the classical dominated
∫ T(
lim E
n→∞
∫ T
S
)2 ( )i F˜αi (t, ω ) − H˜ n α (t, ω ) dt = 0.
The result follows by using a similar development as in the proof of Theorem 3.3. Using the above results, for F˜ ∈ VF (S, T ), the fuzzy Itˆo integral w.r.t. Fuzzy Brownian motion ∫ T S
˜ ω) ˜ ω )d B(t, F(t,
can be defined as. Theorem 3.4. (Fuzzy Itˆo Integral) ˜ ω ) is defined by Let F˜ ∈ VF (S, T ). Then the fuzzy Itˆo integral of F˜ (from S to T ) w.r.t. a Fuzzy Brownian Motion B(t, ∫ T S
˜ ω )d B(t, ˜ ω ) = lim F(t,
∫ T
n→∞ S
˜ n (t, ω )d B(t, ˜ ω ), Φ
(3.29)
{ } 2 ˜n the limit is taken in LF (P) and Φ is a sequence of fuzzy simple functions such that n≥1 c (R) lim E
n→∞
∫ T S
F˜ ⊖H Φ ˜ n (t, ω ) 2 dt = 0.
(3.30)
{ } ˜n satisfying 3.30. Hence we are done. Proof. From the Theorems 2.2, 2.3 and 2.4, there is a sequence Φ n≥1
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Corollary 3.4.
[(∫
T
E S
240
)2 ] [∫ ˜ ˜ F(t, ω )d B(t, ω ) =E
S
T
] 2 ˜ F (t, ω )dt ,
(3.31)
for every F˜ ∈ VF (S, T ) Proof. From Theorem 3.4 and Lemma 3.1, we have the result. ˜ Then Example 3.1. Suppose that B˜ 0 = 0.
∫t
1 ˜2 1 ˜ ˜ 0 B(s)d B(s) = 2 B (t) ⊖H 2 t.
Proof. It is enough to show that for all α ∈ [0,1] , ] [∫ t ] [ 1 1 ˜2 ˜ ˜ B(s)d B(s) = B (t) ⊖H t . 2 2 α 0 α First, for a positive fuzzy Brownian motion [∫ T ] [∫ t ] ∫ t L ˜L U ˜U ˜ ˜ ˜ ˜ = (B(s)) d B (s), ( B(s)) d B (s) B(s)d B(s) α α α α 0 0 0 α [ ] 1 1 1 1 = (B˜ 2 (t))αL − t, (B˜ 2 (t))Uα − t 2 2 2 2 by the definition of Hukuhara difference [19] we have [ ] 1 ˜2 1 B (t) ⊖H t = [min S, max S] , 2 2 α {
where S=
} 1 ˜L 1 1 1 (Bα (t))2 − t, (B˜Uα (t))2 − t . 2 2 2 2
Since (B˜ Lα (t)) ≤ (B˜Uα (t)) for every t ≥ 0 and α ∈ (0, 1] and by the fact of positivity of fuzzy Brownian motion considered, we have (B˜ Lα (t)) > 0. Thus 12 (B˜ αL (t))2 − 12 t < 21 (B˜Uα (t))2 − 12 t, for every t ≥ 0 and α ∈ (0, 1]. Thus min S = 12 (B˜ Lα (t))2 − 21 t and max S = 1 ˜U (Bα (t))2 − 1 t. 2
2
Theorem 3.5. ( Some properties of Fuzzy Itˆo integral) Let F˜ and G˜ ∈ VF (0, T ) and 0 ≤ S < U < T . Then (i)
∫T
(ii)
S
˜ B(t) ˜ = Fd
∫T S
∫U S
˜ B(t) ˜ ⊕ Fd
˜ B(t) ˜ = c˜ ⊗ c˜ ⊗ Fd
∫T
∫T S
∫T U
˜ B(t), ˜ Fd
˜ B(t), ˜ Fd where c˜ ∈ F (R),
∫
˜ B(t) ˜ = ST Fd ˜ B(t) ˜ ⊕ F˜ ⊕ Gd [∫ ] ˜ ˜ B(t) ˜ (iv) E ST Fd = 0, (iii)
(v)
S
∫T S
∫T S
˜ B(t), ˜ Gd
˜ B(t) ˜ is FT -measurable. Fd
Proof. This holds for all simple fuzzy functions, so by taking limits we obtain this for all fuzzy functions F˜ and G˜ ∈ VF (0, T ). We have: ˜ ω ) be a simple fuzzy function given by (i) Let Φ(t, ˜ ω) = Φ(t,
⊕
E˜ j (ω ) ⊗ 1[t j+1 ,t j [ (t).
j≥0
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By definition
∫ T S
−n j2 (n) S tj = tj = T where
241
˜ ω )d B(t, ˜ ω) = Φ(t,
⊕
[ ] ˜ j+1 , ω ) ⊖H B(t ˜ j, ω) , E˜ j (ω ) ⊗ B(t
j≥0
if S ≤ j2−n ≤ T if j2−n < S if j2−n > T (n)
(n)
which is equivalent to S ≤ t1 ≤ · · · ≤ t j ≤ · · · ≤ T . We take now two subdivisions on [S, T ], the first from S ′ ′ k1 2−n if S ≤ k1 2−n ≤ T ′ ′ (n ) to U and the second from U to T as following tk1 = tk1 = S if k1 2−n < S ′ U if k1 2−n > U and k2 2−n” if U ≤ k2 2−n” ≤ T (n”) U if k2 2−n” < U tk2 = tk2 = T if k2 2−n” > T or once′ more (n ) (n′ ) (n”) (n”) S ≤ t1 ≤ · · · ≤ tk1 ≤ · · · ≤ U and U ≤ t1 ≤ · · · ≤ tk2 ≤ · · · ≤ T which are two contiguous infinite subdivisions of interval [S, T ]. Thus ∫ T S
˜ ω )d B(t, ˜ ω) = Φ(t,
⊕
[ ] ˜ j+1 , ω ) ⊖H B(t ˜ j, ω) E˜ j (ω ) ⊗ B(t
j≥0
[ ] ˜ k +1 , ω ) ⊖H B(t ˜ k , ω) ⊕ E˜k1 (ω ) ⊗ B(t 1 1
⊕
=
k1 ≥0
[ ] ˜ k +1 , ω ) ⊖H B(t ˜ k , ω) E˜k2 (ω ) ⊗ B(t 2 2
⊕
k2 ≥0
∫ U
= S
˜ ω )d B(t, ˜ ω) ⊕ Φ(t,
∫ T U
˜ ω )d B(t, ˜ ω ). Φ(t,
(ii) Let c˜ ∈ F (R) ∫ T S
˜ ω )d B(t) ˜ = c˜ ⊗ Φ(t, =
∫ T⊕ S
⊕ j≥0
= c˜ ⊗
˜ c˜ ⊗ E˜ j (ω ) ⊗ 1[t j ,t j+1 [ (t)d B(t)
j≥0
˜ j+1 , ω ) ⊖H B(t ˜ j , ω )] c˜ ⊗ E˜ j (ω ) ⊗ [B(t ⊕
˜ j+1 , ω ) ⊖H B(t ˜ j , ω )] E˜ j (ω ) ⊗ [B(t
j≥0
= c˜ ⊗ = c˜ ⊗ (iii) Let F˜ and G˜ be of the form F˜ =
⊕
∫ T⊕ S
∫ T S
˜ E˜ j (ω ) ⊗ 1[t j ,t j+1 [ (t)d B(t)
j≥0
˜ ω )d B(t). ˜ Φ(t,
E˜i (ω ) ⊗ 1[ti ,ti+1 [ (t)
i≥1
and G˜ =
⊕
E˜ ′ j (ω ) ⊗ 1[t j ,t j+1 [ (t)
j≥1
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Thus F˜ ⊕ G˜ =
242
⊕⊕
E˜i (ω ) ⊕ E˜ ′ j (ω ) ⊗ 1[ti ,ti+1 [∩[t j ,t j+1 [ (t)
i≥1 j≥1
Thus ∫ T S
˜ B(t) ˜ = F˜ ⊕ Gd
⊕⊕
˜ i+1 ∨ t j+1 , ω ) ⊖H B(t ˜ i ∧ t j , ω )] E˜i (ω ) ⊗ [B(t
i≥1 j≥1
⊕
⊕⊕
˜ i+1 ∨ t j+1 , ω ) ⊖H B(t ˜ i ∧ t j , ω )] E˜ ′ i (ω ) ⊗ [B(t
i≥1 j≥1
=
⊕
˜ i+1 , ω ) ⊖H B(t ˜ i , ω )] E˜i (ω ) ⊗ [B(t
i≥1
⊕
⊕
˜ j+1 , ω ) ⊖H B(t ˜ j , ω )] E˜ ′ j (ω ) ⊗ [B(t
j≥1
∫ T
= S
(iv) Let F˜ =
⊕
i≥1 Ei (ω ) ⊗ 1[ti ,ti+1 [ (t),
˜
˜ B(t) ˜ ⊕ Fd
∫ T
˜ B(t). ˜ Gd
S
then
∫ T
E[
⊕
˜ B(t)] ˜ Fd = E[
S
˜ i+1 , ω ) ⊖H B(t ˜ i , ω )] E˜i (ω ) ⊗ [B(t
i≥1
=
⊕
˜ i+1 , ω ) ⊖H B(t ˜ i , ω )] E˜i (ω ) ⊗ E[B(t
i≥1
=
⊕
E˜i (ω ) ⊗ 0˜
i≥1
˜ = 0. (v) Let Let F˜ =
⊕
i≥1 Ei (ω ) ⊗ 1[ti ,ti+1 [ (t),
˜
∫ T S
then
˜ B(t) ˜ = Fd
⊕
˜ i+1 , ω ) ⊖H B(t ˜ i , ω )]. E˜i (ω ) ⊗ [B(t
i≥1
∫
˜ i+1 , ω ) is measurable, so is E˜i (ω ) ⊗ [B(t ˜ i+1 , ω ) ⊖H B(t ˜ i , ω ). Thus ST Fd ˜ B(t) ˜ is measurable for F˜ a Since B(t fuzzy simple function. So by taking the limit we obtain the assertion for all measurable fuzzy functions.
4 Conclusion In this paper, the fuzzy Brownian Motion is used to describe a new fuzzy Itˆo integral of fuzzy valued functions. First we define this fuzzy Itˆo integral for simple fuzzy functions. Then we use the fact that all fuzzy integrable functions can be expressed as a limit of fuzzy simple functions to define their fuzzy Itˆo integral. Acknowledgements We would like to express our gratitude to the Editor in Chief, Prof. Allahviranloo, for valuable assessment of our paper.
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243
References [1] H. Ahmadzade, M. Amini, S. M. Taheri, A. Bozorgnia, Some moment inequalities for fuzzy martingales and their applications, Journal of Uncertainty Analysis and Applications, (2014). http://dx.doi.org/10.1186/2195-5468-2-7 [2] H. Chung Wu, Fuzzy valued integrals based on a constructive methodology, Applications of Mathematics, 52 (2007) 1-23. http://dx.doi.org/10.1007/s10492-007-0001-x [3] H. Chung Wu, Evaluate Fuzzy Riemann integrals using the Monte Carlo method, Journal of Mathematical Analysis and Applications, 264 (2001) 324-343. http://dx.doi.org/10.1006/jmaa.2001.7659 [4] W. Fei, Existence and Uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients, Inform. Sci. 177 (2007) 4329-4337. http://dx.doi.org/10.1016/j.ins.2007.03.004 [5] Y. Feng, Fuzzy Stochastic Differential Systems, Fuzzy Sets and Systems, 115 (2000) 351-363. http://dx.doi.org/10.1016/S0165-0114(98)00389-3 [6] R. Guo, Fuzzy stochastic age processes, The fourth International conference on quality and rehability, August 9 - 11, Beijing, china, (2005). [7] B. K. Kim, J. H. Kim, Stochastic Integrals of Set-valued Processes and Fuzzy Processes, J. Math. Anal. Appl, 236 (1999) 480-502. http://dx.doi.org/10.1006/jmaa.1999.6461 [8] B. K. Kim, J. H. Kim, On Fuzzy Stochastic Equations, J. Korean Math. Soc, 42 (2005) 153-169. http://dx.doi.org/10.4134/JKMS.2005.42.1.153 [9] H. Kwakernaak, Fuzzy random variables, Inform. Sc, 15 (1978) 1-29. http://dx.doi.org/10.1016/0020-0255(78)90019-1 [10] S. Li, L. Guan, Fuzzy set-valued Gaussian processes and Brownian motions, Information Sciences, 177 (2007) 3251-3259. http://dx.doi.org/10.1016/j.ins.2006.11.008 [11] S. Li, A. Ren, Representation Theorems, Set-valued and Fuzzy set-valued Ito Integral, Fuzzy Sets and Systems, 158 (2007) 949-962. http://dx.doi.org/10.1016/j.fss.2006.12.004 [12] M. T. Malinowski, M. Michta, Stochastic fuzzy differential equations with an application, Kybernetika, 47 (2011) 123-143. [13] M. T. Malinowski, M. Michta, Fuzzy stochastic Integral Equations, Dynamic Systems and Applications, 19 (2010) 473-494. [14] M. T. Malinowski, On random fuzzy differential equations, Fuzzy Sets and Systems, 160 (2009) 3152-3165. http://dx.doi.org/10.1016/j.fss.2009.02.003 [15] Y. Ogura, On stochastic differential equations with fuzzy set coefficients. In: Soft Methods for Handling variability and Imprecision, Springer, Berlin, (2008) 263-270. http://dx.doi.org/10.1007/978-3-540-85027-4 32 [16] B. ∅ksendal, Stochastic Differential Equations: An Introduction with Applications, Springer Verlag, Berlin, (1998).
International Scientific Publications and Consulting Services
Journal of Fuzzy Set Valued Analysis 2015 No.3 (2015) 232-244 http://www.ispacs.com/journals/jfsva/2015/jfsva-00256/
244
[17] M. L. Puri, D. A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl, 114 (1986) 406-422. http://dx.doi.org/10.1016/0022-247X(86)90093-4 [18] D. Ralescu, Fuzzy random variables revisited, In: Mathematics of Fuzzy Sets, Handbook Fuzzy Sets Series, Kluwer, Boston, 3 (1999) 701-710. http://dx.doi.org/10.1007/978-1-4615-5079-2 16 [19] L. Stefanini, L. Sorini, M. L. Guerra, Fuzzy numbers Fuzzy arithmetic, Chapter 12 in W. Pedrycz, A. Skowron, V. Kreinovich(eds),Handbook of Granular Computing, John Wiley and Sons, Ltd, (2008). http://dx.doi.org/10.1002/9780470724163.ch12 [20] M. Stojakovic, Fuzzy conditional expectation, Fuzzy Sets and Systems, (1992) 52-60. http://dx.doi.org/10.1016/0165-0114(92)90036-4 [21] M. Stojakovic, Fuzzy martingales-a-simple form of fuzzy processes, Stochastic Anal. Appl, 14 (1996) 355-367. http://dx.doi.org/10.1080/07362999608809443 [22] R. Walo, Limit cicly and existence of periodic solutions of small pertubations of autonomous systems of fuzzy differential equations, SAJPAM, 4 (2009) 1-12. [23] R. Walo, D. Kumwimba, Sur l’existence des solutions de Seikkala des e´ quations aux d´eriv´ees partielles elliptiques et de diffusion floues, Ann. Fac. Sc. Univ. Kinshasa, 1 (2013) 81-91.
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