Existence and Stability of Solutions of Fuzzy Fractional Stochastic Differential Equations with Fractional Brownian Motions
Existence and Stability of Solutions of Fuzzy Fractional Stochastic Differential Equations with...
Arhrrabi, Elhoussain;Elomari, M’hamed;Melliani, Said;Chadli, Lalla Saadia
2021-09-02 00:00:00
Hindawi Advances in Fuzzy Systems Volume 2021, Article ID 3948493, 9 pages https://doi.org/10.1155/2021/3948493 Research Article Existence and Stability of Solutions of Fuzzy Fractional Stochastic Differential Equations with Fractional Brownian Motions Elhoussain Arhrrabi , M’hamed Elomari, Said Melliani, and Lalla Saadia Chadli LMACS, Laboratory of Applied Mathematics and Scientific Calculus, Sultan Moulay Sliman University, P. O. Box 523, Beni Mellal 23000, Morocco Correspondence should be addressed to Elhoussain Arhrrabi; arhrrabi.elhoussain@gmail.com Received 26 May 2021; Revised 17 June 2021; Accepted 11 August 2021; Published 2 September 2021 Academic Editor: Ferdinando Di Martino Copyright © 2021 Elhoussain Arhrrabi et al. &is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. &e existence, uniqueness, and stability of solutions to fuzzy fractional stochastic differential equations (FFSDEs) driven by a fractional Brownian motion (fBm) with the Lipschitzian condition are investigated. Finally, we investigate the exponential stability of solutions. Despite the fact that some research exists on the problem of 1. Introduction the uniqueness and existence of solutions to SDEs and &ere appears to be confusion of various kinds in the FSDEs which are disturbed by Brownian motions or sem- modeling of several real world systems, such as trying to imartingales [4, 11–15], a kind of the FFSDEs driven by an characterize a physical system and opinions on its param- fBm has not been investigated. Agarwal et al. [16, 17] eters. To deal with this ambiguity, the fuzzy set theory will be considered the concept of solution for FDEs with uncer- used [1]. It is able to handle such linguistic statements tainty and some results on FFDEs and optimal control mathematically using this theory, such as “large” and “less.” nonlocal evolution equations. Recently, Zhou et al., in &e capacity to investigate fuzzy differential equations [18–20], gave some important works on the stability analysis (FDEs) in modeling numerous phenomena, including im- of such SFDEs. Our results are inspired by the one in [21] precision, is provided by a fuzzy set. In particular, the fuzzy where the existence and uniqueness results for the FSDEs stochastic differential equations (FSDEs), in instance, might with local martingales under the Lipschitzian conditions are be used to investigate a variety of economics and engineering studied. &e rest of this paper is given as follows. Section 2 problems that involve two types of uncertainty: randomness summarizes the fundamental aspects. In Section 3, existence and fuzziness. and uniqueness of solutions to the FFSDEs are proved. &e fuzzy It o stochastic integral was powered in [2, 3]. In Moreover, the stability of solutions is studied in Section 4. [4, 5], the fuzzy stochastic integral is driven by the Wiener Finally, in Section 5, a conclusion is given. process as a fuzzy adapted stochastic process. In [6], Fei et al. studied the existence and uniqueness of solutions to the 2. Preliminaries (FSDEs) under non-Lipschitzian condition. In [7], Jafari et al. study FSDEs driven by fBm. Jialu Zhu et al., in [8], &is part introduces the notations, definitions, and back- prove existence of solutions to SDEs with fBm. Ding and ground information that will be utilized throughout the Nieto [9] investigated analytical solutions of multitime-scale article. FSDEs driven by fBm. Vas’kovskii et al. [10] prove that the Let K(R ) be the family of nonempty convex and n n pth moments, p≥ 1, of strong solutions of a mixed-type compact subsets of R . In K(R ), the distance d is defined SDEs are driven by a standard Brownian motion and a fBm. by 2 Advances in Fuzzy Systems d (M, N) � maxsup inf ‖m − n‖, sup inf ‖m − n‖, M, N∈ K R . (1) n∈N m∈M m∈M n∈N |‖B‖|≔ d (B, 0) � sup ‖b‖, for B∈ K R . (2) We denote by M(Ω,A; K(R )) the family of A-mea- b∈B surable multifunction, taking value in K(R ). We denote by Definition 1 (see [21, 22]). A multifunction G∈ M(Ω, n p p A; K(R )) is called L -integrably bounded if ∃h∈ L (Ω,A, P; R ) such that |‖G‖|≤ h P-a.e, where p n n p + L Ω,A, P; K R ≔ G∈ MΩ,A; K R :|‖G‖|∈ L Ω,A, P; R . (3) n n n Let E denote the set of the fuzzy x: R ⟶ [0, 1] such Definition 3 (see [23]). Let Df ∈ C ([c, d], E ) ∩ L ([c, d], α α n n that [x] ∈ K(R ), for every α∈ [0, 1], where [x] : � {a E ). &e fuzzy fractional Caputo differentiability of f is n 0 n ∈ R : x(a)≥ α}, for α∈ (0, 1], and [x] : � cl{a∈ R : x given by (a)> 0}. Let the metric be d (x, y)≔ sup d ([x] , u ∞ α∈[0,1] H α 1 C 1−α −α α n D + f(u) � J + (Df)(u) � (u − v) (Df)(v)dv. (5) c c [y] ), in E , a∈ R; we have d (x + z, y + z) � d (x, y), ∞ ∞ Γ(1 − α) d (x + y, z + w)≤ d (x, z) + d (y, w), and d (ax, ay) ∞ ∞ ∞ ∞ Now, we define the Henry–Gronwall inequality [24], � |a|d (x, y). which can be used in the proof of our result. Definition 2 (see [23]). Let f: [c, d]⟶ E ; the fuzzy Lemma 1. Let f, g: [0, T)⟶ R be continuous functions. Riemann–Liouville integral of f is given by If g is nondecreasing and there exists constants K≥ 0 and α> 0 as α α−1 J f (u) � (u − v) f(v)dv. + (4) Γ(α) α− 1 f(u)≤ g(u) + K (u − v) f(v)dv, u∈ [0, T), (6) then (KΓ(α)) nα−1 ⎣ ⎦ ⎡ ⎤ f(u)≤ g(u) + (u − v) g(v) dv, u∈ [0, T). (7) 0 Γ(mα) m�1 p n If g(u) � b is constant on [0, T), the previous inequality Let L (Ω,A, P; E ) denote the set of all fuzzy random is transformed into variables; they are L -integrally bounded. For the notion of an fBm, we referred to [25]. f(u)≤ bE KΓ(α)u , u∈ [0, T), (8) Let us define a sequence of partitions of [a, b] by ψ , m∈ N such that |ψ |⟶ 0 as m⟶∞. If, in m m where E is given by m−1 (m) H (m) H (m) ∞ L (Ω,A, P), ϕ(t )(B (t ) − B (t )) converge i�0 i i+1 i E (z) � . (9) to the same limit for all this sequences ψ , m∈ N, then α m Γ(mα + 1) m�0 this limit is said a Stratonovich-type stochastic integral and noted by ϕ(s)dB (s). Let J: � [0, T], where 0< T<∞. Remark 1 (see [24]). For all u∈ [0, T),∃N > 0 does not depend on b such that f(u)≤ N b. Definition 5 (see [21, 22]). A function f: J ×Ω⟶ E is called fuzzy stochastic Definition 4 (see [21, 22]). process; if ∀t∈ J, f(t, .) � f(t):Ω⟶ E is a fuzzy random variable A function f:Ω⟶ E is said fuzzy random variable if [f] is an A-measurable random variable∀α∈ [0, 1] A fuzzy stochastic process f is continuous; if n n A fuzzy random variable f:Ω⟶ E is said f(., υ): J⟶ E are continuous, and it is p H L -integrably bounded, p≥ 1, if A -adapted if for every α∈ [0, 1] and for all t∈ J, t t∈J α p n α n H [f] ∈ L (Ω,A, P; K(R )),∀α∈ [0, 1] [f(t)] :Ω⟶ K(R ) is A -measurable t Advances in Fuzzy Systems 3 Definition 6 (see [21, 22]). Definition 7 (see [21, 22]). A fuzzy process f: J ×Ω⟶ E p p is said L -integrally bounded if ∃h∈ L (J ×Ω, N; R)/ &e function f is called measurable if d (f(s, υ), 0)≤ h(s, υ). α n p n p [f] : J ×Ω⟶ K(R ) is a B(J)⊗A-measurable, for We denote by L (J ×Ω, N; E ) the set of all L -inte- all α∈ [0, 1] grally bounded and nonanticipating fuzzy stochastic processes. &e function f: J ×Ω⟶ E is said to be non- anticipating if it is A -adapted and measurable t t∈J p n Proposition 1 (see [4]). For f∈ L (J ×Ω, N; E ) and p≥ 1, we have J ×Ω∋ (t, υ)⟶ f(s, υ)ds∈ L (J ×Ω, N; E ) and d -continuous. Remark 2. &e process x is nonanticipating if and only if x is measurable with respect to N: � A∈ B(J)⊗A: A p n H u Proposition 2 (see [4]). For f, g∈ L (J ×Ω, N; E ) and ∈ A , u∈ J}, where, for u∈ J, A � {υ: (u, υ)∈ A}. p≥ 1, we have a a t p p−1 p E sup d f(u)du, g(u)du≤ t Ed (f(u), g(u))du. (10) ∞ ∞ 0 0 0 a∈[0,t] Proposition 3 (see [26]). Let ψ: J⟶ R ; then, for t∈ J, Proposition 4 (see [4]). Assume that the function � � ψ: J⟶ R satisfies ‖ψ(v)‖ dv<∞. 7en, � a � t � � H 2 � � � � sup E ψ(s)dB (s) ≤ c ‖ψ(s)‖ ds. (11) � � t,H � � (i) 7e fuzzy stochastic It o integral 〈 ψ(u)d B 0 0 a∈[0,t] 2 n (u)〉∈ L (J ×Ω, N; E ) n n 2 n Let us define the embedding of R to E as (ii) For x∈ L (J ×Ω, N; E ), we have, for u≤ v∈ J, n n 〈.〉: R ⟶ E : 1, if a � r, 〈r〉(a) � (12) 0, if a≠ r. v v u u v v H H H d x w dw + ψ