Asymptotic behavior of solutions to fractional stochastic multi-term differential equation systems involving non-permutable matrices

In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs for short). Our goal in this paper is to establish results on the global existence and uniqueness and continuity dependence of the initial values of the solutions to Caputo SMTDEs with non-permutable matrices of order $\alpha \in (\frac{1}{2},1)$ and $\beta \in (0,1)$ whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show the asymptotic separation property between two different solutions of Caputo SMTDEs with a more general condition based on $\lambda$. Also, the asymptotic separation rate for the two distinct mild solutions reveals that our asymptotic results are general.


Introduction
Over the years, many results on the theory and applications of stochastic differential equations (SDEs) have been studied [1,2,3]. In particular, fractional stochastic differential equations (FSDEs), which are a generalisation of differential equations by using fractional and stochastic calculus are becoming more popular due to their applications in modeling and financial mathematics. The nonlinear system of FDEs has been studied from various points of view: applications to population dynamics, optimal pricing in economics, and recent COVID-19 epidemics. Recently, FSDEs have been extensively used for modeling mathematical problems in finance [4,5], dynamics of complex systems in engineering [6] and other areas [7,8]. Most results on fractional stochastic dynamical systems are limited to proving the existence and uniqueness of mild solutions using the fixed point theorem [9]- [14]. A study on different types of stability studies for FSDEs can be found in [13,15,16].
Using fractional derivatives instead of integer-order derivatives allows for the modelling of a wider variety of behaviours. But sometimes, SDEs involving one fractional order of differentiation are not sufficient to describe physical processes. Therefore, recently, several authors have studied more general types of fractionalorder stochastic models, such as multi-term equations to get analytical and numerical approximation results. For instance, the authors in [17] have studied Euler-Maruyama scheme for fractional stochastic Langevin multi-term equations and introduced a general form of FSLEs together with strong convergence rate of a numerical mild solution.
Among the many scientific articles on asymptotic behaviour and asymptotic separation of fractional stochastic differential equations, we will mention only a few that motivate this work: • A few works on asymptotic separation of two distinct solutions to fractional stochastic differential equations which can also be found in [18]. Similar work on an exact asymptotic separation rate of two distinct solutions of doubly singular stochastic Volterra integral equations (SVIEs) with two different initial values was discussed in [19].
Hence, the plan of this paper is systematized as follows: Section 2 is an introductory section in which we recall the main definitions, results from fractional calculus, and necessary lemmas from fractional differential equations, and in Section 3 we review to the framework for the main results of the theory. Section 4 is devoted to proving global existence and uniqueness and continuity dependence on the initial values of the solutions of Caputo SMTDEs of order α ∈ ( 1 2 , 1) and β ∈ (0, 1) with non-permutable matrices. In Section 5, we investigate new results on the asymptotic behavior of solutions of the Caputo SMTDE by studying an asymptotic separation between two different solutions. In Section 6 we present an example to verify the results proved in Section 4 and 5. Section 7 is for the conclusion.

Mathematical preliminaries
We embark on this section by briefly introducing the essential structure of fractional calculus and fractional differential operators. For the more salient details on these matters, see the textbooks [27], [28]- [31]. Note that none of the results in this section are new, except Definition 2.3 and 2.4 (they are recently defined in [32] and [33], respectively) at the end, which we place in this section since a representation of solution is the Mittag-Leffler type function and it will be used later in the paper. 27,29]). The Riemann-Liouville integral operator of fractional order α > 0 is defined by The Riemann-Liouville derivative operator of fractional order α > 0 is defined by 27]). Suppose that α > 0, t > 0. The Caputo derivative operator of fractional order α is defined by: In particular, for α ∈ (0, 1) . The Riemann-Liouville fractional integral operator and the Caputo fractional derivative have the following property for α ≥ 0 [27,29]: The relationship between the Riemann-Liouville and Caputo fractional derivatives are as follows: 32]). We define a new Mittag-Leffler type function E A,B α,β,γ (·) : R → R generated by nonpermutable matrices A, B ∈ R n×n as follows: where Q A,B k,m ∈ R n×n , k, m ∈ N 0 ::= N ∪ {0} is given by An explicit representation of Q A,B k,m can be found in Table 1 in [32]. In the case of permutable matrices, i.e. AB = BA, we have Q A,B k,m := k+m m A k B m , k, m ∈ N 0 . Definition 2.4 ( [33]). We consider the Mittag-Leffler type function involving permutable matrices The following results are often used to compute estimations in Section 4 and 5.
Lemma 2.1. For all ω, t > 0, and α ∈ ( 1 2 , 1) the following inequality holds: Proof. Applying the series representation of Mittag-Leffler function and definition of beta function, then by swapping summation and integration, we get the desired result.
The following lemma plays a necessary role on proofs of main results in Section 4 and 5.

Formulation of main problem
In this section, we resort main assumptions which will be used in throughout the Section 4. We consider a Caputo stochastic multi-term differential equation of order α ∈ ( 1 2 , 1) and β ∈ (0, 1) with non-permutable matrices has the following form The coefficients b, σ : [0, T ] × R n → R n are measurable and bounded functions. A, B ∈ R n×n are nonpermutable matrices. We introduce the norm of the matrix which are used throughout this paper. For any matrix A = ( a ij ) n×n ∈ R n×n , the norm of the matrix A, according to the maximum norm on R n is A = max 1≤i≤n n j=1 |a ij |. Moreover, let (W t ) t≥0 denote a standard scalar Brownian motion on a complete probability space (Ω, F , F, P) with filtration F : Let H 2 ([0, T ], R n ) be well-endowed with the weighted maximum norm · ω as Let R n be endowed with the standard Euclidean norm and is complete with respect to the norm · ω and b, σ : [0, T ] × R n → R n are measurable and bounded functions satisfying the following conditions: and σ(·, 0) is essentially bounded i.e. ess sup T ] a.s. and satisfies Volterra integral equation of second kind on t ∈ [0, T ]: To define above integral equation, we apply Riemann-Liouville integral operator I α 0+ to the both side of (3.1), we define

dW (t) dt
Then we use the relationship between Riemann-Liouville integral and Caputo fractional differential operators (2.3) for 1 2 < α ≤ 1, and 0 < β ≤ 1, we get Then, we apply integration by parts formula for (3.4) to get (3.3). Now we can represent our mild solution of (3.1) involving non-permutable matrices. For each initial value η ∈ Ξ 0 , the mild solution X(·) ∈ R n of the Cauchy problem (3.1) can be represented in terms of Mittag-Leffler type functions involving non-permutable matrices as below: As a special case, for each initial value η ∈ Ξ 0 the system (3.1) has a unique mild solution in terms of Mittag-Leffler type functions (2.7) with permutable matrices on [0, T ] as below: These solutions can be derived with the help of variation of constants formula. Then the coincidence between the notion of mild solution and integral equation of (3.1) with permutable and non-permutable matrices can be proved in a similar way depicted in [9]. Therefore, we omit those proofs here.

Existence & uniqueness results and continuity dependence on initial conditions
In Section 5, we will look at the behavior of solutions to multi-order systems as the independent variable goes to infinity. For this purpose, it is important to have an existence and uniqueness result. Therefore, our first aim in this research article is to show the global existence and uniqueness of solution of (3.1). Moreover, we also prove the continuity dependence of solutions on the initial values. For any η ∈ Ξ 0 , we define an operator T η : The following lemma is devoted to showing that T η is well-defined.
From the definition of T η Y as in (4.1) and the Jensen's inequality (2.9) for n = 4, we have for all t ∈ [0, T ]:

(4.2)
Considering M := sup t∈[0,T ] E A,B α−β,α,α (t) and using Cauchy-Schwarz inequality, we obtain the following results: and Therefore, Now using the Itô's isometry, we attain From Assumption 3.1, we also have, Therefore, for all t ∈ [0, T ], we have This together with (4.2) and (4.3)-(4.5) yields that T η Y 2 H 2 < ∞. Hence, the map T η is well-defined. To prove global existence and uniqueness of solutions, we will show that the operator T η is contractive with respect to the weighted maximum norm (3.2). Now, we are in a position to prove Theorem 4.1.
Proof of Theorem 4.1: Let T > 0 be an arbitrary. Choose and fix a positive constant ω such that (i) Choose and fix η ∈ Ξ 0 . By virtue of Lemma 4.1, the operator T η is well-defined. We will prove that the map T η is a contraction with respect to the norm · ω .
For this purpose, let X, Y ∈ H 2 ([0, T ], R n ) be arbitrary. From (4.1) and the inequality (2.9) with n = 3, we derive the following estimations for all t ∈ [0, T ]: By the Cauchy-Schwarz inequality, we have Using Cauchy-Schwarz inequality and Assumption 3.1, we obtain Moreover, by Itô's isometry and Assumption 3.1, we also have Then for all t ∈ [0, T ], we acquire which together with the definition of · ω as in (3.2) implies that By virtue of Lemma 2.1, we have for all t ∈ [0, T ]: As a consequence, By (4.6), we have ζ < 1 and the operator T η is a contractive mapping on H 2 ([0, T ], · ω ). Using the Banach's fixed point theorem, there exists a unique fixed point of this map in H 2 ([0, T ], R n ). This fixed point is also a unique solution of (3.1) with initial conditions X(0) = η. The proof of (i) is complete.
Hence, using the Jensen's inequality (2.9) for n = 5, Assumption 3.1 and 3.2, the Cauchy-Schwarz inequality and Itô's isometry, we obtain By virtue of Lemma 2.1 and the definition of · ω , we have Thus, by (4.6), we have Hence, The proof is complete.
Taking expectation of both sides and using Assumption 3.1, we obtain From (5.1), we derive that lim t→∞ E ϕ(r, η) − ϕ(r, γ) 2 = 0. Thus, to derive contradiction it is enough to show that To show (5.2) and (5.3), choose and fix δ ∈ ( α λ , 1−α). Note that the existence of such a δ comes from the fact that α λ < 1 − α that is equivalent to the assumption λ > α 1−α . For t > max T 1/δ , 1 , using Cauchy-Schwarz inequality and the inequality (5.1), we have α−β,α,α (t − r) ϕ(r, η) − ϕ(r, γ) dr Although the asymptotic separation of two distinct mild solutions have now been constructed, there remain many other interesting open problems to be considered regarding asymptotics of the solution functions which may be studied by methods analogous to those used for computing asymptotics of Mittag-Leffler type functions in the univariate case.
Other related directions of research may include more deeply the various relevant function spaces may be useful in the qualitative theory of fractional stochastic differential equations related to these operators, for example well-posedness and regularity theory [34].