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A variation of constant formula for Caputo fractional stochastic differential equations
Accepted Manuscript A variation of constant formula for Caputo fractional stochastic differential equations P.T. Anh, T.S. Doan, P.T. Huong
PII: DOI: Reference:
S0167-7152(18)30326-2 https://doi.org/10.1016/j.spl.2018.10.010 STAPRO 8349
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Statistics and Probability Letters
Received date : 14 June 2018 Revised date : 25 September 2018 Accepted date : 7 October 2018 Please cite this article as: Anh P.T., et al., A variation of constant formula for Caputo fractional stochastic differential equations. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.10.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A variation of constant formula for Caputo fractional stochastic differential equations P.T. Anh1 , T.S. Doan2 and P.T. Huong3
Abstract We establish and prove a variation of constant formula for Caputo fractional stochastic differential equations whose coefficients satisfy a standard Lipschitz condition. The main ingredient in the proof is to use Ito’s representation theorem and the known variation of constant formula for deterministic Caputo fractional differential equations. As a consequence, for these systems we point out the coincidence between the notion of classical solutions introduced in [13] and mild solutions introduced in [12]. Keywords: Fractional stochastic differential equations, Classical solution, Mild solution, Inhomogeneous linear systems, A variation of constant formula. 1. Introduction Fractional differential equations have recently been received an increasing attention due to their applications in a variety of disciplines such as mechanics, physics, electrical engineering, control theory, etc. We refer the interested reader to the monographs [1, 6, 11] and the references therein for more details. In contrast to the well development in the qualitative theory of deterministic fractional differential equations, there have been only a few papers contributing to the qualitative theory of stochastic differential equations involving with a Caputo fractional time derivative and most of these articles have limited to the existence and uniqueness of solutions, see [13, 7]. 1
Email: [email protected], Le Quy Don Technical University, 236 Hoang Quoc Viet, Ha Noi, Viet Nam 2 Email: [email protected], Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Ha Noi, Viet Nam 3 Email: [email protected], Le Quy Don Technical University, 236 Hoang Quoc Viet, Ha Noi, Viet Nam Preprint submitted to Elsevier
September 25, 2018
It is undoubtable that a variation of constant formula for deterministic fractional systems, see [8], is an important tool in the qualitative theory including the stability theory and the invariant manifold theory built in [3, 4, 5]. In this paper, a stochastic version of variation of constant formula for Caputo fractional systems whose coefficients satisfy a standard Lipschitz condition is established. Roughly speaking, this formula indicates that a solution of nonlinear system can be given as a fixed point of the corresponding Lyapunov-Perron operator. A direct application is that an explicit formula for solutions of inhomogeneous linear fractional stochastic differential equations is formed. Concerning more potential applications, we refer the readers to the conclusion section. It is also worth mentioning that the established variation of constant formula in this paper also points out the coincidence between the notion of classical solutions introduced in [13] and mild solutions introduced in [12] for fractional stochastic differential equations without impulsive effects in a finite-dimensional space. It is interesting to know whether this result can be extended to systems involving impulsive effects and in an infinite dimensional systems. Another question is to weaken the Lipschitz assumption on the coefficients of the systems (cf. [2]). We leave these problems as open questions for the further research. The paper is structured as follows: In Section 2, we introduce briefly about Caputo fractional stochastic differential equations and state the main results of the paper. The first part of Section 3 is devoted to show the result on the existence and uniqueness of mild solutions (Theorem 3.2). The main result (Theorem 2.3) concerning a variation of constants formula for fractional stochastic differential equations is proved in the second part of Section 3. 2. Preliminaries and the statement of the main results 2.1. Fractional calculus and fractional differential equations Let α ∈ (0, 1], [a, b] ⊂ R and x : [a, b] → Rd be a measurable function Rb such that a kx(τ )k dτ < ∞. The Riemann–Liouville integral operator of order α is defined by Z t Z ∞ 1 α (Ia+ x)(t) := (t−τ )α−1 x(τ ) dτ, where Γ(α) := tα−1 exp (−t) dt, Γ(α) a 0 see [6]. The Caputo fractional derivative of order α of a function x ∈ α x(t) := (I 1−α Dx)(t), where D = d is the C 1 ([a, b]) is defined by CDa+ a+ dt 2
usual derivative. Analog to the case of integer derivative, a variation of constant formula is used to derive an explicit solution for inhomogenous linear systems involving fractional derivatives. More concretely, consider an inhomogenous linear fractional differential equation on a bounded interval [0, T ] C α Da+ x(t) = Ax(t) + f (t), x(0) = η, (1) where A ∈ Rd×d and f : [0, T ] → Rd is measurable and bounded. Then, an explicit formula for solution of (1) is given in the following theorem and its proof can be found in [8]. Theorem 2.1 (A variation of constant formula for Caputo fractional differential equations). The unique solution of (1) on [0, T ] is given by α
x(t) = Eα (t A)η +
Z
t
0
where Eα (z) :=
P∞
zk k=0 Γ(kα+1) ,
(t − τ )α−1 Eα,α ((t − τ )α A)f (τ ) dτ, Eα,α (z) :=
P∞
zk k=0 Γ(kα+α) .
2.2. Fractional stochastic differential equation and the main results Consider a Caputo fractional stochastic differential equation (for short Caputo fsde) of order α ∈ ( 12 , 1) on a bounded interval [0, T ] of the following form dWt C α D0+ X(t) = AX(t) + b(t, X(t)) + σ(t, X(t)) , (2) dt where (Wt )t∈[0,∞) is a standard scalar Brownian motion on an underlying complete filtered probability space (Ω, F, F := {Ft }t∈[0,∞) , P), A ∈ Rd×d and b, σ : [0, T ] × Rd → Rd are measurable functions satisfying the following conditions: (H1) There exists L > 0 such that for all x, y ∈ Rd , t ∈ [0, T ] kb(t, x) − b(t, y)k + kσ(t, x) − σ(t, y)k ≤ Lkx − yk. (H2)
RT 0
kb(τ, 0)k2 dτ < ∞,
esssupτ ∈[0,T ] kσ(τ, 0)k < ∞.
For each t ∈ [0, ∞), let Xt := L2 (Ω, Ft , P) denote p the space of all mean square integrable functions f : Ω → Rd with kf kms := E(kf k2 ). A process ξ : [0, ∞) → L2 (Ω, F, P) is said to be F-adapted if ξ(t) ∈ Xt for all t ≥ 0. Now, we recall the notion of classical solution of Caputo fsde, see e.g. [13, p. 209] and [7]. 3
Definition 2.2 (Classical solution of Caputo fsde). For each η ∈ X0 , a F-adapted process X is called a solution of (2) with the initial condition X(0) = η if the following equality holds for t ∈ [0, T ] Rt 1 α−1 (AX(τ ) + b(τ, X(τ ))) dτ X(t) = η + Γ(α) 0 (t − τ ) (3) Rt 1 α−1 σ(τ, X(τ )) dW . + Γ(α) τ 0 (t − τ ) It was proved in [7] that for any η ∈ X0 , there exists a unique solution which is denoted by ϕ(t, η) of (3). In the following main result of this paper, we establish a variation of constant formula for (2) which gives a special presentation of the solution ϕ(t, η).
Theorem 2.3 (A variation of constant formula for Caputo fsde). Let η ∈ X0 arbitrary. Then, the following statement Rt ϕ(t, η) = Eα (tα A)η + 0 (t − τ )α−1 Eα,α ((t − τ )α A)b(τ, ϕ(τ, η)) dτ Rt (4) + 0 (t − τ )α−1 Eα,α ((t − τ )α A)σ(τ, ϕ(τ, η)) dWτ holds for all t ∈ [0, T ].
Remark 2.4. (i) If the noise in (2) vanishes, i.e. σ(t, X(t)) = 0, then (4) becomes the variation of constant formula for deterministic fractional differential equations (cf. Theorem 2.1). (ii) Note that E1 (M ) = E1,1 (M ) = eM for M ∈ Rd×d . Letting α → 1, (4) formally becomes Z t Z t (t−τ )A tA e b(τ, ϕ(τ, η)) dτ + e(t−τ )A σ(τ, ϕ(τ, η)) dWτ , ϕ(t, η) = e η + 0
0
which is a variation of constant formula for solutions of stochastic differential equation dX(t) = (AX(t) + b(t, X(t))) dt + σ(t, X(t)) dWt , see [9, Theorem 3.1].
As an application of the preceding theorem, we obtain an explicit representation of the solution of inhomogeneous linear fsde of the form dWt C α D0+ X(t) = AX(t) + b(t) + σ(t) , X(0) = η. (5) dt Corollary 2.5. Suppose that b ∈ L2 ([0, T ], Rd ), σ ∈ L∞ ([0, T ], Rd ), where T > 0. Then, the explicit solution for (5) on [0, T ] is given by Z t X(t) = Eα (tα A)η + (t − τ )α−1 Eα,α ((t − τ )α A)b(τ ) dτ 0 Z t + (t − τ )α−1 Eα,α ((t − τ )α A)σ(τ ) dWτ . 0
4
3. Proof of the main results We will fix the following notions through this section. Let Rd be endowed with the standard Euclidean norm. For T > 0, let H2 ([0, T ], Rd ) denote the space of all processes ξ which are measurable, FT -adapted, where FT := {Ft }t∈[0,T ] , and satisfies that kξkH2 := sup0≤t≤T kξ(t)kms < ∞. Obviously, (H2 ([0, T ], Rd ), k · kH2 ) is a Banach space. 3.1. Existence and uniqueness of mild solutions We are now recalling the notion of mild solutions of (2), see [12]. Definition 3.1 (Mild solutions of Caputo fsdes). A F-adapted process Y is called a mild solution of (2) with the initial condition Y (0) = η if the following equality holds for t ∈ [0, T ] Rt Y (t) = Eα (tα A)η + 0 (t − τ )α−1 Eα,α ((t − τ )α A)b(τ, Y (τ )) dτ Rt + 0 (t − τ )α−1 Eα,α ((t − τ )α A)σ(τ, Y (τ )) dWτ .
(6)
Now, we establish a result on the existence and uniqueness of mild solutions for equation (2). In this result, we require that the coefficients of the system satisfy (H1) and (H2). The main ingredient of the proof is to introduce a suitable weighted norm (cf. [7]). Note that in [12], a result of the existence and uniqueness of mild solutions for a larger class of systems was also given. However, the assumption of the coefficients of these systems is stronger than (H1) and (H2). Theorem 3.2 (Existence and uniqueness of mild solutions). Suppose that (H1) and (H2) hold. For any η ∈ X0 , there exists a unique mild solution Y of (2) satisfying Y (0) = η, which is denoted by ψ(t, η). Proof. Let H2η ([0, T ], Rd ) := {ξ ∈ H2 ([0, T ], Rd ) : ξ(0) = η}. Define the corresponding Lyapunov-Perron operator Tη : H2η ([0, T ], Rd ) → H2η ([0, T ], Rd ) by Rt Tη Y (t) = Eα (tα A)η + 0 (t − τ )α−1 Eα,α ((t − τ )α A)b(τ, Y (τ )) dτ Rt + 0 (t − τ )α−1 Eα,α ((t − τ )α A)σ(τ, Y (τ )) dWτ . It is easy to see that operator Tη is well-defined. To complete the proof, it is sufficient to show that Tη is contractive with respect to a suitable metric
5
on H2η ([0, T ], Rd ). For this purpose, let H2 ([0, T ], Rd ) be endowed with a weighted norm k · kγ , where γ > 0, defined as follows s E(kξ(t)k2 ) kξkγ := sup for all ξ ∈ H2 ([0, T ], Rd ). (7) E2α−1 (γt2α−1 ) t∈[0,T ] Obviously, two norms k·kH2 and k·kγ are equivalent. Thus, (H2 ([0, T ], Rd ), k· kγ ) is also a Banach space. Therefore, the set H2η ([0, T ], Rd ) with the metric induced by k · kγ is complete. By compactness of [0, T ] and continuity of the function t 7→ Eα,α (tα A), there exists MT := maxt∈[0,T ] kEα,α (tα A)k > 0. Choose and fix a positive constant γ such that 2L2 MT2 (T + 1)
Γ(2α − 1) < 1. γ
(8)
Now, by definition of Tη , (H1), Ito’s isometry and MT we have
2
Z t
α−1 2 2 2 kX(τ ) − Y (τ )k dτ kTη X(t) − Tη Y (t)kms ≤ 2L MT (t − τ )
0
+2L2 MT2
Z
Using H¨older inequality, we obtain that kTη X(t) − Tη Y (t)k2ms ≤ 2L2 MT2 (T + 1) Hence, by definition of k · kγ we have
ms
t
0
Z
(t − τ )2α−2 kX(τ ) − Y (τ )k2ms dτ.
0
t
(t − τ )2α−2 kX(τ ) − Y (τ )k2ms dτ.
kTη X(t) − Tη Y (t)k2ms E2α−1 (γt2α−1 ) Rt (t − τ )2α−2 E2α−1 (γτ 2α−1 ) dτ 2 2 ≤ 2L MT (T + 1) 0 kX − Y k2γ . E2α−1 (γt2α−1 )
Note that for all t > 0 Z t γ (t − τ )2α−2 E2α−1 γτ 2α−1 dτ ≤ E2α−1 γt2α−1 , Γ (2α − 1) 0 see [7, Lemma 5]. Thus,
kTη X − Tη Y kγ ≤
s
2L2 MT2 (T + 1)
Γ(2α − 1) kX − Y kγ , γ
which together with (8) implies that Tη is contractive on H2η ([0, T ], Rd ). By contraction mapping principle, Tη has a unique fixed point and the proof is complete. 6
3.2. Proof of Theorem 2.3 By virtue of Theorem 3.2, to prove Theorem 2.3 it is sufficient to show that ϕ(t, η) = ψ(t, η) for all η ∈ X0 , t ∈ [0, T ]. (9) For a clearer presentation, we give here the motivation and the structure of this proof: Using the Ito’s representation theorem, for any function f ∈ XT there exists a unique adapted process Ξ ∈ H2 ([0, T ], Rd ) such that f = Ef + RT 0 Ξ(τ ) dWτ , see e.g. [10, Theorem 4.3.3]. Then, to prove (9) it is sufficient to show that the following statement Z T Z T ϕ(t, η), C + Ξ(τ ) dWτ = ψ(t, η), C + Ξ(τ ) dWτ 0
0
d holds for all C ∈ Rd and Ξ ∈ H2 ([0, we establish E D T ], R ). To do this, RT in Proposition 3.5 an estimate on ϕ(t, η) − ψ(t, η), C + 0 Ξ(τ ) dWτ . Before going to state and prove this estimate, we need a preparatory result in which we examine the components of the above term, i.e. we estimate
Z T
2
E(ϕ(t, η) − ψ(t, η))(c + ξ(τ ) dWτ )
where c ∈ R, ξ ∈ H ([0, T ], R). 0
Define functions χξ,η,c , κξ,η,c , χ bξ,η,c , κ bξ,η,c : [0, T ] → Rd by
Z
T
χξ,η,c (t) := Eϕ(t, η) c + ξ(τ ) dWτ , 0 Z T κξ,η,c (t) := Eb(t, ϕ(t, η)) c + ξ(τ ) dWτ , 0 Z T χ bξ,η,c (t) := Eψ(t, η) c + ξ(τ ) dWτ , 0 Z T κ bξ,η,c (t) := Eb(t, ψ(t, η)) c + ξ(τ ) dWτ .
(10) (11) (12) (13)
0
Remark 3.3. In the proof of the existence and uniqueness of classical solution and mild solution, we have ϕ(·, η), ψ(·, η) ∈ H2 ([0, T ], Rd ). Thus, χξ,η,c , κξ,η,c , χ bξ,η,c , κ bξ,η,c is measurable and bounded on [0, T ].
7
Lemma 3.4. For all t ∈ [0, T ], the following statements hold: χξ,η,c (t) = c Eα (tα A)Eη Z t + (t − τ )α Eα,α ((t − τ )α A) κξ,η,c (t) + Eξ(τ )σ(τ, ϕ(τ, η)) dτ, (14) 0
χ bξ,η,c (τ ) = c Eα (tα A)Eη Z t bξ,η,c (τ ) + Eξ(τ )σ(τ, ψ(τ, η)) dτ. (15) + (t − τ )α Eα,α ((t − τ )α A) κ 0
Proof. Since ϕ(t, η) is a solution of (2) it follows that Z t 1 ϕ(t, η) = η + (t − τ )α−1 (Aϕ(τ, η) + b(τ, ϕ(τ, η))) dτ Γ(α) 0 Z t 1 + (t − τ )α−1 σ(τ, ϕ(τ, η)) dWτ . Γ(α) 0 RT Taking product of both sides of the preceding equality with c + 0 ξ(τ ) dWτ and then taking the expectation of both sides give that Z t 1 (t − τ )α−1 (Aχξ,η,c (τ ) + κξ,η,c (τ )) dτ χξ,η,c (t) = c Eη + Γ(α) 0 Z t Z T 1 α−1 + (t − τ ) σ(τ, ϕ(τ, η)) dWτ , ξ(τ ) dWτ . Γ(α) 0 0 Using Ito’s isometry, we obtain that χξ,η,c (t) = c Eη Z t 1 + (t − τ )α−1 Aχξ,η,c (τ ) + κξ,η,c (τ ) + Eξ(τ )σ(τ, ϕ(τ, η)) dτ. Γ(α) 0
In the other words, χξ,η,c (t) is the solution of the following fractional differential equation C
α D0+ x(t) = Ax(t) + κξ,η,c (t) + Eξ(t)σ(t, ϕ(t, η)),
x(0) = c Eη.
Then, by virtue of Remark 3.3 and Theorem 2.1, the equality (14) is verified. Next, by Definition 3.1 we have Z t ψ(t, η) = Eα (tα A)η + (t − τ )α−1 Eα,α ((t − τ )α A)b(τ, ψ(τ, η)) dτ 0 Z t + (t − τ )α−1 Eα,α ((t − τ )α A)σ(τ, ψ(τ, η)) dWτ . 0
8
RT Taking product of both sides of the above equality with c + 0 ξ(τ ) dWτ and then taking the expectation of both sides give that Z t α χ bξ,η,c (t) = c Eα (t A)Eη + (t − τ )α Eα,α ((t − τ )α A)b κξ,η,c (τ ) dτ 0 Z t Z T α α ξ(τ ) dWτ . + (t − τ ) Eα,α ((t − τ ) A)σ(τ, ψ(τ, η)) dWτ , 0
0
Thus, by Ito’s isometry (15) is proved. Proposition 3.5. Let MT := maxt∈[0,T ] kEα,α (tα A)k. Then, for any C ∈ Rd and Ξ ∈ H2 ([0, T ], Rd ) we have 2 Z T ϕ(t, η) − ψ(t, η), C + Ξ(τ ) dW τ ≤
2dMT2 L2
T 2α−1
Z
C +
2α − 1
Z
2 2 +2dMT L C +
0
T
0 T
2 Z t
ξ(τ ) dWτ kϕ(τ, η) − ψ(τ, η)k2ms dτ
0 ms 0
2 Z t
ξ(τ ) dWτ (t − τ )2α−2 kϕ(τ, η) − ψ(τ, η)k2ms dτ.
ms
0
Proof. Let C = (c1 , . . . , cd )T and Ξ = (ξ1 , . . . , ξd )T , where ξi ∈ H2 ([0, T ], R), ci ∈ R. Then, Z T ϕ(t, η) − ψ(t, η), C + Ξ(τ ) dW τ 0 v u d 2 Z T u X t ϕi (t, η) − ψi (t, η), ci + ≤ d ξi (τ ) dWτ 0
i=1
v u d Z u X t
E(ϕ(t, η) − ψ(t, η)) ci + ≤ d
0
i=1
v u d u X kχξi ,η,ci (t) − χ bξi ,η,ci (t)k2 . = td
T
2
ξi (τ ) dWτ
(16)
i=1
Next, we are estimating kχξi ,η,ci (t) − χ bξi ,η,ci (t)k. In light of Lemma 3.4, we arrive at Z t kχξi ,η,ci (t) − χ bξi ,η,ci (t)k ≤ MT (t − τ )α−1 kκξi ,η,ci (τ ) − κ bξi ,η,ci (τ )k dτ 0 Z t +MT L (t − τ )α−1 kξi (τ )kms kϕ(τ, η) − ψ(τ, η)kms dτ. 0
9
Consequently, applying H¨older inequality yields that kχξi ,η,ci (t) − χ bξi ,η,ci (t)k Z t 21 Z t 12 2α−2 2 (t − τ ) dτ kκξi ,η,ci (τ ) − κ ≤ MT bξi ,η,ci (τ )k dτ 0
Z
0
t
kξi (τ )k2ms
21 Z
t
2α−2
ψ(τ, η)k2ms
21
+MT L dτ (t − τ ) kϕ(τ, η) − dτ 0 0 s Z t 12 T 2α−1 2 kκξi ,η,ci (τ ) − κ bξi ,η,ci (τ )k dτ ≤ MT 2α − 1 0 Z t 21 Z t 21 2 2α−2 2 kξi (τ )kms dτ +MT L . (t − τ ) kϕ(τ, η) − ψ(τ, η)kms dτ 0
0
On the other hand, by definition of κ and κ b we have for all τ ∈ [0, T ] kκξi ,η,ci (τ ) − κ bξi ,η,ci (τ )k2 Z d X bj (τ, ϕ(τ, η)) − bj (τ, ψ(τ, η)), ci + =
T
ξi (τ ) dWτ
0
j=1
Z
2 2 ≤ L kϕ(τ, η) − ψ(τ, η)kms ci +
0
T
2
ξi (τ ) dWτ
,
2
ms
where we use (H1) to obtain the preceding inequality. Thus, bξi ,η,ci k2 kχξi ,η,ci − χ
2 Z t Z T 2α−1
2 2T
≤ 2MT L c + ξ (τ ) dW kϕ(τ, η) − ψ(τ, η)k2ms dτ i i τ
2α − 1 0 ms 0 Z t Z t 2 2 2 +2MT L kξi (τ )kms dτ (t − τ )2α−2 kϕ(τ, η) − ψ(τ, η)k2ms dτ, 0
0
which together with (16) implies that 2 Z T ϕ(t, η) − ψ(t, η), C + Ξ(τ ) dW τ 0
2 Z t Z T 2α−1
2 2T
≤ 2dMT L C+ ξ(τ ) dWτ kϕ(τ, η) − ψ(τ, η)k2ms dτ
2α − 1 0 0 ms Z t Z t +2dMT2 L2 kξ(τ )k2ms dτ (t − τ )2α−2 kϕ(τ, η) − ψ(τ, η)k2ms dτ. 0
0
10
Furthermore, by Ito’s isometry
Z
C +
0
T
2 Z
2 ξ(τ ) dWτ = kCk +
T
0
ms
kξ(τ )k2ms dτ ≥
Z
0
t
kξ(τ )k2ms dτ,
which completes the proof.
Proof of Theorem 2.3. Let T ∗ := inf{t ∈ [0, T ] : ϕ(t, η) 6= ψ(t, η)}. Then, it is sufficient to show that T ∗ = T . Suppose the contrary, i.e. T ∗ < T . Choose and fix an arbitrary δ > 0 satisfing the following inequality 2dMT2 L2
T 2α−1 δ 2α−1 δ + 2dMT2 L2 < 1. 2α − 1 2α − 1
(17)
To lead a contradiction, we show that ϕ(t, η) = ψ(t, η) for all t ∈ [T ∗ , T ∗ +δ]. For this purpose, choose and fix an arbitrary t ∈ [T ∗ , T ∗ + δ]. Using Ito’s representation theorem, there exists Ct ∈ Rd and ξt∗ ∈ H2 ([0, t], Rd ) R ta unique ∗ such that ϕ(t, η) − ψ(t, η) = Ct + 0 ξt (τ ) dWτ . We extend ξt to the whole interval [0,T] by letting ξt∗ (τ ) = 0 for all τ ∈ (t, T ]. For such a ξt∗ , we have
Z
Ct +
T
0
ξt∗ (τ )
2
2 dWτ
= kϕ(t, η) − ψ(t, η)kms . ms
Thus, using Proposition 3.5 for C = Ct , Ξ = ξt∗ we obtain that kϕ(t, η) −
ψ(t, η)k2ms
Z T 2α−1 t ≤ kϕ(τ, η) − ψ(τ, η)k2ms dτ 2α − 1 T ∗ Z t 2 2 (t − τ )2α−2 kϕ(τ, η) − ψ(τ, η)k2ms dτ. +2dMT L 2dMT2 L2
T∗
Consequently, sup t∈[T ∗ ,T ∗ +δ]
≤ 2dMT2 L2
kϕ(t, η) − ψ(t, η)k2ms
T 2α−1 δ sup kϕ(t, η) − ψ(t, η)k2ms 2α − 1 t∈[T ∗ ,T ∗ +δ]
+2dMT2 L2
δ 2α−1 sup kϕ(t, η) − ψ(t, η)k2ms . 2α − 1 t∈[T ∗ ,T ∗ +δ]
By a choice of δ as in (17), we have supt∈[T ∗ ,T ∗ +δ] kϕ(t, η) − ψ(t, η)kms = 0. This leads to a contradiction and the proof is complete.
11
4. Conclusion In this paper, a variation of constant formula for stochastic fractional differential equations of order α ∈ ( 21 , 1) is established. This formula is a natural extension of the one for fractional differential equations and stochastic differential equations. In the forthcoming paper, we apply this formula to achieve a linearized stability theory for stochastic fractional differential equations. Acknowledgement This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.032017.01. The final work of this paper was done when the second author visited Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank VIASM for hospitality and financial support. The authors would like to thank a referee for his/her constructive comments that lead to an improvement of the paper. References [1] B. Bandyopadhyay and S. Kamal. Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach. Lecture Notes in Electrical Engineering, 317. Springer, 2015. [2] A. Benchaabane and R. Sakthivel. Sobolev-type fractional stochastic differential equations with non-lipschitz coefficients. J. Comput. Appl. Math., 312:65–73, 2017. [3] N. D. Cong, T. S. Doan, H. T. Tuan, and S. Siegmund. Linearized asymptotic stability for fractional differential equations. Electronic Journal of Qualitative Theory of Differential Equations, 39:1–13, 2016. [4] N. D. Cong, T. S. Doan, H. T. Tuan, and S. Siegmund. On stable manifolds for fractional differential equations in high-dimensional spaces. Nonlinear Dynamics, 86:1885–1894, 2016. [5] N. D. Cong, T. S. Doan, H. T. Tuan, and S. Siegmund. An instability theorem for nonlinear fractional differential systems. Discrete and Continuous Dynamical Systems - Series B, 22:3079–3090, 2017.
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[6] K. Diethelm. The Analysis of Fractional Differential Equations. An Application–Oriented Exposition Using Differential Operators of Caputo Type. Lect. Notes Math. 2004, Springer-Verlag, Berlin, 2010. [7] T. S. Doan, P.E. Kloeden, P.T. Huong, and H. T. Tuan. Asymptotic separation between solutions of caputo fractional stochastic differential equations. Stochastic Analysis and Applications, 36:654–664, 2018. [8] K. Li and J. Peng. Laplace transform and fractional differential equations. Applied Mathematics Letters, 24:2019–2023, 2011. [9] X. Mao. Stochastic Differential Equations and Applications. Horwood Publishing Limited, Chichester, 2008. [10] B. Øksendal. Stochastic Differential Equations. An Introduction with Applications. Springer-Verlag, 2000. [11] I. Podlubny. Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, To Methods of Their Solution and Some of Their Applications. Academic Press, Inc., San Diego, CA, 1999. [12] R. Sakthivel, P. Revathi, and Y. Ren. Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal. TMA., 81:70–86, 2013. [13] Y. Wang, J. Xu, and Peter E. Kloeden. Asymptotic behavior of stochastic lattice systems with a caputo fractional time derivative. Nonlinear Anal., 135:205–222, 2016.
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PII: DOI: Reference:
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Statistics and Probability Letters
Received date : 14 June 2018 Revised date : 25 September 2018 Accepted date : 7 October 2018 Please cite this article as: Anh P.T., et al., A variation of constant formula for Caputo fractional stochastic differential equations. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.10.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A variation of constant formula for Caputo fractional stochastic differential equations P.T. Anh1 , T.S. Doan2 and P.T. Huong3
Abstract We establish and prove a variation of constant formula for Caputo fractional stochastic differential equations whose coefficients satisfy a standard Lipschitz condition. The main ingredient in the proof is to use Ito’s representation theorem and the known variation of constant formula for deterministic Caputo fractional differential equations. As a consequence, for these systems we point out the coincidence between the notion of classical solutions introduced in [13] and mild solutions introduced in [12]. Keywords: Fractional stochastic differential equations, Classical solution, Mild solution, Inhomogeneous linear systems, A variation of constant formula. 1. Introduction Fractional differential equations have recently been received an increasing attention due to their applications in a variety of disciplines such as mechanics, physics, electrical engineering, control theory, etc. We refer the interested reader to the monographs [1, 6, 11] and the references therein for more details. In contrast to the well development in the qualitative theory of deterministic fractional differential equations, there have been only a few papers contributing to the qualitative theory of stochastic differential equations involving with a Caputo fractional time derivative and most of these articles have limited to the existence and uniqueness of solutions, see [13, 7]. 1
Email: [email protected], Le Quy Don Technical University, 236 Hoang Quoc Viet, Ha Noi, Viet Nam 2 Email: [email protected], Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Ha Noi, Viet Nam 3 Email: [email protected], Le Quy Don Technical University, 236 Hoang Quoc Viet, Ha Noi, Viet Nam Preprint submitted to Elsevier
September 25, 2018
It is undoubtable that a variation of constant formula for deterministic fractional systems, see [8], is an important tool in the qualitative theory including the stability theory and the invariant manifold theory built in [3, 4, 5]. In this paper, a stochastic version of variation of constant formula for Caputo fractional systems whose coefficients satisfy a standard Lipschitz condition is established. Roughly speaking, this formula indicates that a solution of nonlinear system can be given as a fixed point of the corresponding Lyapunov-Perron operator. A direct application is that an explicit formula for solutions of inhomogeneous linear fractional stochastic differential equations is formed. Concerning more potential applications, we refer the readers to the conclusion section. It is also worth mentioning that the established variation of constant formula in this paper also points out the coincidence between the notion of classical solutions introduced in [13] and mild solutions introduced in [12] for fractional stochastic differential equations without impulsive effects in a finite-dimensional space. It is interesting to know whether this result can be extended to systems involving impulsive effects and in an infinite dimensional systems. Another question is to weaken the Lipschitz assumption on the coefficients of the systems (cf. [2]). We leave these problems as open questions for the further research. The paper is structured as follows: In Section 2, we introduce briefly about Caputo fractional stochastic differential equations and state the main results of the paper. The first part of Section 3 is devoted to show the result on the existence and uniqueness of mild solutions (Theorem 3.2). The main result (Theorem 2.3) concerning a variation of constants formula for fractional stochastic differential equations is proved in the second part of Section 3. 2. Preliminaries and the statement of the main results 2.1. Fractional calculus and fractional differential equations Let α ∈ (0, 1], [a, b] ⊂ R and x : [a, b] → Rd be a measurable function Rb such that a kx(τ )k dτ < ∞. The Riemann–Liouville integral operator of order α is defined by Z t Z ∞ 1 α (Ia+ x)(t) := (t−τ )α−1 x(τ ) dτ, where Γ(α) := tα−1 exp (−t) dt, Γ(α) a 0 see [6]. The Caputo fractional derivative of order α of a function x ∈ α x(t) := (I 1−α Dx)(t), where D = d is the C 1 ([a, b]) is defined by CDa+ a+ dt 2
usual derivative. Analog to the case of integer derivative, a variation of constant formula is used to derive an explicit solution for inhomogenous linear systems involving fractional derivatives. More concretely, consider an inhomogenous linear fractional differential equation on a bounded interval [0, T ] C α Da+ x(t) = Ax(t) + f (t), x(0) = η, (1) where A ∈ Rd×d and f : [0, T ] → Rd is measurable and bounded. Then, an explicit formula for solution of (1) is given in the following theorem and its proof can be found in [8]. Theorem 2.1 (A variation of constant formula for Caputo fractional differential equations). The unique solution of (1) on [0, T ] is given by α
x(t) = Eα (t A)η +
Z
t
0
where Eα (z) :=
P∞
zk k=0 Γ(kα+1) ,
(t − τ )α−1 Eα,α ((t − τ )α A)f (τ ) dτ, Eα,α (z) :=
P∞
zk k=0 Γ(kα+α) .
2.2. Fractional stochastic differential equation and the main results Consider a Caputo fractional stochastic differential equation (for short Caputo fsde) of order α ∈ ( 12 , 1) on a bounded interval [0, T ] of the following form dWt C α D0+ X(t) = AX(t) + b(t, X(t)) + σ(t, X(t)) , (2) dt where (Wt )t∈[0,∞) is a standard scalar Brownian motion on an underlying complete filtered probability space (Ω, F, F := {Ft }t∈[0,∞) , P), A ∈ Rd×d and b, σ : [0, T ] × Rd → Rd are measurable functions satisfying the following conditions: (H1) There exists L > 0 such that for all x, y ∈ Rd , t ∈ [0, T ] kb(t, x) − b(t, y)k + kσ(t, x) − σ(t, y)k ≤ Lkx − yk. (H2)
RT 0
kb(τ, 0)k2 dτ < ∞,
esssupτ ∈[0,T ] kσ(τ, 0)k < ∞.
For each t ∈ [0, ∞), let Xt := L2 (Ω, Ft , P) denote p the space of all mean square integrable functions f : Ω → Rd with kf kms := E(kf k2 ). A process ξ : [0, ∞) → L2 (Ω, F, P) is said to be F-adapted if ξ(t) ∈ Xt for all t ≥ 0. Now, we recall the notion of classical solution of Caputo fsde, see e.g. [13, p. 209] and [7]. 3
Definition 2.2 (Classical solution of Caputo fsde). For each η ∈ X0 , a F-adapted process X is called a solution of (2) with the initial condition X(0) = η if the following equality holds for t ∈ [0, T ] Rt 1 α−1 (AX(τ ) + b(τ, X(τ ))) dτ X(t) = η + Γ(α) 0 (t − τ ) (3) Rt 1 α−1 σ(τ, X(τ )) dW . + Γ(α) τ 0 (t − τ ) It was proved in [7] that for any η ∈ X0 , there exists a unique solution which is denoted by ϕ(t, η) of (3). In the following main result of this paper, we establish a variation of constant formula for (2) which gives a special presentation of the solution ϕ(t, η).
Theorem 2.3 (A variation of constant formula for Caputo fsde). Let η ∈ X0 arbitrary. Then, the following statement Rt ϕ(t, η) = Eα (tα A)η + 0 (t − τ )α−1 Eα,α ((t − τ )α A)b(τ, ϕ(τ, η)) dτ Rt (4) + 0 (t − τ )α−1 Eα,α ((t − τ )α A)σ(τ, ϕ(τ, η)) dWτ holds for all t ∈ [0, T ].
Remark 2.4. (i) If the noise in (2) vanishes, i.e. σ(t, X(t)) = 0, then (4) becomes the variation of constant formula for deterministic fractional differential equations (cf. Theorem 2.1). (ii) Note that E1 (M ) = E1,1 (M ) = eM for M ∈ Rd×d . Letting α → 1, (4) formally becomes Z t Z t (t−τ )A tA e b(τ, ϕ(τ, η)) dτ + e(t−τ )A σ(τ, ϕ(τ, η)) dWτ , ϕ(t, η) = e η + 0
0
which is a variation of constant formula for solutions of stochastic differential equation dX(t) = (AX(t) + b(t, X(t))) dt + σ(t, X(t)) dWt , see [9, Theorem 3.1].
As an application of the preceding theorem, we obtain an explicit representation of the solution of inhomogeneous linear fsde of the form dWt C α D0+ X(t) = AX(t) + b(t) + σ(t) , X(0) = η. (5) dt Corollary 2.5. Suppose that b ∈ L2 ([0, T ], Rd ), σ ∈ L∞ ([0, T ], Rd ), where T > 0. Then, the explicit solution for (5) on [0, T ] is given by Z t X(t) = Eα (tα A)η + (t − τ )α−1 Eα,α ((t − τ )α A)b(τ ) dτ 0 Z t + (t − τ )α−1 Eα,α ((t − τ )α A)σ(τ ) dWτ . 0
4
3. Proof of the main results We will fix the following notions through this section. Let Rd be endowed with the standard Euclidean norm. For T > 0, let H2 ([0, T ], Rd ) denote the space of all processes ξ which are measurable, FT -adapted, where FT := {Ft }t∈[0,T ] , and satisfies that kξkH2 := sup0≤t≤T kξ(t)kms < ∞. Obviously, (H2 ([0, T ], Rd ), k · kH2 ) is a Banach space. 3.1. Existence and uniqueness of mild solutions We are now recalling the notion of mild solutions of (2), see [12]. Definition 3.1 (Mild solutions of Caputo fsdes). A F-adapted process Y is called a mild solution of (2) with the initial condition Y (0) = η if the following equality holds for t ∈ [0, T ] Rt Y (t) = Eα (tα A)η + 0 (t − τ )α−1 Eα,α ((t − τ )α A)b(τ, Y (τ )) dτ Rt + 0 (t − τ )α−1 Eα,α ((t − τ )α A)σ(τ, Y (τ )) dWτ .
(6)
Now, we establish a result on the existence and uniqueness of mild solutions for equation (2). In this result, we require that the coefficients of the system satisfy (H1) and (H2). The main ingredient of the proof is to introduce a suitable weighted norm (cf. [7]). Note that in [12], a result of the existence and uniqueness of mild solutions for a larger class of systems was also given. However, the assumption of the coefficients of these systems is stronger than (H1) and (H2). Theorem 3.2 (Existence and uniqueness of mild solutions). Suppose that (H1) and (H2) hold. For any η ∈ X0 , there exists a unique mild solution Y of (2) satisfying Y (0) = η, which is denoted by ψ(t, η). Proof. Let H2η ([0, T ], Rd ) := {ξ ∈ H2 ([0, T ], Rd ) : ξ(0) = η}. Define the corresponding Lyapunov-Perron operator Tη : H2η ([0, T ], Rd ) → H2η ([0, T ], Rd ) by Rt Tη Y (t) = Eα (tα A)η + 0 (t − τ )α−1 Eα,α ((t − τ )α A)b(τ, Y (τ )) dτ Rt + 0 (t − τ )α−1 Eα,α ((t − τ )α A)σ(τ, Y (τ )) dWτ . It is easy to see that operator Tη is well-defined. To complete the proof, it is sufficient to show that Tη is contractive with respect to a suitable metric
5
on H2η ([0, T ], Rd ). For this purpose, let H2 ([0, T ], Rd ) be endowed with a weighted norm k · kγ , where γ > 0, defined as follows s E(kξ(t)k2 ) kξkγ := sup for all ξ ∈ H2 ([0, T ], Rd ). (7) E2α−1 (γt2α−1 ) t∈[0,T ] Obviously, two norms k·kH2 and k·kγ are equivalent. Thus, (H2 ([0, T ], Rd ), k· kγ ) is also a Banach space. Therefore, the set H2η ([0, T ], Rd ) with the metric induced by k · kγ is complete. By compactness of [0, T ] and continuity of the function t 7→ Eα,α (tα A), there exists MT := maxt∈[0,T ] kEα,α (tα A)k > 0. Choose and fix a positive constant γ such that 2L2 MT2 (T + 1)
Γ(2α − 1) < 1. γ
(8)
Now, by definition of Tη , (H1), Ito’s isometry and MT we have
2
Z t
α−1 2 2 2 kX(τ ) − Y (τ )k dτ kTη X(t) − Tη Y (t)kms ≤ 2L MT (t − τ )
0
+2L2 MT2
Z
Using H¨older inequality, we obtain that kTη X(t) − Tη Y (t)k2ms ≤ 2L2 MT2 (T + 1) Hence, by definition of k · kγ we have
ms
t
0
Z
(t − τ )2α−2 kX(τ ) − Y (τ )k2ms dτ.
0
t
(t − τ )2α−2 kX(τ ) − Y (τ )k2ms dτ.
kTη X(t) − Tη Y (t)k2ms E2α−1 (γt2α−1 ) Rt (t − τ )2α−2 E2α−1 (γτ 2α−1 ) dτ 2 2 ≤ 2L MT (T + 1) 0 kX − Y k2γ . E2α−1 (γt2α−1 )
Note that for all t > 0 Z t γ (t − τ )2α−2 E2α−1 γτ 2α−1 dτ ≤ E2α−1 γt2α−1 , Γ (2α − 1) 0 see [7, Lemma 5]. Thus,
kTη X − Tη Y kγ ≤
s
2L2 MT2 (T + 1)
Γ(2α − 1) kX − Y kγ , γ
which together with (8) implies that Tη is contractive on H2η ([0, T ], Rd ). By contraction mapping principle, Tη has a unique fixed point and the proof is complete. 6
3.2. Proof of Theorem 2.3 By virtue of Theorem 3.2, to prove Theorem 2.3 it is sufficient to show that ϕ(t, η) = ψ(t, η) for all η ∈ X0 , t ∈ [0, T ]. (9) For a clearer presentation, we give here the motivation and the structure of this proof: Using the Ito’s representation theorem, for any function f ∈ XT there exists a unique adapted process Ξ ∈ H2 ([0, T ], Rd ) such that f = Ef + RT 0 Ξ(τ ) dWτ , see e.g. [10, Theorem 4.3.3]. Then, to prove (9) it is sufficient to show that the following statement Z T Z T ϕ(t, η), C + Ξ(τ ) dWτ = ψ(t, η), C + Ξ(τ ) dWτ 0
0
d holds for all C ∈ Rd and Ξ ∈ H2 ([0, we establish E D T ], R ). To do this, RT in Proposition 3.5 an estimate on ϕ(t, η) − ψ(t, η), C + 0 Ξ(τ ) dWτ . Before going to state and prove this estimate, we need a preparatory result in which we examine the components of the above term, i.e. we estimate
Z T
2
E(ϕ(t, η) − ψ(t, η))(c + ξ(τ ) dWτ )
where c ∈ R, ξ ∈ H ([0, T ], R). 0
Define functions χξ,η,c , κξ,η,c , χ bξ,η,c , κ bξ,η,c : [0, T ] → Rd by
Z
T
χξ,η,c (t) := Eϕ(t, η) c + ξ(τ ) dWτ , 0 Z T κξ,η,c (t) := Eb(t, ϕ(t, η)) c + ξ(τ ) dWτ , 0 Z T χ bξ,η,c (t) := Eψ(t, η) c + ξ(τ ) dWτ , 0 Z T κ bξ,η,c (t) := Eb(t, ψ(t, η)) c + ξ(τ ) dWτ .
(10) (11) (12) (13)
0
Remark 3.3. In the proof of the existence and uniqueness of classical solution and mild solution, we have ϕ(·, η), ψ(·, η) ∈ H2 ([0, T ], Rd ). Thus, χξ,η,c , κξ,η,c , χ bξ,η,c , κ bξ,η,c is measurable and bounded on [0, T ].
7
Lemma 3.4. For all t ∈ [0, T ], the following statements hold: χξ,η,c (t) = c Eα (tα A)Eη Z t + (t − τ )α Eα,α ((t − τ )α A) κξ,η,c (t) + Eξ(τ )σ(τ, ϕ(τ, η)) dτ, (14) 0
χ bξ,η,c (τ ) = c Eα (tα A)Eη Z t bξ,η,c (τ ) + Eξ(τ )σ(τ, ψ(τ, η)) dτ. (15) + (t − τ )α Eα,α ((t − τ )α A) κ 0
Proof. Since ϕ(t, η) is a solution of (2) it follows that Z t 1 ϕ(t, η) = η + (t − τ )α−1 (Aϕ(τ, η) + b(τ, ϕ(τ, η))) dτ Γ(α) 0 Z t 1 + (t − τ )α−1 σ(τ, ϕ(τ, η)) dWτ . Γ(α) 0 RT Taking product of both sides of the preceding equality with c + 0 ξ(τ ) dWτ and then taking the expectation of both sides give that Z t 1 (t − τ )α−1 (Aχξ,η,c (τ ) + κξ,η,c (τ )) dτ χξ,η,c (t) = c Eη + Γ(α) 0 Z t Z T 1 α−1 + (t − τ ) σ(τ, ϕ(τ, η)) dWτ , ξ(τ ) dWτ . Γ(α) 0 0 Using Ito’s isometry, we obtain that χξ,η,c (t) = c Eη Z t 1 + (t − τ )α−1 Aχξ,η,c (τ ) + κξ,η,c (τ ) + Eξ(τ )σ(τ, ϕ(τ, η)) dτ. Γ(α) 0
In the other words, χξ,η,c (t) is the solution of the following fractional differential equation C
α D0+ x(t) = Ax(t) + κξ,η,c (t) + Eξ(t)σ(t, ϕ(t, η)),
x(0) = c Eη.
Then, by virtue of Remark 3.3 and Theorem 2.1, the equality (14) is verified. Next, by Definition 3.1 we have Z t ψ(t, η) = Eα (tα A)η + (t − τ )α−1 Eα,α ((t − τ )α A)b(τ, ψ(τ, η)) dτ 0 Z t + (t − τ )α−1 Eα,α ((t − τ )α A)σ(τ, ψ(τ, η)) dWτ . 0
8
RT Taking product of both sides of the above equality with c + 0 ξ(τ ) dWτ and then taking the expectation of both sides give that Z t α χ bξ,η,c (t) = c Eα (t A)Eη + (t − τ )α Eα,α ((t − τ )α A)b κξ,η,c (τ ) dτ 0 Z t Z T α α ξ(τ ) dWτ . + (t − τ ) Eα,α ((t − τ ) A)σ(τ, ψ(τ, η)) dWτ , 0
0
Thus, by Ito’s isometry (15) is proved. Proposition 3.5. Let MT := maxt∈[0,T ] kEα,α (tα A)k. Then, for any C ∈ Rd and Ξ ∈ H2 ([0, T ], Rd ) we have 2 Z T ϕ(t, η) − ψ(t, η), C + Ξ(τ ) dW τ ≤
2dMT2 L2
T 2α−1
Z
C +
2α − 1
Z
2 2 +2dMT L C +
0
T
0 T
2 Z t
ξ(τ ) dWτ kϕ(τ, η) − ψ(τ, η)k2ms dτ
0 ms 0
2 Z t
ξ(τ ) dWτ (t − τ )2α−2 kϕ(τ, η) − ψ(τ, η)k2ms dτ.
ms
0
Proof. Let C = (c1 , . . . , cd )T and Ξ = (ξ1 , . . . , ξd )T , where ξi ∈ H2 ([0, T ], R), ci ∈ R. Then, Z T ϕ(t, η) − ψ(t, η), C + Ξ(τ ) dW τ 0 v u d 2 Z T u X t ϕi (t, η) − ψi (t, η), ci + ≤ d ξi (τ ) dWτ 0
i=1
v u d Z u X t
E(ϕ(t, η) − ψ(t, η)) ci + ≤ d
0
i=1
v u d u X kχξi ,η,ci (t) − χ bξi ,η,ci (t)k2 . = td
T
2
ξi (τ ) dWτ
(16)
i=1
Next, we are estimating kχξi ,η,ci (t) − χ bξi ,η,ci (t)k. In light of Lemma 3.4, we arrive at Z t kχξi ,η,ci (t) − χ bξi ,η,ci (t)k ≤ MT (t − τ )α−1 kκξi ,η,ci (τ ) − κ bξi ,η,ci (τ )k dτ 0 Z t +MT L (t − τ )α−1 kξi (τ )kms kϕ(τ, η) − ψ(τ, η)kms dτ. 0
9
Consequently, applying H¨older inequality yields that kχξi ,η,ci (t) − χ bξi ,η,ci (t)k Z t 21 Z t 12 2α−2 2 (t − τ ) dτ kκξi ,η,ci (τ ) − κ ≤ MT bξi ,η,ci (τ )k dτ 0
Z
0
t
kξi (τ )k2ms
21 Z
t
2α−2
ψ(τ, η)k2ms
21
+MT L dτ (t − τ ) kϕ(τ, η) − dτ 0 0 s Z t 12 T 2α−1 2 kκξi ,η,ci (τ ) − κ bξi ,η,ci (τ )k dτ ≤ MT 2α − 1 0 Z t 21 Z t 21 2 2α−2 2 kξi (τ )kms dτ +MT L . (t − τ ) kϕ(τ, η) − ψ(τ, η)kms dτ 0
0
On the other hand, by definition of κ and κ b we have for all τ ∈ [0, T ] kκξi ,η,ci (τ ) − κ bξi ,η,ci (τ )k2 Z d X bj (τ, ϕ(τ, η)) − bj (τ, ψ(τ, η)), ci + =
T
ξi (τ ) dWτ
0
j=1
Z
2 2 ≤ L kϕ(τ, η) − ψ(τ, η)kms ci +
0
T
2
ξi (τ ) dWτ
,
2
ms
where we use (H1) to obtain the preceding inequality. Thus, bξi ,η,ci k2 kχξi ,η,ci − χ
2 Z t Z T 2α−1
2 2T
≤ 2MT L c + ξ (τ ) dW kϕ(τ, η) − ψ(τ, η)k2ms dτ i i τ
2α − 1 0 ms 0 Z t Z t 2 2 2 +2MT L kξi (τ )kms dτ (t − τ )2α−2 kϕ(τ, η) − ψ(τ, η)k2ms dτ, 0
0
which together with (16) implies that 2 Z T ϕ(t, η) − ψ(t, η), C + Ξ(τ ) dW τ 0
2 Z t Z T 2α−1
2 2T
≤ 2dMT L C+ ξ(τ ) dWτ kϕ(τ, η) − ψ(τ, η)k2ms dτ
2α − 1 0 0 ms Z t Z t +2dMT2 L2 kξ(τ )k2ms dτ (t − τ )2α−2 kϕ(τ, η) − ψ(τ, η)k2ms dτ. 0
0
10
Furthermore, by Ito’s isometry
Z
C +
0
T
2 Z
2 ξ(τ ) dWτ = kCk +
T
0
ms
kξ(τ )k2ms dτ ≥
Z
0
t
kξ(τ )k2ms dτ,
which completes the proof.
Proof of Theorem 2.3. Let T ∗ := inf{t ∈ [0, T ] : ϕ(t, η) 6= ψ(t, η)}. Then, it is sufficient to show that T ∗ = T . Suppose the contrary, i.e. T ∗ < T . Choose and fix an arbitrary δ > 0 satisfing the following inequality 2dMT2 L2
T 2α−1 δ 2α−1 δ + 2dMT2 L2 < 1. 2α − 1 2α − 1
(17)
To lead a contradiction, we show that ϕ(t, η) = ψ(t, η) for all t ∈ [T ∗ , T ∗ +δ]. For this purpose, choose and fix an arbitrary t ∈ [T ∗ , T ∗ + δ]. Using Ito’s representation theorem, there exists Ct ∈ Rd and ξt∗ ∈ H2 ([0, t], Rd ) R ta unique ∗ such that ϕ(t, η) − ψ(t, η) = Ct + 0 ξt (τ ) dWτ . We extend ξt to the whole interval [0,T] by letting ξt∗ (τ ) = 0 for all τ ∈ (t, T ]. For such a ξt∗ , we have
Z
Ct +
T
0
ξt∗ (τ )
2
2 dWτ
= kϕ(t, η) − ψ(t, η)kms . ms
Thus, using Proposition 3.5 for C = Ct , Ξ = ξt∗ we obtain that kϕ(t, η) −
ψ(t, η)k2ms
Z T 2α−1 t ≤ kϕ(τ, η) − ψ(τ, η)k2ms dτ 2α − 1 T ∗ Z t 2 2 (t − τ )2α−2 kϕ(τ, η) − ψ(τ, η)k2ms dτ. +2dMT L 2dMT2 L2
T∗
Consequently, sup t∈[T ∗ ,T ∗ +δ]
≤ 2dMT2 L2
kϕ(t, η) − ψ(t, η)k2ms
T 2α−1 δ sup kϕ(t, η) − ψ(t, η)k2ms 2α − 1 t∈[T ∗ ,T ∗ +δ]
+2dMT2 L2
δ 2α−1 sup kϕ(t, η) − ψ(t, η)k2ms . 2α − 1 t∈[T ∗ ,T ∗ +δ]
By a choice of δ as in (17), we have supt∈[T ∗ ,T ∗ +δ] kϕ(t, η) − ψ(t, η)kms = 0. This leads to a contradiction and the proof is complete.
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4. Conclusion In this paper, a variation of constant formula for stochastic fractional differential equations of order α ∈ ( 21 , 1) is established. This formula is a natural extension of the one for fractional differential equations and stochastic differential equations. In the forthcoming paper, we apply this formula to achieve a linearized stability theory for stochastic fractional differential equations. Acknowledgement This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.032017.01. The final work of this paper was done when the second author visited Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank VIASM for hospitality and financial support. The authors would like to thank a referee for his/her constructive comments that lead to an improvement of the paper. References [1] B. Bandyopadhyay and S. Kamal. Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach. Lecture Notes in Electrical Engineering, 317. Springer, 2015. [2] A. Benchaabane and R. Sakthivel. Sobolev-type fractional stochastic differential equations with non-lipschitz coefficients. J. Comput. Appl. Math., 312:65–73, 2017. [3] N. D. Cong, T. S. Doan, H. T. Tuan, and S. Siegmund. Linearized asymptotic stability for fractional differential equations. Electronic Journal of Qualitative Theory of Differential Equations, 39:1–13, 2016. [4] N. D. Cong, T. S. Doan, H. T. Tuan, and S. Siegmund. On stable manifolds for fractional differential equations in high-dimensional spaces. Nonlinear Dynamics, 86:1885–1894, 2016. [5] N. D. Cong, T. S. Doan, H. T. Tuan, and S. Siegmund. An instability theorem for nonlinear fractional differential systems. Discrete and Continuous Dynamical Systems - Series B, 22:3079–3090, 2017.
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