Impulsive stochastic fractional differential equations driven by fractional Brownian motion

In this research, we study the existence and uniqueness results for a new class of stochastic fractional differential equations with impulses driven by a standard Brownian motion and an independent fractional Brownian motion with Hurst index $1/2< H<1$1/2<H<1 under a non-Lipschitz condition with the Lipschitz one as a particular case. Our analysis depends on an approximation scheme of Carathéodory type. Some previous results are improved and extended.


Introduction
Traditionally, a dominant interest in practical applications is the existence of solutions to deterministic fractional differential equations and fractional stochastic differential equations (FSDEs) driven by Brownain motion due to their role for helping candidates explore the hidden properties of the dynamics of complex systems in viscoelasticity, diffusion, me-Sobolev-type stochastic fractional control systems with fBm by using semigroup theory, fractional calculus, stochastic analysis, and Banach's fixed point theorem. Pei and Xu [28] derived the unique solution for non-Lipschitz SDEs with fBm by using successive approximations. Moreover, for a massive body of published studies covering the existence and uniqueness of FSDEs driven by fBm; see [23,24,35] and references therein.
On the other hand, the past recent years have seen a rapid development of the theory of impulsive effects in many evolutionary processes such as telecommunications, finance, electronics, economics, and mechanics, in which states are often subject to abrupt and short changes in discrete moments of time and can be neglected throughout the whole duration of the intended process [22]. In light of recent developments in the theory of SDEs, it is becoming extremely difficult to ignore the existence of impulsive effects. Therefore several studies have documented the effect of impulses in studying the SDEs driven by Brownian motion [18] and fBm; see [10,11,13,14] and references therein.
To the best of our knowledge, there is no work yet reported in the literature on impulsive fractional stochastic differential equations driven by fBm. Therefore, motivated by this fact and in order to close this gap, in this paper, we initiate a research on one of such equations. The specific objective of this study is to prove the existence and uniqueness of solutions to the following impulsive stochastic fractional differential equations (ISFDEs) driven by a standard Brownian motion and an independent fBm of the form: where T ≥ 0 is a fixed horizon, W H is an m-dimensional fBm with 1/2 < H < 1 independent of an m-dimensional standard Gaussian process W (t), t ∈ [0, T]. In what follows, (Ω, F, P) is a complete probability space with probability measure P on Ω, and the filtration {F t } t≥0 refers to the σ -field generated by {W H (s), W (s), s ∈ [0, t]} and satisfying the usual conditions, that is, it is right continuous, and F 0 contains all P-null sets. Assume that b, . . , m) are bounded functions with fixed times t j satisfying 0 = t 0 < t 1 < t 2 < · · · < t m < T, and X(t + j ) and X(tj ) represent the right and left limits of The class of Eqs. (1) has attracted our attention because of their applications in complex dynamic processes in sciences and engineering and modeling many phenomena in ecological and epidemiological processes of population dynamic perturbed by unavoidable noises under multitime scales [27]. Moreover, Eqs. (1) can be used as a model of many evolutionary processes where the noises are correlated and can be modeled by fBm.
To summarize, our contribution here is the first attempt to consider the existence and uniqueness of solutions to ISFDEs driven by fBm. We obtained our results on Eqs. (1) by using Carathéodory approximation [2,3] under non-Lipschitz (Taniguchi [34]) condition with Lipschitz one as a particular case. Moreover, the results are still new even when the coefficients of (1) satisfy the Lipschitz condition and under the non-Lipschitz condition used in [4], which is a particular case of our conditions. Finally, the obtained results extend and improve some published results of [1,4,28,36]. This paper is outlined as follows. In Sect. 2, we provide necessary notions and preliminaries on the pathwise integrals with respect to fBm and hypotheses needed throughout the paper. We give our main results on the existence and uniqueness theorem for ISFDEs driven by a standard Brownian motion and an independent fBm given by (1) followed by some remarks and corollaries in Sect. 3.

Preliminaries
In this section, we review some basic notions and notations on the backward stochastic integral with respect to fBm, and for more details, we refer to [9,16,25]. The fBm with the Hurst index H ∈ ( 1 2 , 1) is a centered Wiener process W H = {W H (t)} 0≤t≤T with the covariance function where H ∈ ( 1 2 , 1), and define the space of Borel-measurable functions h : which is a separable Hilbert space under the inner product For any integer n ≥ 1, denote by S the set of smooth cylindrical random variables of the form , h and its partial derivatives of all orders are bounded), φ i ∈ H (i = 1, 2, . . . , n), H is a Hilbert space [7] defined as the completion of measurable functions The Malliavin ψ-derivative of a smooth and cylindrical random variable F ∈ S is defined as the H-valued random variable where

Definition 2.1 ([30]) Let η(t), t ∈ [0, T], be a stochastic process with integrable trajectories. The backward stochastic integral
provided that the limit exists in probability.
According to Remark 1 and Lemma 2 in [36], the following lemma comes: be an fBm with Hurst index H > 1 2 , and let a stochastic process The following definition defines the integration with respect to (dt) β , and the reader is referred to [17] for the proof.

Definition 2.2 Let g(t) be a continuous function. Then its integral with respect to
Similar to Definition 2.2 in [1], the definition of the unique solution to Eq. (1) can be given as follows.
(ii) For every t ∈ [0, T], X(t) satisfies the following integral equation: To attain the main results, the following assumptions are imposed on the coefficients b, σ 1 , g, and σ 2 .
where R(t, v) : [0, +∞) × R + − → R + is a function locally integrable in t for any fixed v ≥ 0 and continuous, nondecreasing, and concave in v for any fixed where G : [0, +∞) × R + − → R + is a function locally integrable in t for any fixed v ≥ 0 and continuous, nondecreasing, and concave in v for any fixed t for all X, Y ∈ L ψ [0, T] ∩ D 1,2 (|H|) and |I j (0)| = 0.

Main results
In this section, we present the existence and uniqueness of solutions to Eq. (1). Proof To begin with, we introduce the Carathéodory approximation as follows. For any integer n ≥ 1, define X n (t) = X(0) = X 0 for all -1 ≤ t ≤ 0 and We split the proof into the following three parts. Thus by conditions (H1) and (H3) and the Jensen inequality we have where

Now, by condition (H1) there exists a solution u(t), t ∈ [0, T], satisfying
Comparing this above equation and Eq. (4), we have which shows the uniform boundedness of {X n (t)} n≥1 .
Part 2. For 0 ≤ s < t ≤ T and integer n ≥ 1, we claim that E X n (t) -X n (s) 2 ≤ C 4 (ts) + C 6 (ts) 2α + C 5 where C 4 , C 5 , C 6 will be defined further in the proof, and the constant C comes from Part 1.
Taking the expectation to Eq.
By the plus and minus technique and assumption (H2) this yields In terms of Part 2, we have Similarly to (10), by the B-D-G inequality, Lemma 2.1, and condition (H2) we have Finally, for J 5 , by condition (H3) we obtain Combining Eqs. (9)-(12), we conclude Then Eqs. (13) and (14), together with Fatou's lemma, yield s-1/n<t j <s-1/m C ds where we have used the facts that G(s, 0) = 0 and s-1/n<t j <s-1/m C → 0 as n, m → ∞. Lastly, through Eq. (15) and condition (H1), we immediately get indicating that {X n (t)} n≥1 is a Cauchy sequence. The Borel-Cantelli lemma shows that, as n → ∞, X n (t) → X(t) uniformly for t ∈ [0, T]. Hence taking limits on both sides of Eq. (3), we obtain that X(t), t ∈ [0, T], is a solution to Eq. (1) with the property E(sup 0≤s≤t |X(s)| 2 ) < ∞ for all t ∈ [0, T], and this completes the proof of the existence. Now the uniqueness of solution can be obtained by the same procedure as Part 3. Therefore the proof of Theorem 3.1 is completed.