Existence, uniqueness and continuous dependence of solutions to conformable stochastic differential equations

Partial Stability of Conformable Stochastic Systems

“…Suppose that h δ 0 and ϕ δ 0 are smooth enough, i.e., satisfying the Lipschitz and the growth condition to ensure the existence and the uniqueness of a global solution of system (7) (for more details, see eorem 4.3 in [6]). Definition 4.…”

“…In particular, the asymptotic behaviour and the stability theory of the solution of ordinary stochastic systems, conformable fractional-order stochastic systems, and deterministic systems were investigated by many researchers (see [2][3][4][5][6]).…”

“…In [2,3,12], the authors studied the stability of the solution of conformable deterministic systems. In the literature, there is a few work about the stability of the solution of conformable stochastic systems (see [5,6]). On the other hand, in many real-world phenomena, such stability is sometimes too difficult to be satisfied.…”

Stability of conformable stochastic systems depending on a parameter

“…Let $$$ {\mathcal{W}}^{\varepsilon}\left(t,\lambda \right)\in {C}^{1,2}\left(\right[{t}_0,+\infty \left)\times {\mathbb{R}}^n,{\mathbb{R}}_{+}\right) $$$. By Itô formula (see [13]), one has: for t > t 0 $$$$ {\displaystyle \begin{array}{cc}\hfill {T}_{t_0}^{\beta }{\mathcal{W}}^{\varepsilon}\left(t,\lambda \right)=& \kern0.2em {\mathcal{L}}_{\beta}^{t_0}{\mathcal{W}}^{\varepsilon}\left(t,\lambda \right)\hfill \\ {}\hfill & +{\mathcal{W}}_{\lambda}^{\varepsilon}\left(t,\lambda \right){\psi}_{t_0}\left(t,\lambda, \varepsilon \right)\frac{dW(t)}{dt},\hfill \end{array}} $$$$ and …”

“…In particular, the stability theory of the solution of conformable fractional-order nonlinear deterministic and stochastic systems has attracted much more attention (see [6], [7], [8], and [9]). Recently, it has established its effectiveness as an important tools in many fields applications, such as control theory [6,7,[9][10][11][12][13][14][15], physics [16], or biology [8,17,18]. In fact, Neirameh in [8] suggested a method to find exact solitary wave solutions of extended biological population model.…”