On the stability of solutions to conformable stochastic differential equations

Abstract

In this paper, we study the stability of solutions to conformable stochastic differential equations. Firstly, we show the trivial solution are stochastially stable, stochastically asymptotically stable and almost surely exponentially stable, respectively. Secondly, we show the nontrivial solution are Ulam's type stable in the sense of probabilities. Finally, two examples are given to present the theoretically results.

Stability of conformable stochastic systems depending on a parameter

“…Different from the previous results in [9], taking into account the influence of the stochastic perturbed term and the conformable fractional derivative in the stability theory makes our results more general.…”

“…In this sense, our results present a complete generalization of the work [7]. The main contributions of our paper are as follows: Study the almost sure exponential stability ( A.S.E.S ) and the exponential stability in p‐th moment ( E.S.P.M ) of CSSP under some new conditions. Different from the previous results in [9], taking into account the influence of the stochastic perturbed term and the conformable fractional derivative in the stability theory makes our results more general. Generalize the work in [9] to the CSSP taking into account the parameter ε in system (). …”

Partial Stability of Conformable Stochastic Systems

“…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 the last decades, it has established its effectiveness as an essential tool in many real-world applications, such as physics [7], biology [4,8,9], and control theory [2,3,5,[10][11][12][13][14]. In fact, Neirameh in [4] proposed a new method to find exact solutions of the general biological population model.…”

“…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.…”