This paper presents a new type of Gronwall-Bellman inequality, which arises from a class of integral equations with a mixture of nonsingular and singular integrals. The new idea is to use a binomial function to combine the known Gronwall-Bellman inequalities for integral equations having nonsingular integrals with those having singular integrals. Based on this new type of Gronwall-Bellman inequality, we investigate the existence and uniqueness of the solution to a fractional stochastic differential equation (SDE) with fractional order 0 < α < 1. This result generalizes the existence and uniqueness theorem related to fractional order 1 2 < α < 1 appearing in [1]. Finally, the fractional type Fokker-Planck-Kolmogorov equation associated to the solution of the fractional SDE is derived using Itô's formula.
In this paper, the stability behaviors of stochastic differential equations (SDEs) driven by timechanged Brownian motions are discussed. Based on the generalized Lyapunov method and stochastic analysis, necessary conditions are provided for solutions of time-changed SDEs to be stable in different senses, such as stochastic stability, stochastically asymptotic stability and globally stochastically asymptotic stability. Also, a connection between the stability of the solution to the time-changed SDEs and that to their corresponding non-time-changed SDEs is revealed by applying the duality theorem. Finally, two examples are provided to illustrate the theoretical results.
This paper investigates the stability of a class of differential systems time-changed by E t which is the inverse of a β-stable subordinator. In order to explore stability, a time-changed Gronwall's inequality and a generalized Itô formula related to both the natural time t and the time-change E t are developed. For different time-changed systems, corresponding stability behaviors such as exponential sample-path stability, pth moment asymptotic stability and pth moment exponential stability are investigated. Also a connection between the stability of the time-changed system and that of its corresponding non-time-changed system is revealed.
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