Razumikhin-type Theorems on Exponential Stability of Stochastic Functional Differential Equations with Infinite Delay

Abstract

The main aim of this paper is to study the stability of the stochastic functional differential equations with infinite delay. We establish several Razumikhin-type theorems on the exponential stability for stochastic functional differential equations with infinite delay. By applying these results to stochastic differential equations with distributed delay, we obtain some sufficient conditions for both pth moment and almost surely exponentially stable. Finally, some examples are presented to illustrate our theory.

Stability of a Class of Hybrid Neutral Stochastic Differential Equations with Unbounded Delay

“…For example, in the neural network model, stochastic unbounded variable 2 Discrete Dynamics in Nature and Society delay differential system must be considered to model transmission and transformation of the signal in a better fashion (see [19]). Some related works on unbounded delay can be found in [20][21][22][23][24]. References [22,24] investigated the existence and uniqueness, as well as the pathwise stability of the global solutions to SDEs and NSDEs with unbounded delay, respectively.…”

“…References [22,24] investigated the existence and uniqueness, as well as the pathwise stability of the global solutions to SDEs and NSDEs with unbounded delay, respectively. Reference [23] studied the stability of SFDEs with unbounded delay. In Section 4.2 of [21], the stability and boundedness of nonlinear hybrid SDEs were discussed when the delay function ( ) was given as (0 < < 1).…”

Razumikhin-Type Theorems on General Decay Stability of Impulsive Stochastic Functional Differential Systems with Markovian Switching

“…Mao incorporated the Razumikhin approach in stochastic functional differential equations [9] and in neutral stochastic functional differential equations [10] to investigate both p-th moment and almost sure exponential stability of these systems (see also [11]- [13], for instance). Later, this technique was appropriately developed and extended to some other stochastic functional differential systems, especially important in applications, such as stochastic functional differential systems with infinite delay [14]- [16], hybrid stochastic delay interval systems [17] and impulsive stochastic delay differential systems [18]- [20]. Recently, some researchers have introduced ψ-type function and extended the stability results to the general decay stability, including the exponential stability as a special case in [21]- [23], which has a wide applicability.…”

Stability and Boundedness of Stochastic Volterra Integrodifferential Equations with Infinite Delay

“…So it is vital to seek some antidisturbance systems or present the intensity of stochastic perturbation that stable system can tolerate without losing the property of stability [9]. In recent years, many meaningful works on this topic have come out; see, for example, [10][11][12][13][14][15][16][17][18][19][20][21].…”