Pathwise Estimation of Stochastic Differential Equations with Unbounded Delay and Its Application to Stochastic Pantograph Equations

Abstract

The existence and uniqueness of the global solution of stochastic differential equations with discrete variable delay is investigated in this paper, and the pathwise estimation is also done by using Lyapunov function method and exponential martingale inequality. The results can be used not only in the case of bounded delay but also in the case of unbounded delay. As the applications, this paper considers the pathwise estimation of solutions of stochastic pantograph equations.

Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation

“…The paper did not find a ready home at the time, but subsequently has steadily attracted citations through its online preprint incarnation. These are included in the bibliography below [4,5,19,23,24,27,33,44,45,48,49,50,51,52,55,56]. Since a number of other works quote [6], we feel it best that the paper be subject to formal review, and as the first author's introduction to the subject came through a paper in the EJQTDE [38], we felt it fitting, after a long (but not unbounded) delay, to submit a revised version of it here.…”

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.…”

“…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. Reference [23] studied the stability of SFDEs with unbounded delay.…”

Almost Surely Exponential Stability of Numerical Solutions for Stochastic Pantograph Equations

“…Stochastic unbounded delay systems play an important role in a variety of application areas, including biology, epidemiology, mechanic, economics, and finance. The systems provide powerful models, such as infinite delay Kolmogorov-type systems in mathematic biology [1][2][3][4], stochastic neural networks [5][6][7][8], and stochastic pantograph equations in science and engineering. The pantograph equation which is a very special unbounded delay equation was used by Ockendon and Tayler [9] in 1971 to study how the electric current is collected by the pantograph of an electric locomotive, from where it gets its name.…”