Mixed Caputo Fractional Neutral Stochastic Differential Equations with Impulses and Variable Delay

In this manuscript, a new class of impulsive fractional Caputo neutral stochastic differential equations with variable delay (IFNSDEs, in short) perturbed by fractional Brownain motion (fBm) and Poisson jumps was studied. We utilized the Carathéodory approximation approach and stochastic calculus to present the existence and uniqueness theorem of the stochastic system under Carathéodory-type conditions with Lipschitz and non-Lipschitz conditions as special cases. Some existing results are generalized and enhanced. Finally, an application is offered to illustrate the obtained theoretical results.

Controllability of impulsive neutral stochastic integro-differential systems driven by fractional Brownian motion with delay and Poisson jumps

In this paper the controllability of a class of impulsive neutral stochastic integro-differential systems driven by fractional Brownian motion and Poisson process in a separable Hilbert space with infinite delay is studied. The controllability result is obtained by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.

The Truncated Theta-EM Method for Nonlinear and Nonautonomous Hybrid Stochastic Differential Delay Equations with Poisson Jumps

In this paper, we study a class of nonlinear and nonautonomous hybrid stochastic differential delay equations with Poisson jumps (HSDDEwPJs). The convergence rate of the truncated theta-EM numerical solutions to HSDDEwPJs is investigated under given conditions. An example is shown to support our theory.