In this paper, we investigate the exponential mean-square stability for both the solution of n-dimensional stochastic delay integro-differential equations (SDIDEs) with Poisson jump, as well for the split-step θ-Milstein (SSTM) scheme implemented of the proposed model. First, by virtue of Lyapunov function and continuous semi-martingale convergence theorem, we prove that the considered model has the property of exponential mean-square stability. Moreover, it is shown that the SSTM scheme can inherit the exponential mean-square stability by using the delayed difference inequality established in the paper. Eventually, three numerical examples are provided to show the effectiveness of the theoretical results.
In this paper, we discuss about the boundedness and convergence analysis of the fractional Brownian motion (FBM) with Hurst parameter H. By the simple analysis and using the mean value theorem for stochastic integrals we conclude that in case of decreasing diffusion function, the solution of FBM is bounded for any H ∈ (0,1). Also, we derive the convergence rate which shows efficiency and accuracy of the computed solutions.
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