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In this paper we examine the convolution of a compound distribution with another given distribution. For this, new general upper and lower bounds on the tail probabilities of the convolution of a compound geometric distribution are derived. Also, such bounds are obtained under the truncated Cramér-Lundberg condition for the heavy-tailed case. Applications to the ruin probabilities of compound Poisson claims processes perturbed by diffusion as well as some new upper bounds are given. Also, several numerical examples to illustrate the effectiveness of the bounds are given. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly 1 for the approximate solution. The proof is based on new Hölder continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain H 1 stability, higher moment H 1 stability, L 2 stability, and higher moment L 2 stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract We provide a construction of a method based on path-dependent adaptive step-size control for global approximation of jump–diffusion SDEs. The discretization points and the number of them are chosen in adaptive way with respect to trajectories of the driving Poisson and Wiener processes. The resulting algorithm is in many cases much more efficient than methods based on nonuniform meshes obtained from adaptive but path-independent sampling. [ABSTRACT FROM AUTHOR]

In this paper, a gradually deteriorating system with imperfect repair is considered. The deterioration is modeled by a stationary stochastic process. The system fails once the deterioration level exceeds a given threshold L. At failure, an imperfect repair is performed and the deterioration level is reduced to a fixed value r , say. The system can be repaired n − 1 times and will be replaced after the n th failure. The article aims to estimate the parameters of the proposed deterioration process based on the observed failures. To this end we consider the Wiener and Gamma processes which are the most common used stochastic process models. In Wiener process, an explicit expression for the estimators is obtained. Birnbaum–Saunders approximation is extended to estimate the parameters in Gamma process. An optimal replacement policy is also discussed. Finally, a Monte-Carlo simulation is conducted to investigate the performance of estimators. [ABSTRACT FROM AUTHOR]

Abstract: We study linear stochastic evolution partial differential equations driven by additive noise. We present a general and flexible framework for representing the infinite dimensional Wiener process, which drives the equation. Since the eigenfunctions and eigenvalues of the covariance operator of the process are usually not available for computations, we propose an expansion in an arbitrary frame. We show how to obtain error estimates when the truncated expansion is used in the equation. For the stochastic heat and wave equations, we combine the truncated expansion with a standard finite element method and derive a priori bounds for the mean square error. Specializing the frame to biorthogonal wavelets in one variable, we show how the hierarchical structure, support and cancelation properties of the primal and dual bases lead to near sparsity and can be used to simplify the simulation of the noise and its update when new terms are added to the expansion. [Copyright &y& Elsevier]

Abstract: Stabilizing unstable periodic orbits of a deterministic chaotic system which is perturbed by a stochastic process is studied in this paper. The stochastic chaos is modeled by exciting a deterministic chaotic system with a white noise obtained from derivative of a Wiener process which eventually generates an Ito differential equation. It is also assumed that the chaotic system being studied has some model uncertainties which are not random. The sliding mode controller with some modifications is used for stochastic chaos suppression. It is shown that the system states converge to the desired orbit in such a way that the error covariance converges to an arbitrarily small bound around zero. As some case studies, the stabilization of 1-cycle and 2-cycle orbits of chaotic Duffing and Van der Pol systems is investigated by applying the proposed method to their corresponding stochastically perturbed systems. Simulation results show the effectiveness of the method and the accuracy of the statements proved in the paper. [Copyright &y& Elsevier]

Abstract: In this paper we prove the convergence of stochastic Navier–Stokes equations driven by white noise. A linearized version of the implicit Crank–Nicolson scheme is considered for the approximation of the solutions to the N–S equations. The noise is defined as the distributional derivative of a Wiener process and approximated by using the generalized -projection operator. Optimal strong convergence error estimates in the norm are obtained. [Copyright &y& Elsevier]