Modified averaged vector field methods preserving multiple invariants for conservative stochastic differential equations

Under a non-Lipschitz condition, the averaging principle for the general stochastic differential delay equations (SDDEs) is established. The solutions of convergence in mean square and convergence in probability between standard SDDEs and the corresponding averaged SDDEs are considered.MSC:34K50, 34C29.

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

Convergence and stability of the backward Euler method for jump–diffusion SDEs with super-linearly growing diffusion and jump coefficients

We present a novel approach to the mean square exponential stability of stochastic delay differential equations. Consequently, some new explicit criteria for the mean square exponential stability of general nonlinear stochastic delay differential equations are derived. Both delay-independent and delay-dependent stability criteria are given, and illustrative examples are also provided.

General population growth models with Allee effects in a random environment

This article is devoted to the existence of strong solutions to stochastic differential equations (SDEs). Compared with Ito's theory, we relax the assumptions on the volatility term and replace the global Lipschitz continuity condition with a local Lipschitz continuity condition and a Hoelder continuity condition. In particular, our general SDE covers the Cox–Ingersoll–Ross SDE as a special case. We note that the general weak existence theory presumably extends to our general SDE (although the explicit time dependence of the drift term and the volatility term might require some extra considerations). However, avoiding weak existence theory we prove the existence of a strong solution directly using a priori estimates (the so-called energy estimates) derived from the SDE. The benefit of this approach is that the argument only requires some basic knowledge about stochastic and functional analysis. Moreover, the underlying principle has developed to become one of the cornerstones of the modern theory of partial differential equations (PDEs). In this sense, the general goal of this article is not just to establish the existence of a strong solution to the SDE under consideration but rather to introduce a new principle in the context of SDEs that has already proven to be successful in the context of PDEs.

The method of characteristics (the averaging over the characteristic formula) and the weak-sense numerical integration of ordinary stochastic differential equations together with the Monte Carlo technique are used to propose numerical methods for linear stochastic partial differential equations (SPDEs). Their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. A variance reduction technique for the Monte Carlo procedures is considered. Layer methods for linear and semilinear SPDEs are constructed and the corresponding convergence theorems are proved. The approach developed is supported by numerical experiments.

By a novel approach, we present several new criteria for the mean square exponential stability of general non-linear stochastic delay differential equations. Both delay-independent and delay-dependent stability criteria are provided. A discussion of the obtained results is given.

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh. © 2012 Springer Science + Business Media B.V.

We present a numerical algorithm that allows the approximation of optimal controls for stochastic reaction-diffusion equations with additive noise by first reducing the problem to controls of feedback form and then approximating the feedback function using finitely based approximations. Using structural assumptions on the finitely based approximations, rates for the approximation error of the cost can be obtained. Our algorithm significantly reduces the computational complexity of finding controls with asymptotically optimal cost. Numerical experiments using artificial neural networks as well as radial basis function networks illustrate the performance of our algorithm. Our approach can also be applied to stochastic control problems for high dimensional stochastic differential equations and more general stochastic partial differential equations.

Stochastic exponential integrators for a finite element discretisation of SPDEs with additive noise

Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients

The authors describe a method of random generation of the integrals \[ A_{1,2} ( {t,t + h} ) = \int _t^{t + h} \int _t^s dw_1 ( r )dw_2 ( s ) - \int _t^{t + h} \int _{t}^s dw_2 ( r )dw_1 ( s ), \] together with the increments$w_1 ( {t + h} ) - w_1 ( t )$ and $w_2 ( {t + h} ) - w_2 ( t )$ of a two-dimensional Brownian path $( {w_1 ( t ),w_2 ( t )} )$. The method chosen is based on Marsaglia's “rectangle-wedge-tail” method, generalised to higher dimensions. The motivation is the need for a numerical scheme for simulation of strong solutions of general multidimensional stochastic differential equations with an order of convergence $O( h )$, where h is the stepsize. Previously, no method has obtained an order of convergence better than $O( {\sqrt h } )$ in the general case.

A numerical method for some stochastic differential equations with multiplicative noise

Asymptotic stability of fractional order (1,2] stochastic delay differential equations in Banach spaces