Discrete maximal regularity of an implicit Euler--Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equations
An implicit Euler--Maruyama method with non-uniform step-size applied to a class of stochastic partial differential equations is studied. A spectral method is used for the spatial discretization and the truncation of the Wiener process. A discrete analogue of maximal $L^2$-regularity of the scheme and the discretised stochastic convolution is established, which has the same form as their continuous counterpart.
In this paper, we extend the logarithmic Euler-Maruyama scheme for stochastic delay differential equation in one dimension to the part where we propose a scheme for a system of stochastic delay differential equations. We then show that the scheme always maintains positivity subject to initial conditions. We then show the convergence of the proposed Euler-Maruyama scheme. With this scheme, all the approximate solutions are positive and the rate of convergence of this scheme is 0.5.
The truncated Euler-Maruyama (EM) method is proposed to approximate a class of non-autonomous stochastic differential equations (SDEs) with the Hölder continuity in the temporal variable and the super-linear growth in the state variable. The strong convergence with the convergence rate is proved. Moreover, the strong convergence of the truncated EM method for a class of highly non-linear time-changed SDEs is studied.
In this paper, we consider the equivalence of the $p$th moment exponential stability for stochastic differential equations (SDEs), stochastic differential equations with piecewise continuous arguments (SDEPCAs) and the corresponding Euler-Maruyama methods EMSDEs and EMSDEPCAs. We show that if one of the SDEPCAs, SDEs, EMSDEs and EMSDEPCAs is $p$th moment exponentially stable, then any of them is $p$th moment exponentially stable for a sufficiently small step size $h$ and $τ$ under the global Lipschitz assumption on the drift and diffusion coefficients
In this paper, we derive error estimates of the backward Euler-Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable, but assumed to be convex. We show that the backward Euler-Maruyama method is well-defined and convergent of order at least $1/4$ with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in [Nochetto, Savaré, and Verdi, Comm.\ Pure Appl.\ Math., 2000]. We verify tha…
This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler--Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same cond…
In this paper, we study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.
In this paper, we propose a class of explicit positivity preserving numerical methods for general stochastic differential equations which have positive solutions. Namely, all the numerical solutions are positive. Under some reasonable conditions, we obtain the convergence and the convergence rate results for these methods. The main difficulty is to obtain the strong convergence and the convergence rate for stochastic differential equations whose coefficients are of exponential growth. Some numerical experiments are provided to illustrate the theoretical results for our schemes.
The weak rate of convergence for the Euler-Maruyama approximation of one-dimensional stochastic differential equations involving the local times of the unknown process
We study the strong convergence order of the Euler-Maruyama scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev-Slobodeckij-type regularity of order $κ\in (0,1)$ for the non-smooth part of the drift, our analysis of the quadrature problem yields the convergence order $\min\{3/4,(1+κ)/2\}-ε$ for the equidistant Euler-Maruyama scheme (for arbitrarily small $ε>0$). The cut-off of the convergence …
Classical approximation results for stochastic differential equations analyze the $L^p$-distance between the exact solution and its Euler-Maruyama approximations. In this article we measure the error with temporal-spatial Hölder-norms. Our motivation for this are multigrid approximations of the exact solution viewed as a function of the starting point. We establish the classical strong convergence rate $0.5$ with respect to temporal-spatial Hölder-norms if the coefficient functions have bounded derivatives of first and second order.
We will introduce Euler-Maruyama approximations given by an orthogonal system in $L^{2}[0,1]$ for high dimensional SDEs, which could be finite dimensional approximations of SPDEs. In general, the higher the dimension is, the more one needs to generate uniform random numbers at every time step in numerical simulation. The schemes proposed in this paper, in contrast, can deal with this problem by generating very few uniform random numbers at every time step. The schemes save time in the simulation of very high dimensional SDEs. In particular, we conclude that an Euler-Maruyama approximation base…
The backward Euler-Maruyama (BEM) method is employed to approximate the invariant measure of stochastic differential equations, where both the drift and the diffusion coefficient are allowed to grow super-linearly. The existence and uniqueness of the invariant measure of the numerical solution generated by the BEM method are proved and the convergence of the numerical invariant measure to the underlying one is shown. Simulations are provided to illustrate the theoretical results and demonstrate the application of our results in the area of system control.
In this article, we consider numerical schemes for polynomial diffusions on the unit ball, which are solutions of stochastic differential equations with a diffusion coefficient of the form $\sqrt{1-|x|^{2}}$. We introduce a semi-implicit Euler--Maruyama scheme with the projection onto the unit ball and provide the $L^{2}$-rate of convergence. The main idea to consider the numerical scheme is the transformation argument introduced by Swart for proving the pathwise uniqueness for some stochastic differential equation with a non-Lipschitz diffusion coefficient.
This work deals with the Euler-Maruyama (EM) scheme for stochastic differential equations with Markovian switching (SDEwMSs). We focus on the Lp-convergence rate (p is greater than or equal to 2) of the EM method given in this paper. As far as we know, the skeleton process of the Markov chain is used in the continuous numerical methods in most papers. By contrast, the continuous EM method in this paper is to use the Markov chain directly. To the best of our knowledge, there are only two papers that consider the rate of Lp-convergence, which is no more than 1/p (p is greater than or equal to 2)…
A class of linear parabolic equations are considered. We give a posteriori error estimates in the maximum norm for a method that comprises extrapolation applied to the backward Euler method in time and finite element discretisations in space. We use the idea of elliptic reconstructions and certain bounds for the Green's function of the parabolic operator.
The semi-implicit Euler-Maruyama (EM) method is investigated to approximate a class of time-changed stochastic differential equations, whose drift coefficient can grow super-linearly and diffusion coefficient obeys the global Lipschitz condition. The strong convergence of the semi-implicit EM is proved and the convergence rate is discussed. When the Bernstein function of the inverse subordinator (time-change) is regularly varying at zero, we establish the mean square polynomial stability of the underlying equations. In addition, the numerical method is proved to be able to preserve such an asy…
Tamed-adaptive Euler-Maruyama approximation for SDEs with superlinearly growing and piecewise continuous drift, superlinearly growing and locally Hölder continuous diffusion
The present study utilizes the Girsanov transformation based framework for solving a nonlinear stochastic dynamical system in an efficient way in comparison to other available approximate methods. In this approach, a rejection sampling is formulated to evaluate the Radon-Nikodym derivative arising from the change of measure due to Girsanov transformation. The rejection sampling is applied on the Euler Maruyama approximated sample paths which draw exact paths independent of the diffusion dynamics of the underlying dynamical system. The efficacy of the proposed framework is ensured using more ac…
An explicit numerical method is developed for a class of non-autonomous time-changed stochastic differential equations, whose coefficients obey Hölder's continuity in terms of the time variables and are allowed to grow super-linearly in terms of the state variables. The strong convergence of the method in the finite time interval is proved and the convergence rate is obtained. Numerical simulations are provided.
The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $α$-Hölder drift in the recent literature the rate $α/2$ was proved in many related situations. By exploiting the regularising effect of the noise more efficiently, we show that the rate is in fact arbitrarily close to $1/2$ for all $α>0$. The result extends to Dini continuous coefficients, while in $d=1$ also to all bounded measurable coefficients.
The well-known stochastic SIS model characterized by highly nonlinear in epidemiology has a unique positive solution taking values in a bounded domain with a series of dynamical behaviors. However, the approximation methods to maintain the positivity and long-time behaviors for the stochastic SIS model, while very important, are also lacking. In this paper, based on a logarithmic transformation, we propose a novel explicit numerical method for a stochastic SIS epidemic model whose coefficients violate the global monotonicity condition, which can preserve the positivity of the original stochast…
Central limit theorem for temporal average of backward Euler--Maruyama method
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology（华中科技大学工程建模与科学计算湖北省重点实验室）
In this paper, we introduce adaptive Euler-Maruyama schemes for McKean-Vlasov stochastic differential equations (SDEs) assuming only a standard monotonicity condition on the drift and diffusion coefficients but no global Lipschitz continuity in the state variable for either, while global Lipschitz continuity is required for the measure component only. We prove moment stability of the discretised processes and a strong convergence rate of $1/2$. Several numerical examples, centred around a mean-field model for FitzHugh-Nagumo neurons, illustrate that the standard uniform scheme fails and that t…
Strong convergence rate of Euler-Maruyama method for stochastic differential equations with Hölder continuous drift coefficient driven by symmetric $α$-stable process
We study the numerical approximation of stochastic evolution equations with a monotone drift driven by an infinite-dimensional Wiener process. To discretize the equation, we combine a drift-implicit two-step BDF method for the temporal discretization with an abstract Galerkin method for the spatial discretization. After proving well-posedness of the BDF2-Maruyama scheme, we establish a convergence rate of the strong error for equations under suitable Lipschitz conditions. We illustrate our theoretical results through various numerical experiments and compare the performance of the BDF2-Maruyam…
Strong Convergence for Euler-Maruyama and Milstein Schemes with Asymptotic Method
Graduate School of Science and Engineering, Ritsumeikan University（立命馆大学理工学研究科）；Graduate School of Economics, The University of Tokyo（东京大学经济学研究科）；Mitsubishi UFJ Trust Investment Technology Institute Co., Ltd. (MTEC)（三菱UFJ信托投资技术研究所有限公司）
We present new discretisation of the moist compressible Euler equations, using the compatible finite element framework identified in Cotter and Shipton (2012). The discretisation strategy is described and details of the parametrisations of moist processes are presented. A procedure for establishing hydrostatic balance for moist atmospheres is introduced, and the model's performance is demonstrated through several test cases, two of which are new.
Numerical approximations to invariant measures of hybrid stochastic differential equations with superlinear coefficients via the backward Euler-Maruyama method
A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise
Heriot Watt University（赫瑞-瓦特大学）；The African Institute for Mathematical Sciences(AIMS) of South Africa and Stellenbosh University, Center for Research in Computational and Applied Mechanics (CERECAM), and Department of Mathematics and Applied Mathematics, University of Cape Town（南非非洲数学科学研究所（AIMS）和斯泰伦博斯大学，计算与应用力学研究中心（CERECAM），以及开普敦大学数学与应用数学系）
