High order local linearization methods: an approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise
type: Item
instance of: scholarly article
title: High order local linearization methods: an approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise
zbMATH Open document ID: 1205.65027
author: H. de la Cruz Cancino
author: Juan Carlos Jimenez
author: Felix Carbonell
author: Rolando Biscay
author: Tohru Ozaki
published in: BIT
publication date: 2010-10-13T00:00:00Z
review text: New explicit numerical methods called high order local linear (HOLL) methods are introduced for the purpose of strong approximation of the solution of the \(d\)-dimensional stochastic differential equation with additive noise \[ dX(t)= f(t,X(t))\,dt+ \sum^m_{j=1} g_j(t)\,dW^j_t,\quad t\in [t_0,T],\quad X(t_0)= x_0, \] where \(W_t\) is an \(m\)-dimensional standard Wiener process. Theorems are proved that establish the global order of convergence and the A-stability of the methods. The methods are shown, for linear equations with stationary solutions, to produce a random attractor that converges to the actual one as the stepsize approaches zero, a property dubbed random A-stability. Two types of HOLL methods, one employing Taylor schemes and the other employing Runge-Kutta schemes, are presented and then shown in examples to have faster convergence with less computation cost than some comparable known methods.
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H35
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
zbMATH DE Number: 5799390
zbMATH Keywords: local linearization method
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: additive noise
zbMATH Keywords: numerical integration
zbMATH Keywords: A-stability
zbMATH Keywords: Wiener process
zbMATH Keywords: convergence
zbMATH Keywords: Taylor schemes
zbMATH Keywords: Runge-Kutta schemes
reviewed by: Melvin D. Lax
describes a project that uses: RODAS
describes a project that uses: Expokit
MaRDI profile type: Publication
full work available at URL: s10543-010-0272-6
OpenAlex ID: W1967726010
cites work: Q4210478
cites work: Q4210940
cites work: Approximation of the Zakaï Equation by the Splitting up Method
cites work: Local linearization method for the numerical solution of stochastic differential equations
cites work: Time series: theory and methods.
cites work: Adams-type methods for the numerical solution of stochastic ordinary differential equations
cites work: Q4839865
cites work: Order conditions of stochastic Runge--Kutta methods by B-series
cites work: Numerical methods for strong solutions of stochastic differential equations: an overview
cites work: Stiffly accurate Runge-Kutta methods for stiff stochastic differential equations
cites work: Numerical Solution of Stochastic Differential Equations with Constant Diffusion Coefficients
cites work: The local linearization method for numerical integration of random differential equations
cites work: Weak local linear discretizations for stochastic differential equations: convergence and numerical schemes
cites work: Local Linearization-Runge Kutta (LLRK) Methods for Solving Ordinary Differential Equations
cites work: A higher order local linearization method for solving ordinary differential equations
cites work: Adams methods for the efficient solution of stochastic differential equations with additive noise
cites work: A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting
cites work: Diffusions for Global Optimization
cites work: Q5707650
cites work: Q4714145
cites work: Geometric ergodicity of discrete-time approximations to multivariate diffusions
cites work: Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations
cites work: Almost Sure and Moment Exponential Stability in the Numerical Simulation of Stochastic Differential Equations
cites work: Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations
cites work: \(A\)-stability of Runge-Kutta methods for systems with additive noise
cites work: Step size control for the uniform approximation of systems of stochastic differential equations with additive noise.
cites work: A simple algebraic expression to evaluate the local linearization schemes for stochastic differential equations
cites work: Approximation of continuous time stochastic processes by the local linearization method revisited
cites work: A numerical method for the computation of the Lyapunov exponents of nonlinear ordinary differential equations
cites work: Rate of convergence of local linearization schemes for initial-value problems
cites work: Local Linear Approximations of Jump Diffusion Processes
cites work: Local Linearization Method for Numerical Integration of Delay Differential Equations
cites work: Local linearization filters for non-linear continuous-discrete state space models with multiplicative noise
cites work: An Approximate Innovation Method For The Estimation Of Diffusion Processes From Discrete Data
cites work: Simulation of stochastic differential equations through the local linearization method. A comparative study
cites work: Q4004325
cites work: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise.
cites work: Q4826106
cites work: Q4319807
cites work: A Lie algebraic approach to numerical integration of stochastic differential equations
cites work: Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later
cites work: Q4263372
cites work: Q4839955
cites work: A local linearization approach to nonlinear filtering
cites work: A General Implicit Splitting for Stabilizing Numerical Simulations of Itô Stochastic Differential Equations

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High order local linearization methods: an approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise (DOI)
DOI: 10.1007/S10543-010-0272-6

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