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20
2011), A comparative linear meansquare stability analysis of
 Maruyama and Milsteintype methods, Math. Comput. Simulation
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Asymptotic meansquare stability of twostep methods for stochastic ordinary differential equations
 BIT Numerical Mathematics
"... We deal with linear multistep methods for SDEs and study when the numerical approximation shares asymptotic properties in the meansquare sense of the exact solution. As in deterministic numerical analysis we use a linear timeinvariant test equation and perform a linear stability analysis. Standar ..."
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We deal with linear multistep methods for SDEs and study when the numerical approximation shares asymptotic properties in the meansquare sense of the exact solution. As in deterministic numerical analysis we use a linear timeinvariant test equation and perform a linear stability analysis. Standard approaches used either to analyse deterministic multistep methods or stochastic onestep methods do not carry over to stochastic multistep schemes. In order to obtain sufficient conditions for asymptotic meansquare stability of stochastic linear twostepMaruyama methods we construct and apply Lyapunovtype functionals. In particular we study the asymptotic meansquare stability of stochastic counterparts of twostep AdamsBashforth and AdamsMoultonmethods, the MilneSimpson method and the BDF method.
Onestep approximations for stochastic functional differential equations
, 2004
"... We consider the problem of strong approximations of the solution of Ito stochastic functional differential equations (SFDEs). We develop a general framework for the convergence of driftimplicit onestep schemes to the solution of SFDEs. We provide examples to illustrate the applicability of the fr ..."
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We consider the problem of strong approximations of the solution of Ito stochastic functional differential equations (SFDEs). We develop a general framework for the convergence of driftimplicit onestep schemes to the solution of SFDEs. We provide examples to illustrate the applicability of the framework.
SDELab: stochastic differential equations with MATLAB
, 2006
"... We introduce SDELab, a package for solving stochastic differential equations (SDEs) within MATLAB. SDELab features explicit and implicit integrators for a general class of Ito and Stratonovich SDEs, including Milstein's method, sophisticated algorithms for iterated stochastic integrals, and fle ..."
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We introduce SDELab, a package for solving stochastic differential equations (SDEs) within MATLAB. SDELab features explicit and implicit integrators for a general class of Ito and Stratonovich SDEs, including Milstein's method, sophisticated algorithms for iterated stochastic integrals, and flexible plotting facilities.
A practical splitting method for stiff SDEs with applications to problems with small noise, Multiscale Modeling and Simulation
"... Abstract. We present an easy to implement drift splitting numerical method for the approximation of stiff, nonlinear stochastic differential equations (SDEs). The method is an adaptation of the SBDF multistep method for deterministic differential equations and allows for a semiimplicit discretiza ..."
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Abstract. We present an easy to implement drift splitting numerical method for the approximation of stiff, nonlinear stochastic differential equations (SDEs). The method is an adaptation of the SBDF multistep method for deterministic differential equations and allows for a semiimplicit discretization of the drift term to remove high order stability constraints associated with explicit methods. For problems with small noise, of amplitude , we prove that the method converges strongly with order O(∆t2 + ∆t+ 2∆t1/2) and thus exhibits second order accuracy when the time step is chosen to be on the order of or larger. We document the performance of the scheme with numerical examples and also present as an application a discretization of the stochastic CahnHilliard equation which removes the high order stability constraints for explicit methods. Key words. stochastic differential equations, meansquare convergence, weak convergence, multistep methods, IMEXmethods, CahnHilliard equation, conservative phase field models, Langevin
Meansquare convergence of stochastic multistep methods with variable stepsize
 J. Comput. Appl. Math
"... Abstract. We study meansquare consistency, stability in the meansquare sense and meansquare convergence of driftimplicit linear multistep methods with variable stepsize for the approximation of the solution of Itô stochastic differential equations. We obtain conditions that depend on the steps ..."
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Abstract. We study meansquare consistency, stability in the meansquare sense and meansquare convergence of driftimplicit linear multistep methods with variable stepsize for the approximation of the solution of Itô stochastic differential equations. We obtain conditions that depend on the stepsize ratios and that ensure meansquare convergence for the special case of adaptive twostep Maruyama schemes. Further, in the case of small noise we develop a local error analysis with respect to the h−ε approach and we construct some stochastic linear multistep methods with variable stepsize that have order 2 behavior if the noise is small enough. Key words. Stochastic linear multistep methods, Adaptive methods, Meansquare convergence, Meansquare numerical stability, Meansquare consistency, Small noise, Twostep Maruyama methods.
The ΘMaruyama scheme for stochastic functional differential equation with distributed memory term
, 2004
"... We consider the problem of strong approximations of the solution of Itostochastic functional differential equations involving a distributed delay term. The meansquare consistency of a class of schemes, the ΘMaruyama methods, is analysed, using an appropriate Itôformula. In particular, we investi ..."
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We consider the problem of strong approximations of the solution of Itostochastic functional differential equations involving a distributed delay term. The meansquare consistency of a class of schemes, the ΘMaruyama methods, is analysed, using an appropriate Itôformula. In particular, we investigate the consequences of the choice of a quadrature formula. Numerical examples illustrate the theoretical results.
Multistep Maruyama methods for stochastic delay differential equations
, 2004
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Twosided error estimates for stochastic onestep methods of higher order
, 2009
"... Abstract. This paper presents a unifying theory for the numerical analysis of stochastic onestep and multistep methods. In addition to wellknown results on the error of strong convergence we prove a twosided error estimate. This is characterized by Dahlquist’s strong root condition and is used to ..."
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Abstract. This paper presents a unifying theory for the numerical analysis of stochastic onestep and multistep methods. In addition to wellknown results on the error of strong convergence we prove a twosided error estimate. This is characterized by Dahlquist’s strong root condition and is used to determine the maximum order of convergence. In particular, we apply our theory to the stochastic theta method, BDF2Maruyama and higher order ItôTaylor schemes. The main ingredient of the stability analysis is a stochastic version of Spijker’s norm. Key words. SODE, ItôTaylor schemes, BDF2Maruyama, stochastic multistep method, twosided error estimate, consistency, bistability, stochastic Spijker norm
Efficient Transient Noise Analysis in Circuit Simulation
"... Transient noise analysis means time domain simulation of noisy electronic circuits. We consider mathematical models where the noise is taken into account by means of sources of Gaussian white noise that are added to the deterministic network equations, leading to systems of stochastic differential a ..."
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Transient noise analysis means time domain simulation of noisy electronic circuits. We consider mathematical models where the noise is taken into account by means of sources of Gaussian white noise that are added to the deterministic network equations, leading to systems of stochastic differential algebraic equations (SDAEs). A crucial property of the arising SDAEs is the large number of small noise sources that are included. As efficient means of their integration we discuss adaptive linear multistep methods, in particular stochastic analogues of the trapezoidal rule and the twostep backward differentiation formula, together with a new stepsize control strategy. Test results including reallife problems illustrate the performance of the presented methods. 1 Transient noise analysis in circuit simulation The increasing scale of integration, high clock frequencies and low supply voltages cause smaller signaltonoise ratios. Reduced signaltonoise ratio means that the difference between the wanted signal and noise is getting smaller. A