Multistep methods for SDEs and their application to problems with small noise

In this article the numerical approximation of solutions of Ito stochastic differential equations is considered, in particular for equations with a small parameter $\epsilon$ in the noise coefficient. We construct stochastic linear multistep methods and develop the fundamental numerical analysis concerning their mean-square consistency, numerical stability in the mean-square sense and mean-square convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency. Further, for the small noise case we obtain expansions of the local error in terms of the step size and the small parameter $\epsilon$. Simulation results using several explicit and implicit stochastic linear $k$-step schemes, $k=1,\;2$, illustrate the theoretical findings.

[1]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[2]  Evelyn Buckwar,et al.  On Two-step Schemes for SDEs with Small Noise , 2004 .

[3]  Rózsa Horváth Bokor.  On Two-Step Methods for Stochastic Differential Equations , 1997, Acta Cybern..

[4]  Pamela Burrage,et al.  Runge-Kutta methods for stochastic differential equations , 1999 .

[5]  G. Mil’shtein.  A Theorem on the Order of Convergence of Mean-Square Approximations of Solutions of Systems of Stochastic Differential Equations , 1988 .

[6]  J. B. Walsh,et al.  An introduction to stochastic partial differential equations , 1986 .

[7]  M. V. Tretyakov,et al.  Stochastic Numerics for Mathematical Physics , 2004, Scientific Computation.

[8]  W. Gautschi.  Numerical analysis: an introduction , 1997 .

[9]  R. Winkler.  Stochastic differential algebraic equations of index 1 and applications in circuit simulation , 2003 .

[10]  R D Richtmyek,et al.  Survey of the Stability of Linear Finite Difference Equations , 2022 .

[11]  Kevin Burrage,et al.  General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems , 1998 .

[12]  R. Plato.  Numerische Mathematik kompakt , 2000 .

[13]  G. N. Milstein,et al.  Mean-Square Numerical Methods for Stochastic Differential Equations with Small Noises , 1997, SIAM J. Sci. Comput..

[14]  M. Wiktorsson.  Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions , 2001 .

[15]  Roger Temam,et al.  Numerical Analysis of Stochastic Schemes in Geophysics , 2004, SIAM J. Numer. Anal..

[16]  C. Penski,et al.  A new numerical method for SDEs and its application in circuit simulation , 2000 .

[17]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[18]  G. Milstein.  Numerical Integration of Stochastic Differential Equations , 1994 .

[19]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[20]  K. Burrage,et al.  Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations , 2000 .

[21]  Germund Dahlquist,et al.  33 years of numerical instability, Part I , 1985 .

[22]  Stefan Schäffler,et al.  Adams methods for the efficient solution of stochastic differential equations with additive noise , 2007, Computing.

[23]  R. Seydel.  Numerical Integration of Stochastic Differential Equations , 2004 .

[24]  Renate Winkler.  Stochastic DAEs in Transient Noise Simulation , 2004 .

[25]  Georg Denk,et al.  Modelling and simulation of transient noise in circuit simulation , 2007 .

[26]  Walter Gautschi,et al.  Numerical Analysis , 1978, Mathemagics: A Magical Journey Through Advanced Mathematics.

[27]  P. Rentrop,et al.  Modeling, Simulation, and Optimization of Integrated Circuits , 2003 .

[28]  Evelyn Buckwar,et al.  NUMERICAL ANALYSIS OF EXPLICIT ONE-STEP METHODS FOR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS , 1975 .

[29]  Werner Römisch,et al.  Stochastic DAEs in Circuit Simulation , 2003 .

[30]  T. Rydén,et al.  On the simulation of iterated Itô integrals , 2001 .

[31]  R. Bokor.  Stochastically stable one-step approximations of solutions of stochastic ordinary differential equations , 2003 .

[32]  Evelyn Buckwar,et al.  Improved linear multi-step methods for stochastic ordinary differential equations , 2007 .