Pathwise Taylor schemes for random ordinary differential equations

Random ordinary differential equations (RODEs) are ordinary differential equations which contain a stochastic process in their vector fields. They can be analyzed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable. Traditional numerical schemes for ordinary differential equations thus do not achieve their usual order of convergence when applied to RODEs. Nevertheless, deterministic calculus can still be used to derive higher order numerical schemes for RODEs by means of a new kind of integral Taylor expansion. The theory is developed systematically here, applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes and compared with other numerical schemes for RODEs in the literature.

[1]  H. Sussmann.  On the Gap Between Deterministic and Stochastic Ordinary Differential Equations , 1978 .

[2]  Peter Deuflhard,et al.  Scientific Computing with Ordinary Differential Equations , 2002 .

[3]  M. Röckner,et al.  A Concise Course on Stochastic Partial Differential Equations , 2007 .

[4]  Rolando J. Biscay,et al.  The Local Linearization Method for Numerical Integration of Random Differential Equations , 2005 .

[5]  L. Arnold.  Random Dynamical Systems , 2003 .

[6]  P. Kloeden,et al.  Time-discretised Galerkin approximations of parabolic stochastic PDE's , 1996, Bulletin of the Australian Mathematical Society.

[7]  Arnulf Jentzen,et al.  A random Euler scheme for Carathéodory differential equations , 2009 .

[8]  Lars Grüne,et al.  Pathwise Approximation of Random Ordinary Differential Equations , 2001 .

[9]  G. Stengle.  Numerical methods for systems with measurable coefficients , 1990 .

[10]  K. Sobczyk.  Stochastic Differential Equations: With Applications to Physics and Engineering , 1991 .

[11]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[12]  T. T. Soong,et al.  Random differential equations in science and engineering , 1974 .

[13]  P. Kloeden,et al.  Pathwise convergent higher order numerical schemes for random ordinary differential equations , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[14]  Lars Grüne,et al.  Higher order numerical schemes for affinely controlled nonlinear systems , 2001, Numerische Mathematik.

[15]  Alberto Isidori,et al.  Nonlinear control systems: an introduction (2nd ed.) , 1989 .

[16]  A. Isidori.  Nonlinear Control Systems , 1985 .

[17]  Peter Imkeller,et al.  The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors , 2001 .

[18]  F. Weidenhammer,et al.  H. Bunke, Gewöhnliche Differentialgleichungen mit zufälligen Parametern. (Mathematische Lehrbücher und Monographien, Band XXXI). VI + 210 S. Berlin 1972. Akademie-Verlag. Preis geb. 40,—M , 1973 .

[19]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[20]  H. Bunke,et al.  Gewöhnliche Differentialgleichungen mit zufälligen Parametern , 1972 .