Abstract
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.
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References
Arnold, L.: Random Dynamical Systems. Springer, Heidelberg (1998)
Bunke, H.: Gewöhnliche Differentialgleichungen mit zufälligen Parametern. Akademie, Berlin (1972)
Carbonell, F., Jimenez, J.C., Biscay, R.J., de la Cruz, H.: The local linearization method for numerical integration of random differential equations. BIT 45, 1–14 (2005)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Deuflhard, P., Bornemann, V.: Scientific Computing with Ordinary Differential Equations. Springer, Berlin (2002)
Grecksch, W., Kloeden, P.E.: Time-discretised Galerkin approximation of parabolic stochastic PDEs. Bull. Austral. Math. Soc. 54, 79–85 (1996)
Grüne, L., Kloeden, P.E.: Higher order numerical schemes for affinely controlled nonlinear systems. Numer. Math. 89, 669–690 (2001)
Grüne, L., Kloeden, P.E.: Pathwise approximation of random ordinary differential equations. BIT 41, 710–721 (2001)
Imkeller, P., Schmalfuß, B.: The conjugacy of stochastic and random differential equations and the existence of global attractors. J. Dyn. Differ. Equ. 13, 215–249 (2001)
Isidori, A.: Nonlinear Control Systems. An Introduction, 2nd edn. Springer, Heidelberg
Jentzen, A.: Numerische Verfahren hoher Ordnung für zufällige Differentialgleichungen. Diplomarbeit, J.W. Goethe Universität, Frankfurt am Main, February 2007
Jentzen, A., Kloeden, P.E.: Pathwise convergent higher order numerical schemes for random ordinary differential equations. Proc. R. Soc. Lond. A 463(2087), 2929–2944 (2007)
Jentzen, A., Neuenkirch, A.: A random Euler scheme for Carathéodory differential equations. J. Comput. Appl. Math. 224(1), 346–359 (2009)
Kloeden, P.E., Platen, E.: Numerical Solutions of Stochastic Differential Equations. Springer, Berlin (1992)
Prévot, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Springer, Berlin (2007)
Sobczyk, K.: Stochastic Differential Equations with Applications to Physics and Engineering. Kluwer Academic, Dordrecht (1991)
Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press, San Diego (1973)
Stengle, G.: Numerical methods for systems with measurable coefficients. Appl. Math. Lett. 3, 25–29 (1990)
Sussmann, H.J.: On the gap between deterministic and stochastic differential equations. Ann. Probab. 6, 590–603 (1977)
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Communicated by Anders Szepessy.
Partially supported by the DFG project “Pathwise numerics and dynamics of stochastic evolution equations”.
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Jentzen, A., Kloeden, P.E. Pathwise Taylor schemes for random ordinary differential equations. Bit Numer Math 49, 113–140 (2009). https://doi.org/10.1007/s10543-009-0211-6
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DOI: https://doi.org/10.1007/s10543-009-0211-6
Keywords
- Random ordinary differential equations
- Integral Taylor expansion
- One-step numerical scheme
- Pathwise convergence
- Brownian motion
- Fractional Brownian motion