Abstract
Numerical schemes for random ordinary differential equations, abbreviated RODEs, with an affine structure can be derived in a similar way as for affine control systems using Taylor expansions that resemble stochastic Taylor expansions for Stratonovich stochastic differential equations. The driving noise processes can be quite general, such as Wiener processes or fractional Brownian motions with continuous sample paths or compound Poisson processes with piecewise constant sample paths, and even more general noises. Such affine-Taylor schemes of arbitrarily high order are constructed here. It is shown how their structure simplifies when the noise terms are additive or commutative. A derivative free counterpart is given and multi-step schemes are derived too. Numerical comparisons are provided for various explicit one-step and multi-step schemes in the context of a toggle switch model from systems biology.
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Asai, Y., Kloeden, P.E. Numerical schemes for random ODEs with affine noise. Numer Algor 72, 155–171 (2016). https://doi.org/10.1007/s11075-015-0038-y
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DOI: https://doi.org/10.1007/s11075-015-0038-y