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Splitting Methods in Communication, Imaging, Science, and Engineering pp 499–539Cite as

Splitting Methods for SPDEs: From Robustness to Financial Engineering, Optimal Control, and Nonlinear Filtering

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Abstract

In this survey chapter we give an overview of recent applications of the splitting method to stochastic (partial) differential equations, that is, differential equations that evolve under the influence of noise. We discuss weak and strong approximations schemes. The applications range from the management of risk, financial engineering, optimal control, and nonlinear filtering to the viscosity theory of nonlinear SPDEs.

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Fig. 15.1
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Notes

  1. 1.

    Named after the botanist Robert Brown who observed in 1827 that pollen grains suspended in water execute continuous but jittery motions. The physical explanation was given by Albert Einstein in 1905 (his “annus mirabilis”: small water molecules hit the pollen) and a little earlier Marian Smoluchowski had already emphasized the importance of this process for physics. Further important contributions are due to Louis Bachelier, Andrey Kolmogorov, Paul Lévy, Joseph Doob, Norbert Wiener, and finally Kyoshi Ito

  2. 2.

    More precisely, the relevant property of the Brownian motion here is that B is a martingale. Geometrically, this is an orthogonality relation between the increments B t − B s and the path up to time s. Hence, the construction works in a geometric L 2(Ω) sense which allows to take advantage of this structure.

  3. 3.

    There are some subtle measure-theoretic issues which we gloss over but refer the reader to [1] for more details.

  4. 4.

    For example \(E\left [\int _{0}^{t}\vert h(X_{s})\vert ^{2}ds\right ] < \infty \),\(E\left [\int _{0}^{t}Z_{s}\vert h(X_{s})\vert ^{2}ds\right ] < \infty \) and \(\tilde{\mathbb{P}}\left [\int _{0}^{t}[\rho _{s}(\vert h\vert )]^{2}ds < \infty \right ] = 1\) is sufficient where \(Z_{s} =\exp \left (-\sum _{i}\int _{0}^{s}h^{i}(X_{r})dB_{r}^{i} -\frac{1} {2}\int _{0}^{s}h^{i}(X_{r})^{2}dr\right )\) ; see [1, Chapter 3]

  5. 5.

    That is, even without any numerical error, it is generally not possible to obtain a perfect fit to market prices, due to the model limitations.

  6. 6.

    By which we do not mean difficult-to-evaluate series expansions, Bessel functions or similar solutions. Instead, we mean formulas with comparable complexity to the vector fields themselves.

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Acknowledgements

Harald Oberhauser is grateful for the support of the ERC (grant agreement No.291244 Esig) and the Oxford-Man Institute of Quantitative finance.

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Authors and Affiliations

  1. Weierstrass Institute, Mohrenstr. 39, 10117, Berlin, Germany

    Christian Bayer

  2. Department of Mathematics, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, UK

    Harald Oberhauser

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  1. Department of Mathematics, University of Houston, Houston, Texas, USA

    Roland Glowinski

  2. Department of Mathematics, UCLA, Los Angeles, California, USA

    Stanley J. Osher

  3. Department of Mathematics, UCLA, Los Angeles, California, USA

    Wotao Yin

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Bayer, C., Oberhauser, H. (2016). Splitting Methods for SPDEs: From Robustness to Financial Engineering, Optimal Control, and Nonlinear Filtering. In: Glowinski, R., Osher, S., Yin, W. (eds) Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-41589-5_15

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