Abstract
To solve a stochastic linear evolution equation numerically, finite dimensional approximations are commonly used. If one uses the well-known Galerkin scheme, one might end up with a sequence of ordinary stochastic linear equations of high order. To reduce the high dimension for practical computations we consider balanced truncation as a model order reduction technique. This approach is well-known from deterministic control theory and successfully employed in practice for decades. So, we generalize balanced truncation for controlled linear systems with Levy noise, discuss properties of the reduced order model, provide an error bound, and give some examples.
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Notes
\((\fancyscript{F}_t)_{t\ge 0}\) shall be right continuous and complete.
Cadlag means that \(\mathbb {P}\)-almost all paths are right continuous and the left limits exist.
This means that \(\mathbb {P}\)-almost all paths are of bounded variation.
The process \(\left\langle M, N\right\rangle \) is measurable with respect to \(\fancyscript{P}\), which we characterize below Definition 2.10.
We assume that \(\left( \fancyscript{F}_t\right) _{t\ge 0}\) is right continuous and that \(\fancyscript{F}_0\) contains all \(\mathbb {P}\) null sets.
By Theorem VI.21 in Reed, Simon [23], \(\fancyscript{Q}\) is a compact operator such that this property follows by the spectral theorem.
Curtain, Ichikawa and Haussmann stated these conditions for exponential mean square stability for the Wiener case, which can be easily generalized for the case of square integrable Levy process with mean zero.
The theory regarding this method can be found in Kloeden and Platen [16].
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Acknowledgments
The authors would like to thank Tobias Damm for his comments and advice and Tobias Breiten for providing Example 4.3.
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Benner, P., Redmann, M. Model reduction for stochastic systems. Stoch PDE: Anal Comp 3, 291–338 (2015). https://doi.org/10.1007/s40072-015-0050-1
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DOI: https://doi.org/10.1007/s40072-015-0050-1