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On the radius of convergence of the logarithmic signature. (English) Zbl 1103.60060

From the authors’ summary: It has recently been proved that a continuous path of bounded variation in \(\mathbb R^d\) can be characterised in terms of its transform into a sequence of iterated integrals called the signature of the path. The signature takes its values in an algebra and always has a logarithm. The authors study the radius of convergence of the series corresponding to this logarithmic signature for the path. This convergence can be interpreted in control theory and can provide efficient numerical approximations to solutions of SDEs. They give a simple lower bound for the radius of convergence of this series in terms of the length of the path. However, the main result of the paper is that the radius of convergence of the full log signature is finite for two wide classes of paths.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34A34 Nonlinear ordinary differential equations and systems
93C35 Multivariable systems, multidimensional control systems
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