On the radius of convergence of the logarithmic signature
On the radius of convergence of the logarithmic signature
复制标题
关于对数签名的收敛半径
DOI：
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发表时间：
2006
影响因子：
0.6
通讯作者：
N. Sidorova
中科院分区：
文献类型：
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作者：
Terry Lyons;N. Sidorova
最近已经证明，在R-d中有界变差的连续路径可以通过将其变换为称为路径签名的迭代积分序列来表征。签名在代数中取值并且总是具有对数。在本文中，我们研究了与路径的对数签名相对应的级数的收敛半径，这种收敛可以在控制理论中得到解释（特别是，该级数可以用于有效地计算时不变向量场，其幂运算产生与时间非齐次流相同的自同构），并且可以提供有效的数值逼近。我们给出了一个简单的下界的收敛半径的这一系列的长度的路径。然而，本文的主要结果是，收敛半径的全日志签名是有限的两个广泛的类的路径（我们猜想，这适用于所有的路径不同的直线）。
It has recently been proved that a continuous path of bounded variation in 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. In this paper we study the radius of convergence of the series corresponding to this logarithmic signature for the path. This convergence can be interpreted in control theory (in particular, the series can be used for effective computation of time invariant vector fields whose exponentiation yields the same diffeomorphism as a time inhomogeneous flow) and can provide efficient numerical approximations to solutions of SDEs. We 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 (and we conjecture that this holds for all paths different from straight lines).