## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 50, Number 1-4 (2006), 763-790.

### On the radius of convergence of the logarithmic signature

Terry J. Lyons and Nadia Sidorova

#### Abstract

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).

#### Article information

**Source**

Illinois J. Math., Volume 50, Number 1-4 (2006), 763-790.

**Dates**

First available in Project Euclid: 12 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258059491

**Digital Object Identifier**

doi:10.1215/ijm/1258059491

**Mathematical Reviews number (MathSciNet)**

MR2247845

**Zentralblatt MATH identifier**

1103.60060

**Subjects**

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Secondary: 34A34: Nonlinear equations and systems, general 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03] 93C35: Multivariable systems

#### Citation

Lyons, Terry J.; Sidorova, Nadia. On the radius of convergence of the logarithmic signature. Illinois J. Math. 50 (2006), no. 1-4, 763--790. doi:10.1215/ijm/1258059491. https://projecteuclid.org/euclid.ijm/1258059491