# Iterated integrals: stochastic analysis, algebra and machine learning

This seminar at TU Chemnitz for the winter semester 2018 is geared towards Bachelor and Master students. Below I sketch some of the possible topics, with samples of the literature.
Interested students please write me a short email, with their status (Bachelor/Master), background and maybe even with a subtopic they are interested in: joscha.diehl@gmail.com
The first meeting has taken place. If you still want to get a topic, please contact me via email.

Iterated integrals of the form $\ \int_0^T \int_0^{r_n} \int_0^{r_2} dX^{(i_1)}_{r_1} .. dX^{(i_n)}_{r_n}, %= %\int_0^T \int_0^{r_n} \int_0^{r_2} \dot X^{i_1}_{r_1} .. \dot X^{i_n}_{r_n} dr_1 .. dr_n$ appear in a plethora of mathematical situations. Here, $$X:[0,T]\to\mathbb{R}^d$$ is a curve, $$n \ge 1$$ and $$i_1, .., i_n \in \{1, .., d\}$$.

If $$X$$ is differentiable, these integrals are classically well-defined and encode a lot of information about the curve. Indeed, it is (almost) possible to reconstruct the curve $$X$$ given the values of all these integrals ($$T$$ is fixed!)

Boedihardjo, Horatio, et al. "The signature of a rough path: uniqueness." Advances in Mathematics 293 (2016): 720-737.

These integrals play an important role in control theory

Brockett, R. W., and Liyi Dai. "Non-holonomic kinematics and the role of elliptic functions in constructive controllability." Nonholonomic motion planning. Springer, Boston, MA, 1993. 1-21.

There are many interesting algebraic aspects; commutative and non-commutative

various Sections of Reutenauer, Christophe. "Free lie algebras." Handbook of algebra. Vol. 3. North-Holland, 2003. 887-903.

What can we do if $$X$$ is not differentiable?

In the case where $$X$$ is $$\alpha$$-Hölder continous, with $$\alpha \in (1/2,1]$$, the integration theory of Young applies.

Young, Laurence C. "An inequality of the Hölder type, connected with Stieltjes integration." Acta Mathematica 67.1 (1936): 251.

Section 4.3 - Friz, Peter K., and Martin Hairer. A course on rough paths: with an introduction to regularity structures. Springer, 2014.

In stochastic analysis, the most fundamental stochastic process is the Brownian motion (or Wiener process)

It turns out that its sample paths are only Hölder continous for $$\alpha < 1/2$$, so that Young’s theory is not applicable. Even more can be said: there is no linear space carrying Brownian motion that allows for a sensible integration theory

Section 1.5.1/1.5.2 - Lyons, Terry J. "Differential equations driven by rough signals." Revista Matemática Iberoamericana 14.2 (1998): 215-310.

There is nonetheless a stochastic way to define integration against a Brownian motion $$X$$: via Itôs stochastic calculus. Following a slightly different approach (Stratonovich integration) a larger class of Gaussian processes can also be integrated

Chapter 10 - Friz, Peter K., and Martin Hairer. A course on rough paths: with an introduction to regularity structures. Springer, 2014.

Not only does this procedure define these iterated integrals. We actually get a rough path lift of the Gaussian process. This rough path can now be used to solve differential equations (which in this setting corresponds to stochastic differential equations

various Sections - Friz, Peter K., and Martin Hairer. A course on rough paths: with an introduction to regularity structures. Springer, 2014.

Recently, iterated integrals have found application in the analysis of time series and more generally in machine learning. They are used as a feature extraction mechanism (think: "nonlinear Fourier transform")

Chevyrev, Ilya, and Andrey Kormilitzin. "A primer on the signature method in machine learning." arXiv preprint arXiv:1603.03788 (2016).

Diehl, Joscha, and Jeremy Reizenstein. "Invariants of multidimensional time series based on their iterated-integral signature." arXiv preprint arXiv:1801.06104 (2018).

Pfeffer, Max, Anna Seigal, and Bernd Sturmfels. "Learning Paths from Signature Tensors." arXiv preprint arXiv:1809.01588 (2018).

Kiraly, Franz J., and Harald Oberhauser. "Kernels for sequentially ordered data." arXiv preprint arXiv:1601.08169 (2016).