Computing and inverting the signatures of rough paths

A path models the evolution of a variable in a certain state space. The state space could represent physical quantities, such as the position of a gas particle, or data such as future sea levels. A common feature in these examples is that they are random processes.

Since at each time a random path could move in any direction, its trajectory would be erratic and not smooth in general. Remarkable theories of calculus have been developed to describe how these oscillatory paths affect each other. A first major success was Itô's theory which applies to systems driven by Brownian motion, a canonical mathematical model for random particle motion. Another breakthrough occurred in the late 1990s with the advent of rough path theory. Unlike Itô's construction, rough path theory is able to handle paths that move in much more irregular directions than Brownian motion. It has also led to breakthroughs on the modelling of surface growth, an achievement recognized by the award of the Fields Medal to Martin Hairer in 2014.

Meanwhile, many successful applications of rough path theory have been established, ranging from new numerical and statistical methods to an international award-winning algorithm for Chinese handwriting recognition. Most of these applications use a tool, known as the signature, to analyze irregular paths. The signature is purpose-built to describe paths that move so randomly in for example, a square, that they can fill the entire square. The first term of the signature captures the one dimensional aspects of the path, such as the displacement. The second term represents two dimensional aspects such as the area, and so on. Successive terms in the signature will tell us higher and higher dimensional information about the path. The signature has a complex structure and this means that many fundamental problems have remained unresolved. For example:

Problem 1: How do we calculate the average values of signatures of random paths?
Problem 2: How is the signature related to the other key features of paths?

As rough path-based methods demonstrate their initial promise, these problems have emerged as the main challenges hindering further development. This state of affairs is the main motivation for our current proposal.

Instead of studying the signature directly, we will first examine the properties of functions on signatures. Crucially, most recent advances on signatures have used the qualitative properties of these functions. Their quantitative aspects have remained underused, possibly due to their complex structure. We will develop new methods for understanding these structures, making novel use of important tools from other areas of mathematics, including Lie algebra, hyperbolic geometry and stochastic analysis.

The study of Problems 1 and 2 is expected to reveal the deep relationship between the signature and other important ideas in mathematics, such as the notion of length. This is a worthwhile pursuit because many mathematical breakthroughs were born out of linking two hitherto unrelated ideas, with the proof of Fermat's Last Theorem being a famous example. A key element of this project is to disseminate our new results in rough path theory beyond our usual audience in probability theory, as the biggest gains will come from reaching those who have not been aware of rough path theory and its potential relevance to their work.

There will also be impact beyond academia. Scientists have observed that many real-world random processes, such as river flow and stock prices, have rough path behaviour. If we can resolve Problem 1, it will extend the existing applications of signatures to these real-world processes. For Problem 2, any progress will provide crucial insights into why signature-based methods work and could lead to tangible improvements to the efficiency of, for instance, recognition methods that use the signature.

The first grant will help the PI establish an independent program to tackle significant open problems in rough path theory. While the immediate impact of the project will be academic - providing a new prospective for studying the classical notions of length and quadratic variation as well as stochastic differential equations (SDEs), wider impact is expected to follow in the medium term. This research will provide new methods for numerically and statistically analyzing anomalous SDEs, which are expected to have wide-ranging applications across a number of different sectors. As a new investigator, the PI will focus on profile raising activities as well as identifying opportunities for engagement with potential end users from a number of different sectors, building on preliminary discussions with stakeholders in the following areas:

1. Modelling of financial risks

Stochastic differential equations (SDEs) have frequently been used to quantify risks and uncertainty, for example in the pricing of financial derivatives and insurance premiums. The methods arising from the outputs of this project will improve the numerical and statistical analysis of anomalous SDEs, which in turn will lead to more robust pricing of financial derivatives and insurance premiums. These improvements would ultimately help businesses to better manage their risks, as well as to help investors assess their portfolios. There are also implications for the general public. For instance, if insurance companies were better able to understand and model risks, this would allow for them to better design and provide insurance policies tailored to the need of car drivers.

2. Model future climate

Stochastic models have increasingly been used to model and forecast climate. Typically such a model will have a deterministic long-term trend, with an additive random noise that describes the day-to-day or seasonal fluctuations. Although these models are usually simplified descriptions of the reality, they can be a highly informative way to understand certain qualitative phenomena. The results of this project can in the long term improve the stability of numerical approximation schemes for stochastic models. These improvements could be crucial for risk management for example, from risks to the agriculture industry caused by flooding, to risks to the general public arising from weather uncertainty.

3. Monitoring of anaesthesia

Clinically, anaesthesia monitors are often used to track the progression of anaesthesia during a surgeries or dental operations. To design the monitors, neuro-scientists have to model ion-channels that open and close stochastically. Although the states of these ion-channels are discrete, SDEs can provide good approximations. The methods arising from results in this project are expected to improve the statistical analysis of these neuron-dynamics, which will in turn improve the design of anaesthesia monitors.

4. Chinese handwriting recognition.

The HCII-SCUT laboratory has developed a signature-based smartphone application gPen. gPen translates the user's onscreen handwriting to digital Chinese characters. Without active marketing, the software has now been downloaded over one million times from Android's software store alone. Given the high costs of computing signatures, one important problem to address in this area is how many terms of the signature would be enough to facilitate recognition. This is currently being addressed on an ad-hoc basis. The outputs from the present project will improve understanding for why the empirical solutions work and therefore provide a theoretical basis for analyzing decisions made within the algorithms. This would ultimately lead to a faster and more accurate experience for the users of gPen or any other future recognition software that uses signature as features.