EPSRC Reference: 
EP/R008205/1 
Title: 
Computing and inverting the signatures of rough paths 
Principal Investigator: 
Boedihardjo, Dr H 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics and Statistics 
Organisation: 
University of Reading 
Scheme: 
First Grant  Revised 2009 
Starts: 
01 October 2017 
Ends: 
31 May 2019 
Value (£): 
100,853

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
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 awardwinning algorithm for Chinese handwriting recognition. Most of these applications use a tool, known as the signature, to analyze irregular paths. The signature is purposebuilt 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 signaturrelated to the other key features of paths?
As rough pathbased 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 realworld 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 realworld processes. For Problem 2, any progress will provide crucial insights into why signaturebased methods work and could lead to tangible improvements to the efficiency of, for instance, recognition methods that use the signature.

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