Scientific Research

Probability and Stochastic Analysis, Rough Paths Theory, Data Science

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty.

Stochastic analysis is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes.

Probability theory and Stochastic analysis have been widely applied in everyday life in modelling and risk assessment. Many consumer products, such as automobiles and consumer electronics, use reliability theory in product design to reduce the probability of failure. Governments apply probabilistic methods in environmental regulation, entitlement analysis, and financial regulation. Stochastic analysis also has profound impacts on financial industry, where they use stochastic modelling to determine pricing and make trading decisions.

As a branch of Probability and Stochastic analysis, the theory of rough paths aims to create the appropriate mathematical framework for expressing the relationships between evolving systems, by extending classical calculus to the natural models for noisy evolving systems, which are often far from differentiable.

By considering an effective lifting of the original driving process, the theory of rough paths defines and solves differential equations driven by a wide class of stochastic processes, which are not restricted to Brownian motion and semi-martingales. The theory of rough paths has wide applications in financial mathematics and data science, e.g. in stochastic volatility models and analysing financial data streams.

In recent years, with the advent of the era of deep learning, machine learning has seen dramatic breakthroughs in natural language processing, image recognition and autonomous driving etc. The theory of rough paths, especially the Signature transform of data streams, has found far-reaching applications in modelling the effects of time series data. It is possible to treat Signature as a feature transform, and it is also possible to incorporate Signature as a layer in deep neural networks, with backpropagation and GPU accelerations. Combined with deep learning, the Signature transform has achieved start-of-art performance in character recognition, medical data analysis and image recognition etc. This is a fast-evolving research field with great potential.