PhD project 1: Exchangeable Particle Filter for time-discretised Markov jump processes
Supervisors: Prof. Chris Sherlock and Dr. Lloyd Chapman
Inference in stochastic reaction-network models, such as the susceptible-exposed-infectious-recovered (SEIR) epidemic model, is crucial for understanding the dynamics of interacting systems in epidemiology, ecology, and systems biology. These models are typically represented as Markov jump processes (MJPs), with discrete, noisy observations and intractable likelihoods. Particle Markov chain Monte Carlo (PMCMC) offers a natural mechanism for inference on these hidden Markov models, but particle degeneracy, even in moderate dimensional state spaces, typically leads to poor mixing of the PMCMC algorithm.
This project focuses on improving the efficiency of particle Gibbs for inference on reaction networks by addressing the degeneracy problem. Building on recent work on the exchangeable particle Gibbs (xPG) sampler for continuous-state-space stochastic processes, this paper develops a novel version of xPG tailored to discrete-state-space reaction networks, where randomness is driven by Poisson processes rather than Brownian motion. We focus on the tau-leap discretisation of the MJPs. The proposed method retains the exchangeability framework of xPG while adapting it to the structural and computational challenges specific to reaction networks. The new method is applied to several examples, including four coupled SEIR models, where it substantially improves mixing efficiency compared with standard particle Gibbs. In this example, xPG improves sample efficiency by more than an order of magnitude relative to standard particle Gibbs, making inference feasible in hours rather than weeks. As part of our tuning strategy, we provide a mechanism for tuning the number of particles that is applicable more generally, to any particle Gibbs algorithm.
PhD project 2: Exchangeable Particle Filter for continuous-time Markov jump processes
Supervisors: Prof. Chris Sherlock and Dr. Lloyd Chapman
My second PhD project develops an exchangeable Particle Gibbs method for continuous-time Markov jump processes based on exact simulation. This project builds on my earlier work on tau-leap xPG, but avoids the tau-leap approximation by working directly with exactly simulated MJP paths.
The aim is to construct a sampler that retains the advantages of exchangeable Particle Gibbs while targeting continuous-time models more directly. I am currently developing the theoretical justification for the method, studying its key properties, and implementing the algorithm in Rcpp.
MSc project: Deep Pricing in the CEV model
Supervisor: Dr. John Armstrong
The application of machine learning in finance receives more and more attention. In this paper, we price American put options under the CEV model with finite difference method. Then we use a neural network to approximate the pricing map from model parameters to option prices, which realizes rapid computation of option price. We also use market data downloaded from Bloomberg to calibrate the CEV model. In addition, we estimate value at risk of a portfolio containing American put options and train another neural network mapping model parameters to value at risk.
MSc project: Predictable Forward Performance Process in Binomial Tree Model with Robo-Advising Application
Supervisor: Dr. Liang Gechun
we derive the discrete-time predictable m-forward performance processes in the case of logarithmic utility function and exponential utility function. Next, we compare the solutions for the single-period investment problem using two different approaches: the classical expectation maximization method and the method involving the forward performance process. We also discuss the Robo-advising application of predictable m-forward performance processes.