Methodologies used to iinfer the dynamics of high dimensional time-series data.

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This project falls within the EPSRC Statistics and Applied Probability Research Area. My research focuses on methodologies used to infer the dynamics of high dimensional time-series data. Alongside Nikolaos Kantas and George Deligiannidis, we aim to perform model calibration on state space models with non-Markovian, potentially long-memory latent states. Current state-of-the-art approaches involve particle MCMC, where information on filtering density is stored in a finite set of particles. However, these methods suffer from the curse of dimensionality, path degeneracy, and scale poorly with the size of the dataset. To overcome these issues, I am investigating the application of diffusion models in time-series, initially with a look towards time-series synthesis of Fractional Brownian Motion. Our future aims include to investigate how well diffusion models can infer the drift and diffusion coefficients of SDE-driven time-series, and, ultimately, how they can be applied in particle filtering. On the other hand, alongside Mihai Cucuringu and Guy Nason, I am focusing on modelling the covariance structure of high dimensional time-series, with a specific focus on financial time-series and portfolio construction. Previous literature on covariance forecasting often focuses on M-GARCH models, which suffer from the curse of dimensionality. In recent years, a growing body of literature has applied network science methods for both modelling and forecasting. Building on previous research by both supervisors, I am investigating how to incorporate autoregressive and network effects in models for correlation and covariance structures, with flexibility in choosing the maximum autoregressive lag and the influence of the network at each lag. In the long-term, I aim to use these models to build better measures of cross-asset risk, and use graph clustering techniques to build portfolios that deliver better returns.