Investigating Properties of Non-Markov Stochastic Processes with Application to Modelling the Dynamics of Financial Markets

The very recent studies conducted by my PhD advisor, Dr Luca Giuggioli, have cleared a path for a systematic approach to develop models for the dynamics of non-Markov stochastic systems. The approach exploits the well-known connection between the description via an ordinary differential equation with additive noise (Langevin equation) and the associated partial differential equation for the associated probability distribution (Fokker-Planck equation). While this connection is well-known for Markov systems, it has been modified to deal with the case of a linear time non-local Langevin equation, e.g. the linear delayed Langevin equation. Previously, in my Master's thesis, I have shown that this procedure allows one to study first-passage properties of random delayed systems, as well as situations where absorbing or reflecting boundaries are imposed on the dynamics. In the case of a non-linear delayed Langevin equation, approximate methodologies for moment-hierarchy truncation employed in Markov cases can be borrowed and used with the Fokker-Planck equations associated with the non-Markov cases. The PhD thus presents the opportunity to build on the knowledge accumulated during my Master's thesis on first-passage properties and boundary effects and investigate a plethora of non-linear non-Markov systems. The modelling applications for non-Markov stochastic processes are boundless. While Markov models are widely used to represent stochastic processes across the sciences and engineering, non-Markov models bring about a greater accuracy for those systems in which history plays an important role. While there are plenty of opportunities in employing non-Markov models to study processes across the sciences, I am interested in modelling financial systems in general, and market dynamics in particular. Markov models have a long history in the study of financial systems and have also been relatively successful at making predictions. However, in an arena where the smallest of margins can be the difference between a profit and a loss, there is an emerging awareness that more accurate models are necessary. Non-Markov models will certainly leap the predictive abilities forward. As an example, the celebrated Black-Scholes formula for calculating options pricing has been shown to produce some awry results when compared to market data. It has been postulated that the cause of this is that the Black-Scholes formula assumes constant volatility, when in some cases it is time dependent. This formula is based on the underlying asset price dynamics undergoing geometric Brownian motion (a Markov process), where evidence now suggests there is some history dependence affecting future asset prices. This calls for the introduction of a non-Markov model to represent the underlying asset price dynamics, such as using variants of the delayed Langevin equation. In addition, first passage properties and bounded dynamics correspond to financial applications, e.g. the optimal time to sell an asset can be interpreted as the first-passage time to a certain price, and trading in the presence of some asset price caps can be modelled as the process in the presence of boundaries. To conclude, the mathematical theory required for modelling financial processes is certainly a direct application of the formalism that is being studied during this PhD. This project falls within the EPSRC Mathematical Sciences research area, specifically largely in Statistics and Applied Probability theory.