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

我的博士导师Luca Giuggioli博士最近进行的研究，为系统地开发非马尔可夫随机系统动力学模型扫清了道路。该方法利用了众所周知的通过带有加性噪声的常微分方程（Langevin方程）和相关概率分布的相关偏微分方程（Fokker-Planck方程）的描述之间的联系。虽然这种联系在马尔可夫系统中是众所周知的，但它已经被修改为处理线性时间非局部朗之万方程的情况，例如线性延迟朗之万方程。之前，在我的硕士论文中，我已经证明了这个过程允许人们研究随机延迟系统的第一通道性质，以及在动力学上施加吸收或反射边界的情况。在非线性延迟朗格万方程的情况下，可以借鉴马尔可夫情况下矩层截断的近似方法，并将其用于与非马尔可夫情况相关的福克-普朗克方程。因此，博士学位为我提供了一个机会，可以在我的硕士论文中积累的关于首通道特性和边界效应的知识基础上，研究大量的非线性非马尔可夫系统。非马尔可夫随机过程的建模应用是无限的。当马尔可夫模型被广泛用于表示科学和工程领域的随机过程时，非马尔可夫模型为那些历史起重要作用的系统带来了更高的准确性。虽然利用非马尔可夫模型研究各科学领域的过程有很多机会，但我感兴趣的是一般金融系统的建模，尤其是市场动态。马尔可夫模型在研究金融系统方面有着悠久的历史，在预测方面也相对成功。然而，在一个哪怕是最小的利润也可能决定盈亏的领域，人们逐渐意识到，有必要建立更准确的模型。非马尔可夫模型肯定会使预测能力向前飞跃。例如，与市场数据相比，计算期权定价的著名的Black-Scholes公式已被证明会产生一些错误的结果。据推测，造成这种情况的原因是布莱克-斯科尔斯公式假设恒定波动，而在某些情况下，它与时间有关。该公式基于经历几何布朗运动（马尔科夫过程）的潜在资产价格动态，现在有证据表明，存在一些影响未来资产价格的历史依赖。这需要引入非马尔可夫模型来表示潜在资产价格动态，例如使用延迟朗格万方程的变体。此外，第一通道属性和有界动态对应于金融应用，例如，出售资产的最佳时间可以解释为到达某个价格的第一通道时间，并且存在某些资产价格上限的交易可以建模为存在边界的过程。综上所述，建模金融过程所需的数学理论当然是在博士期间正在研究的形式主义的直接应用。该项目属于EPSRC数学科学研究领域，主要研究统计和应用概率论。

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.