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科夫随机系统的动力学模型扫清了道路。该方法利用了通过带有加性噪声的常微分方程组(朗之万方程)和相关概率分布的相关偏微分方程式(福克-普朗克方程)来描述的众所周知的联系。虽然这种联系对于马尔可夫系统是众所周知的，但它已经被修改以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。以前，在我的硕士论文中，我已经证明，这个过程允许一个人研究随机时滞系统的首次通过性质，以及在动力学上施加吸收或反射边界的情况。对于非线性时滞的朗之万方程，可以借用马尔可夫情形下的阶矩截断的近似方法，并将其用于非马尔可夫情形下的福克-普朗克方程。因此，博士学位提供了一个机会来建立在我的硕士论文期间积累的关于首次通过性质和边界效应的知识，并研究过多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。虽然马尔可夫模型被广泛用于表示科学和工程中的随机过程，但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。尽管利用非马尔可夫模型研究跨学科的过程有很多机会，但我对总体上的金融系统建模感兴趣，尤其是市场动态。马尔可夫模型在研究金融系统方面有很长的历史，在预测方面也相对成功。然而，在一个利润率可能是盈亏之差的领域，人们逐渐意识到，更准确的模型是必要的。非马尔可夫模型肯定会使预测能力向前飞跃。例如，著名的布莱克-斯科尔斯期权定价计算公式在与市场数据相比时，已被证明产生了一些错误的结果。有人推测，这是因为布莱克-斯科尔斯公式假定波动性不变，而在某些情况下，它是时间相关的。这个公式是基于潜在的资产价格动态经历几何布朗运动(马尔可夫过程)，其中证据现在表明，有一些历史相关性影响未来的资产价格。这就需要引入一个非马尔可夫模型来表示潜在的资产价格动态，例如使用延迟朗之万方程的变体。此外，首次通过性质和有界动态对应于金融应用，例如，出售资产的最优时间可以被解释为到某一特定价格的首次通过时间，而在存在某些资产价格上限的情况下的交易可以被建模为存在边界的过程。总而言之，为金融过程建模所需的数学理论肯定是本博士期间正在研究的形式主义的直接应用。这个项目属于EPSRC数学科学研究领域，特别是统计学和应用概率论。

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.