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

研究非马尔可夫随机过程的性质及其在金融市场动态建模中的应用

• 批准号: 2328227
• 金额: --
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2328227
• 关键词: Investigating; Properties; Non; Markov; Stochastic

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.

我的博士导师Luca Giuggioli博士最近进行的研究为系统方法开发非马尔可夫随机系统的动态模型扫清了道路。该方法利用了众所周知的连接之间的描述通过一个常微分方程加性噪声（朗之万方程）和相关的偏微分方程的相关概率分布（福克-普朗克方程）。虽然这种连接是众所周知的马尔可夫系统，它已被修改，以处理的情况下，线性时间非局部朗之万方程，例如线性延迟朗之万方程。以前，在我的硕士论文中，我已经表明，这个过程允许一个研究随机延迟系统的首次通过性能，以及吸收或反射边界施加在动态的情况。在非线性延迟朗之万方程的情况下，近似的方法，用于在马尔可夫情况下的时刻层次截断可以借用和使用与非马尔可夫情况下的福克-普朗克方程。因此，博士学位提供了一个机会，可以利用我在硕士论文中积累的关于首次通过性质和边界效应的知识，并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。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扮演重要角色的系统带来了更高的准确性。虽然有很多机会采用非马尔可夫模型来研究跨学科的过程，但我对一般的金融系统建模感兴趣，特别是市场动态。马尔可夫模型在研究金融系统方面有着悠久的历史，并且在预测方面也相对成功。然而，在一个竞技场中，最小的利润率可能是利润和亏损之间的差异，人们逐渐意识到需要更准确的模型。非马尔可夫模型肯定会使预测能力向前飞跃。作为一个例子，著名的布莱克-斯科尔斯公式计算期权定价已被证明产生一些错误的结果相比，市场数据。据推测，其原因是布莱克-斯科尔斯公式假设波动率恒定，而在某些情况下波动率是时间依赖的。该公式基于经历几何布朗运动（马尔可夫过程）的基础资产价格动态，现在有证据表明存在影响未来资产价格的历史依赖性。这就需要引入一个非马尔可夫模型来表示基础资产价格动态，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被解释为达到某个价格的首次通过时间，并且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，金融过程建模所需的数学理论无疑是本博士研究的形式主义的直接应用。该项目属于EPSRC数学科学研究领域的福尔斯，特别是在统计和应用概率论。