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.

我的博士顾问Luca Giuggioli博士进行的最新研究已清除了一种系统的方法，以开发非马科夫随机系统动力学的模型。该方法通过具有添加噪声（Langevin方程）的普通微分方程和相关的偏微分方程（Fokker-Planck方程）利用了描述之间的众所周知的连接。尽管该连接是Markov系统众所周知的，但已对其进行修改以处理线性时间非本地langevin方程的情况，例如线性延迟的langevin方程。以前，在我的硕士论文中，我已经表明，此过程允许一个人研究随机延迟系统的第一通道特性，以及在动力学上施加吸收或反射边界的情况。在非线性延迟的langevin方程中，可以借用Markov案例中使用的矩层截断的近似方法，并与与非马科夫案件相关的Fokker-Planck方程一起使用。因此，博士学位提供了建立在我的硕士论文中对第一学期特性和边界效应的知识的机会，并研究了许多非线性非线性非马尔科夫系统。非马科夫随机过程的建模应用是无限的。尽管马尔可夫模型被广泛用于代表科学和工程学的随机过程，但非马科夫模型为历史上起着重要作用的系统带来了更高的准确性。尽管采用非马科夫模型来研究整个科学的过程，但我对一般的金融系统，尤其是市场动态感兴趣。马尔可夫模型在金融系统的研究中具有悠久的历史，并且在做出预测方面也相对成功。但是，在一个最小的利润可能是利润和损失之间的区别的舞台上，人们意识到需要更准确的模型。非马尔科夫模型肯定会跨越预测能力。例如，与市场数据相比，已显示出著名的黑色chcholes公式用于计算选项定价。据推测，其原因是黑 - choles公式假设恒定波动率，在某些情况下是依赖时间的。该公式基于正在进行几何布朗尼运动（马尔可夫流程）的基本资产价格动态，现在有证据表明存在一些历史依赖，影响未来的资产价格。这要求引入非马尔科夫模型来表示基本的资产价格动态，例如使用延迟的langevin方程的变体。此外，第一通道属性和有限的动态对应于财务应用，例如销售资产的最佳时间可以解释为一定价格的第一步时间，在存在一些资产价格上限的情况下进行交易可以在界限存在下建模为该过程。总而言之，建模财务过程所需的数学理论无疑是该博士学位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，特别是在统计和应用概率理论中。