Optimal Trading in Financial Markets: a multi-pronged mathematical approach

学生将研究最优交易算法在金融市场中的作用。他们将需要确定新的和新兴的领域，例如本地化能源市场或加密货币，这些模型适用（并且确实需要）。我们主要感兴趣的是市场参与者之间的相互作用，以及个人策略如何相互影响或更广泛的市场。此外，该模型的结果可用于对上述市场形成有效的未来监管。学生需要发展和确定市场的主要参与者，并能够描述他们扮演的不同角色。对于每个参与者，学生将需要确定影响市场行为的关键变量（然后是参数），并酌情建立随机和/或确定性模型。此外，学生需要将参与者的行为框架化为最优交易问题的解决方案。给定输入变量和模型，问题可能最终导致偏微分方程（PDEs），我们希望学生必须同时使用解析和数值解技术。由于做出的交易决定发生在很短的时间尺度上，而投资或监管市场的决定发生在很长的时间尺度上，学生需要能够在短时间尺度上解决这些问题，并通过各种场景来检查它们在长期内的影响，也就是说，这些可能是非常多尺度的过程。因此，可能需要某些类型的模拟技术来探索模型，至少在第一个实例中，以指导确定性模型的发展。为了使他们能够建立适当的模型并解决由此产生的问题，学生将需要使用各种技术。例如，模拟方法将首先在一些非标准随机微分方程上进行测试，以展示它们如何建立未来事件的概率情景。我们相信，这将指导一种新的有限差分方法的发展，这种方法可以根据方程进行定制，使它们比标准模拟技术效率高出许多倍。学生将被要求解决由此产生的复杂和新颖的偏微分方程，这可能会带来新的和有趣的挑战。很可能需要调整或扩展现有的技术，以便及时地为这些复杂的问题提供准确的结果，以便能够产生各种各样的未来情景

The student will be investigating the role of Optimal Trading Algorithms in financial markets. They will need to identify new and emerging areas such as those in localised energy markets or crypto- currencies, where such models are applicable (and indeed required). Primarily we will be interested in the interactions between market participants and how individual strategies affect each other or the wider market. Furthermore, the results from the model could be used to form effective future regulation of said markets. The student will need to develop and identify the major participants in the market and be able to describe the different roles they play. For each of the participants, the student will need to determine the key variables (and then parameters) affecting the market behaviour and build stochastic and/or deterministic models as appropriate. Further, the student will need to frame the behaviour of participants as the solution to optimal trading problems. Given the input variables and models, the problems will likely ultimately lead to Partial Differential Equations (PDEs), for which we expect the student must employ both analytical and numerical solution techniques. Since the decisions made to trade happen over very short timescales, whereas decisions to invest or regulate a market happen over much longer timescales, the student will need to be able to solve these problems over both short timescales and also examine their effect over the long term through a variety of scenarios, i.e. these are likely to be very much multi-scaled processes. Therefore some types of simulation techniques are likely to be required to explore the models, at least in the first instance, in order to guide the development of deterministic models. In order that they will be able to build appropriate models and solve the resulting problems, the student will need to employ a variety of techniques. For example, simulation methods will first be tested on some classes of non-standard Stochastic Differential Equations to show how they can build up probabilistic scenarios of future events. We believe this should then guide the development of a novel finite-difference approach, that can be tailored to the equations to make them many times more efficient than standard simulation techniques. The student will be required to solve the resulting complex and novel PDEs, that could present new and interesting challenges. It is likely that existing techniques will need to be adapted or extended to provide accurate results for these complex problems in a timely manner, so that a full range of future scenarios can be generated