Stochastic Modelling of Big Data in Finance, Insurance and Energy Markets

大数据现在已经成为包括金融、保险和能源市场在内的多个领域的建模和分析的驱动力。该提案致力于对这些领域产生的大数据进行随机建模和分析。在金融学中，我们引入了不同的一般复合Hawkes过程(GCHP)和马尔可夫更新过程(MRP)来建模极限订单书(LOB)的动力学。为了处理大数据，我们在更长的时间尺度上考虑我们的动态，即秒或分钟，而不是毫秒，然后应用渐近方法研究日内价格波动与LOB订单流之间的联系，即大数定律(LLN)和函数中心极限定理(FCLT)。我们使用真实数据来证明和实施我们的结果。通过定量和比较分析，找出哪种模型最能描述LOB的真实动态。研究了描述多只股票中间价动态变化的多元广义复合霍克斯过程。MRP和GCHP模型都考虑了最优清算、收购和做市问题。在保险业，特别是在风险理论中，一个中心问题是如何对描述大量索赔发生的随机过程进行建模。研究了基于GCHP的索赔到达风险模型。结果表明，该模型适用于保险数据的实证建模。利用该模型的渐近方法，如LLN和FCLT，我们首先推导出净利润条件，然后分别给出一个纯扩散近似，它允许分析计算有限时间和无限时间破产概率。应用这个近似，我们还将研究保险公司在不完全市场中的最优投资策略。在能源市场中，我们也面临着处理大数据的问题，例如，大量的现货价格变化。为了避免气候变化的最坏后果，全球经济的能源链必须彻底脱碳，例如，通过引入碳税来减少温室气体排放。我们基于来自不同能源市场的大数据，研究了正确的碳定价方法。我们将碳价格定义为激励发电商从煤炭转向碳密集度较低的天然气，然后最终从天然气转向风能、太阳能、水电或其他清洁和可再生能源的必要税收。我们将考虑几种类型的随机模型，包括基于Levy的OU模型，并对哪种模型进行比较分析。我们使用GCHP来模拟能源市场现货价格的集聚效应和长记忆特性。一条从化石燃料能源转向清洁和可再生能源的道路绝对是可能的：一组美国工程师计算出，如果我们决定这样做，加拿大完全可以由可再生能源提供动力。在这项提议中，我们特别将展示如何使用我们的随机模型、分析和方法来实现这一点。

Big data has now become a driver of models building and analysis in a number of areas, including finance, insurance and energy markets. The proposal is devoted to stochastic modelling and analyzing of big data arising in these areas.  In finance, we introduce different general compound Hawkes processes (GCHP) and Markov renewal processes (MRP) to model the dynamics of limit order book (LOB). To deal with big data, we consider our dynamics on a longer time scale, seconds or minutes, instead of milliseconds, and then applying the asymptotic methods to study the link between intraday price volatilities and order flows in LOB, i.e., law of large numbers (LLN) and functional central limit theorems (FCLT). We use real data to justify and implement our results. Quantitative and comparative analyses are performed to find out which model is the best in describing the real dynamics of LOB. Multivariate general compound Hawkes process describing the dynamics of the mid-price of many stocks is studied as well. Optimal liquidation, acquisition and market making problems consider for both MRP and GCHP models. In insurance, in particular in risk theory, a central question is how to model the random process describing a big number of claim occurrences. We study a risk model with claim arrivals based on GCHP. We show that it is suitable to model empirical insurance data. Using asymptotic methods, such as LLN and FCLT for this model, we derive net profit condition first, and then present a pure diffusion approximation, respectively, which allow analytical calculation of finite-time and infinite-time ruin probabilities. Applying this approximation, we will also study an optimal investment strategies for an insurer in an incomplete market. In energy markets, we also have a problem of dealing with big data, e.g., a big number of spot price changes. To avoid the worst consequences of climate change, the energy chain of the global economy must be drastically decarbonized, e.g., by introducing a carbon tax to reduce greenhouse gas emissions. We study the correct approach to carbon pricing based on big data from different energy markets. We define the carbon price as the necessary tax to incite electricity producers to switch from coal to natural gas, which is less carbon intensive, and then ultimately switching from natural gas to wind, solar, hydro, or other clean and renewable energy. We will consider several types of stochastic models, including Levy-based OU models, and give comparative analyses which model is the best. We use GCHP to model clustering effects and long memory properties of spot prices in energy markets. A path out of fossil fuel energy into the clean and renewable energy is definitely possible: a group of US engineering has calculated that Canada could be completely powered by renewable energy if we just decide to do it. In this proposal, in particular, we will show how it can be done using our stochastic models, analyses and methodologies.