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

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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.