This book provides a rigorous overview and exploration of stochastic modelling of big data in finance (BDF). The book describes various stochastic models, including multivariate models, to deal with big data in finance. This includes data in high-frequency and algorithmic trading, specifically in limit order books (LOB).

1. A Brief Introduction: Stochastic Modelling of Big Data in Finance. 1.1. Introduction. 1.2. Big Data in Finance: Limit Order Books. 1.3. Stochastic Modelling of Big Data in Finance: Limit Order Books (LOB). 1.4 Illustration and Justification of Our Method to Study Big Data in Finance. 1.5. Methodological Aspects of Using the Models. 1.6. Conclusion. I. Semi-Markovian Modelling of Big Data in Finance. 2. A Semi-Markovian Modelling of Big Data in Finance. 2.1. Introduction. 2.2. A Semi-Markovian Modeling of Limit Order Markets. 2.3. Main Probabilistic Results. 2.4. Diffusion Limit of the Price Process. 2.5. Numerical Results. 2.6. More Big Data. 2.7. Conclusion. 3. General Semi-Markovian Modelling of Big Data in Finance. 3.1. Introduction. 3.2. Reviewing the Assumptions with Our New Data Sets. 3.3. General Semi-Markov Model for the Limit Order Book with Two States. 3.4. General Semi-Markov Model for the Limit Order Book with arbitrary number of states. 3.5. Discussion on Price Spreads. 3.6. Conclusion. II. Modelling of Big Data in Finance with Hawkes Processes. 4. A Brief Introduction to Hawkes Processes. 4.1. Introduction. 4.2. Definition of Hawkes Processes (HPs). 4.3. Compound Hawkes Processes. 4.4. Limit Theorems for Hawkes Processes: LLN and FCLT. 4.5. Limit Theorems for Poisson Processes: LLN and FCLT. 4.6. Stylized Properties of Hawkes Process. 4.7. Conclusion. 5. Stochastic Modelling of Big Data in Finance with CHP. 5.1. Introduction. 5.2. Definitions of HP, CHP and RSCHP. 5.3. Diffusion Limits and LLNs for CHP and RSCHP in Limit Order Books. 5.4. Numerical Examples and Parameters Estimations. 5.5. Conclusion. 6. Stochastic Modelling of Big Data in Finance with GCHP. 6.1. A Brief Introduction and Literature Review. 6.2. Diffusion Limits and LLNs. 6.3. Empirical Results. 6.4. Conclusion. 7. Quantitative and Comparative Analyses of Big Data with GCHP. 7.1. Introduction. 7.2. Theoretical Analysis. 7.3. Application. 7.4. Hawkes Process and Models Calibrations. 7.5. Error Measurement. 7.6. Conclusion. III. Multivariate Modelling of Big Data in Finance. 8. Multivariate General Compound Hawkes Processes in BDF. 8.1. Introduction. 8.2. Hawkes Processes and Limit Theorems. 8.3. Multivariate General Compound Hawkes Processes (MGCHP) and Limit Theorems. 8.4. FCLT II for MGCHP: Deterministic Centralization. 8.5. Numerical Example. 8.6. Conclusion. 9. Multivariate General Compound Point Processes in BDF. 9.1. Introduction. 9.2. Definition of Multivariate General Compound Point Process (MGCPP). 9.3. LLNs and Diffusion Limits for MGCPP. 9.4. Diffusion Limit for the MGCPP: Deterministic Centralization. 9.5. Conclusion. IV. Appendix: Basics in Stochastic Processes