Stochastic Modelling of Big Data in Finance

Foreword

    Preface

    Symbols

    Acknowledgements

    1 A Brief Introduction: Stochastic Modelling of Big Data in Finance

        1.1 Introduction

        1.2 Big Data in Finance: Limit Order Books

            1.2.1 Description of Limit Order Books Mechanism

            1.2.2 Big Data in Finance: Lobster Data

            1.2.3 More Big Data in Finance: Xetra and Frankfurt Markets (Deutsche Boerse Group), on September 23, 2013 and CISCO Data on November 3, 2014

        1.3 Stochastic Modelling of Big Data in Finance: Limit Order Books (LOB)

            1.3.1 Semi-Markov Modelling of LOB

            1.3.2 General Semi-Markov Modelling of LOB

            1.3.3 Modelling of LOB with a Compound Hawkes Processes

            1.3.4 Modelling of LOB with a General Compound Hawkes Processes

            1.3.5 Modelling of LOB with a Non-linear General Compound Hawkes Processes

            1.3.6 Modelling of LOB with a Multivariable General Compound Hawkes Processes

        1.4 Illustration and Justification of Our Method to Study Big Data in Finance

            1.4.1 Numerical Results: Lobster Data (Apple, Google and Microsoft Stocks)

            1.4.2 Numerical Results: Xetra and Frankfurt Markets stocks (Deutsche Boerse Group), on September 23, 2013

            1.4.3 Numerical Results: CISCO Data, November 3, 2014

        1.5 Methodological Aspects of Using the Models

        1.6 Conclusion

        Bibliography

    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 Modelling of Limit Order Markets

            2.2.1 Markov Renewal and Semi-Markov Processes

            2.2.2 Semi-Markovian Modelling of Limit Order Books

        2.3 Main Probabilistic Results

            2.3.1 Duration until the next price change

            2.3.2 Probability of Price Increase

            2.3.3 The stock price seen as a functional of a Markov renewal process

        2.4 Diffusion Limit of the Price Process

            2.4.1 Balanced Order Flow case: Pa(1,1)=Pa(−1,−1) and Pb(1,1)=Pb(−1,−1)

            2.4.2 Other cases: either Pa(1,1)