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A statement of the research problem can be expressed, simply as, can bi-directional grid trading (BGT) be distilled into a mathematical model that can then be used to demonstrate that the corresponding grid trading problem (GTP) leads to ruin at a much slower rate than the well known gambler's ruin problem (GRP)? The methods and procedures utilized in this thesis examine various probability theory and stochastic process concepts and techniques, and apply them to mathematical finance, in particular to model the BGT strategies of algorithmic trading. My original contribution to knowledge is to take Ito diffusions and constrain them via various geometric schemes so as to keep them bounded within two (bi-directional) hidden barriers per distance dimension. I have called this novel research 'bi-directional grid constrained' (BGC) stochastic processes (BGCSPs). The BGCSP research is related to and compliments the Langevin, Ornstein-Ulhenbeck and multi-skew Brownian motion (M-SBM) stochastic differential equations (SDEs). A summary of findings includes the publishing of three research papers in applied probability theory for BGCSPs. The research also contributes to knowledge by implementing the theory into algorithms, which allow one to undertake extensive simulations which help visualise the resulting geometry. Armed with the insights of this theory, the GTP has been stated and modelled mathematically, allowing one to better simulate the stochastic dynamics of grid trading and its parallels with dynamic portfolio management and optimization. This has resulted in additional four published research papers in applied mathematical finance. The thesis concludes by examining a number of related areas that provide additional avenues for further research in this field. |
