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Sequential Modelling and Inference of High-frequency Limit Order Book with State-space Models and Monte Carlo Algorithms
The high-frequency limit order book (LOB) market has recently attracted increasing research attention from both the industry and the academia as a result of expanding algorithmic trading. However, the massive data throughput and the inherent complexity of high-frequency market dynamics also present challenges to some classic statistical modelling approaches. By adopting powerful state-space models from the field of signal processing as well as a number of Bayesian inference algorithms such as particle filtering, Markov chain Monte Carlo and variational inference algorithms, this thesis presents my extensive research into the high-frequency limit order book covering a wide scope of topics.
Chapter 2 presents a novel construction of the non-homogeneous Poisson process to allow online intensity inference of limit order transactions arriving at a central exchange as point data. Chapter 3 extends a baseline jump diffusion model for market fair-price process to include three additional model features taken from real-world market intuitions. In Chapter 4, another price model is developed to account for both long-term and short-term diffusion behaviours of the price process. This is achieved by incorporating multiple jump-diffusion processes each exhibiting a unique characteristic. Chapter 5 observes the multi-regime nature of price diffusion processes as well as the non-Markovian switching behaviour between regimes. As such, a novel model is proposed which combines the continuous-time state-space model, the hidden semi-Markov switching model and the non-parametric Dirichlet process model. Additionally, building upon the general structure of the particle Markov chain Monte Carlo algorithm, I further propose an algorithm which achieves sequential state inference, regime identification and regime parameters learning requiring minimal prior assumptions. Chapter 6 focuses on the development of efficient parameter-learning algorithms for state-space models and presents three algorithms each demonstrating promising results in comparison to some well-established methods.
The models and algorithms proposed in this thesis not only are practical tools for analysing high-frequency LOB markets, but can also be applied in various areas and disciplines beyond finance
Crossing-Free Acyclic Hamiltonian Path Completion for Planar st-Digraphs
In this paper we study the problem of existence of a crossing-free acyclic
hamiltonian path completion (for short, HP-completion) set for embedded upward
planar digraphs. In the context of book embeddings, this question becomes:
given an embedded upward planar digraph , determine whether there exists an
upward 2-page book embedding of preserving the given planar embedding.
Given an embedded -digraph which has a crossing-free HP-completion
set, we show that there always exists a crossing-free HP-completion set with at
most two edges per face of . For an embedded -free upward planar digraph
, we show that there always exists a crossing-free acyclic HP-completion set
for which, moreover, can be computed in linear time. For a width-
embedded planar -digraph , we show that we can be efficiently test
whether admits a crossing-free acyclic HP-completion set.Comment: Accepted to ISAAC200
Efficient Monte Carlo methods for continuum radiative transfer
We discuss the efficiency of Monte Carlo methods in solving continuum
radiative transfer problems. The sampling of the radiation field and
convergence of dust temperature calculations in the case of optically thick
clouds are both studied. For spherically symmetric clouds we find that the
computational cost of Monte Carlo simulations can be reduced, in some cases by
orders of magnitude, with simple importance weighting schemes. This is
particularly true for models consisting of cells of different sizes for which
the run times would otherwise be determined by the size of the smallest cell.
We present a new idea of extending importance weighting to scattered photons.
This is found to be useful in calculations of scattered flux and could be
important for three-dimensional models when observed intensity is needed only
for one general direction of observations. Convergence of dust temperature
calculations is studied for models with optical depths 10-10000. We examine
acceleration methods where radiative interactions inside a cell or between
neighbouring cells are treated explicitly. In optically thick clouds with
strong self-coupling between dust temperatures the run times can be reduced by
more than one order of magnitude. The use of a reference field was also
examined. This eliminates the need for repeating simulation of constant sources
(e.g., background radiation) after the first iteration and significantly
reduces sampling errors. The applicability of the methods for three-dimensional
models is discussed.Comment: submitted to A&A, 19 page
Comparative study of semiclassical approaches to quantum dynamics
Quantum states can be described equivalently by density matrices, Wigner
functions or quantum tomograms. We analyze the accuracy and performance of
three related semiclassical approaches to quantum dynamics, in particular with
respect to their numerical implementation. As test cases, we consider the time
evolution of Gaussian wave packets in different one-dimensional geometries,
whereby tunneling, resonance and anharmonicity effects are taken into account.
The results and methods are benchmarked against an exact quantum mechanical
treatment of the system, which is based on a highly efficient Chebyshev
expansion technique of the time evolution operator.Comment: 32 pages, 8 figures, corrected typos and added references; version as
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Unsteady two dimensional airloads acting on oscillating thin airfoils in subsonic ventilated wind tunnels
The numerical calculation of unsteady two dimensional airloads which act upon thin airfoils in subsonic ventilated wind tunnels was studied. Neglecting certain quadrature errors, Bland's collocation method is rigorously proved to converge to the mathematically exact solution of Bland's integral equation, and a three way equivalence was established between collocation, Galerkin's method and least squares whenever the collocation points are chosen to be the nodes of the quadrature rule used for Galerkin's method. A computer program displayed convergence with respect to the number of pressure basis functions employed, and agreement with known special cases was demonstrated. Results are obtained for the combined effects of wind tunnel wall ventilation and wind tunnel depth to airfoil chord ratio, and for acoustic resonance between the airfoil and wind tunnel walls. A boundary condition is proposed for permeable walls through which mass flow rate is proportional to pressure jump
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