Modelling the stock market using a multi-scale approach

Mathematical modelling is one of the fundamental elements in the modern financial industry, playing vital roles in terms of decision making, risk management, financial innovation, and government regulation. The financial market has attracted extensive research interest from academics due to its strategic importance to the global economy. With the increased understanding of the market, the financial industry has developed towards a direction of a structured and refined system, thanks to a large number of financial instruments carefully engineered to meet the demand from the investors. On the other hand, fundamental research such as risk modelling remains challenging. The advancement of financial models does not prevent the market from financial crisis or mitigate the consequence of crash. Bearing in mind that modelling is an abstract of reality, the current research takes a step back to examine one of the corner stones of financial modelling: the efficient market hypothesis, the Gaussian statistics and the Brownian motion, as well as the process of data analyses for the modelling inputs. Using the context of the stock market, a systematic approach is adopted based on a variety of data from different stock markets during different periods. Some interesting statistical findings are presented in a quantitative manner, providing both the confirmation of the non-Gaussian statistics and empirical understandings to the market movements. Meanwhile, two different research methodologies are adopted to model the empirical findings: 1) macroscopic and phenomenological modelling based on analysing statistical data, and 2) microscopic and mechanism-based modelling based on understanding the behaviours of the market players. By taking advantage of both modelling methodologies, a multi-scale modelling approach is proposed in the current research. A step by step method is used to pin down the essential mechanisms that lead to the market inefficiency and non-Gaussian statistics. It is shown that the proposed approach requires a small number of input parameters by maximising the information obtained from market performance data and market microstructure. It is also shown that the multi-scale modelling approach, facilitated by a systematic empirical study, will greatly enhance both our understandings on the micro-foundations of the stock market and the applicability of the classical models widely used by the modern financial industry

Interactions Between Fine Particles.

Computer simulation using the Discrete Element Method (DEM) has emerged as a powerful tool in studying the behaviour of particulate systems during powder flow and compaction. Contact law between particles is the most important input to the Discrete Element simulation. However, most of the present simulations employ over-simplistic contact laws which cannot capture the real behaviour of particulate systems. For example, plastic yielding, material brittleness, sophisticated particle geometry, surface roughness, and particle adhesion are all vitally important factors affecting the behaviour of particle interactions, but have been largely ignored in most of the DEM simulations. This is because it is very difficult to consider these factors in an analytical contact law which has been the characteristic approach in DEM simulations. This thesis presents a strategy for obtaining the contact laws numerically and a comprehensive study of all these factors using the numerical approach. A numerical method, named as the Material Point Method (MPM) in the literature, is selected and shown to be ideal to study the particle interactions. The method is further developed in this work in order to take into account all the factors listed above. For example, to study the brittle failure during particle impact, Weibull’s theory is incorporated into the material point method; to study the effect of particle adhesion, inter-atomic forces are borrowed from the Molecular Dynamic model and incorporated into the method. These developments themselves represent a major progress in the numerical technique, enabling the method to be applied to a much wider range of problems. The focus of the thesis is however on the contact laws between extremely fine particles. Using the numerical technique as a tool, the entire existing theoretical framework for particle contact is re-examined. It is shown that, whilst the analytical framework is difficult to capture the real particle behaviour, numerical contact laws should be used in its place

Pattern Selection in Network of Coupled Multi-Scroll Attractors

<div><p>Multi-scroll chaotic attractor makes the oscillator become more complex in dynamic behaviors. The collective behaviors of coupled oscillators with multi-scroll attractors are investigated in the regular network in two-dimensional array, which the local kinetics is described by an improved Chua circuit. A feasible scheme of negative feedback with diversity is imposed on the network to stabilize the spatial patterns. Firstly, the Chua circuit is improved by replacing the nonlinear term with Sine function to generate infinite aquariums so that multi-scroll chaotic attractors could be generated under appropriate parameters, which could be detected by calculating the Lyapunov exponent in the parameter region. Furthermore, negative feedback with different gains (<i>D</i>₁, <i>D</i>₂) is imposed on the local square center area A₂ and outer area A₁ of the network, it is found that spiral wave, target wave could be developed in the network under appropriate feedback gain with diversity and size of controlled area. Particularly, homogeneous state could be reached after synchronization by selecting appropriate feedback gain and controlled size in the network. Finally, the distribution for statistical factors of synchronization is calculated in the two-parameter space to understand the transition of pattern region. It is found that developed spiral waves, target waves often are associated with smaller factor of synchronization. These results show that emergence of sustained spiral wave and continuous target wave could be effective for further suppression of spatiotemporal chaos in network by generating stable pacemaker completely.</p></div

Multi-scrolls attractors are generated within different simulation time.

<p>For (<b>a)</b> <i>t</i> = 50 time units, for (<b>b)</b> = 200 time units, for (c) <i>t</i> = 300 time units, for (<b>d)</b> <i>t</i> = 500 time units, <i>α</i> = 10.814, <i>β</i> = 14, <i>a</i> = 0.2, <i>b</i> = 0.15.</p

Bifurcation diagram.

<p>The bifurcation diagram for the system as a function of feedback gains <i>D</i> with three variables (<i>x</i>,<i>y</i>,<i>z</i>) feedback, the random initial values are chosen in the simulation.</p

Distribution of factor of synchronization.

<p><b>(a)</b><i>D</i>₁ = 0.1, the distribution of factor of synchronization <i>R</i> in the two-parameter space <i>A</i>₂ and <i>D</i>₂; <b>(b)</b> <i>D</i>₁ = 0.2 the distribution of factor of synchronization R in the two-parameter space <i>A</i>₂ and <i>D</i>₂; <b>(c)</b> <i>D</i>₁ = 0.1, the distribution of different patterns in the two-parameter space <i>A</i>₂ and <i>D</i>₂; <b>(d)</b> <i>D</i>₁ = 0.2, the distribution of different patterns in the two-parameter space <i>A</i>₂ and <i>D</i>₂. (unstable pattern for 0-navy blue; target wave for 1-sky blue; broken patterns for 2-yellow; spiral wave for 3-red).</p

Distribution for the largest Lyapunov exponent for Eq 1 in the two-parameter space.

<p>The snapshots are plotted in color scale at <i>α</i> = 10.814, <i>β</i> = 14.</p

The basin for the system with different feedback gains <i>D</i>.

<p>For (a) <i>D</i> = 0.1, (b) <i>D</i> = 0.2, (c) <i>D</i> = 0.3, (d) <i>D</i> = 0.4, the variables <i>x</i> and <i>y</i> are selected uniformly in the box [−20,20] and [−2,2].</p

sj-pdf-1-ctj-10.1177_17407745241243308 – Supplemental material for Causal interpretation of the hazard ratio in randomized clinical trials

Supplemental material, sj-pdf-1-ctj-10.1177_17407745241243308 for Causal interpretation of the hazard ratio in randomized clinical trials by Michael P Fay and Fan Li in Clinical Trials</p

Multi-scroll attractors and basion.

<p>(a) The chaotic attractor is developed at <i>t</i> = 300 time units, and <i>D</i> = 0.0. (b) The basin for the system with <i>D</i> = 0.0, initial values of variable <i>x</i>, <i>y</i> are chosen uniformly in the <i>x-y</i> plane box, <i>x</i>∈[−20,20], <i>y</i>∈[−2,2], z = 0.03.</p