Stochastic modelling of financial time series with memory and multifractal scaling

Financial processes may possess long memory and their probability densities may display heavy tails. Many models have been developed to deal with this tail behaviour, which reflects the jumps in the sample paths. On the other hand, the presence of long memory, which contradicts the efficient market hypothesis, is still an issue for further debates. These difficulties present challenges with the problems of memory detection and modelling the co-presence of long memory and heavy tails. This PhD project aims to respond to these challenges. The first part aims to detect memory in a large number of financial time series on stock prices and exchange rates using their scaling properties. Since financial time series often exhibit stochastic trends, a common form of nonstationarity, strong trends in the data can lead to false detection of memory. We will take advantage of a technique known as multifractal detrended fluctuation analysis (MF-DFA) that can systematically eliminate trends of different orders. This method is based on the identification of scaling of the q-th-order moments and is a generalisation of the standard detrended fluctuation analysis (DFA) which uses only the second moment; that is, q = 2. We also consider the rescaled range R/S analysis and the periodogram method to detect memory in financial time series and compare their results with the MF-DFA. An interesting finding is that short memory is detected for stock prices of the American Stock Exchange (AMEX) and long memory is found present in the time series of two exchange rates, namely the French franc and the Deutsche mark. Electricity price series of the five states of Australia are also found to possess long memory. For these electricity price series, heavy tails are also pronounced in their probability densities. The second part of the thesis develops models to represent short-memory and longmemory financial processes as detected in Part I. These models take the form of continuous-time AR(∞) -type equations whose kernel is the Laplace transform of a finite Borel measure. By imposing appropriate conditions on this measure, short memory or long memory in the dynamics of the solution will result. A specific form of the models, which has a good MA(∞) -type representation, is presented for the short memory case. Parameter estimation of this type of models is performed via least squares, and the models are applied to the stock prices in the AMEX, which have been established in Part I to possess short memory. By selecting the kernel in the continuous-time AR(∞) -type equations to have the form of Riemann-Liouville fractional derivative, we obtain a fractional stochastic differential equation driven by Brownian motion. This type of equations is used to represent financial processes with long memory, whose dynamics is described by the fractional derivative in the equation. These models are estimated via quasi-likelihood, namely via a continuoustime version of the Gauss-Whittle method. The models are applied to the exchange rates and the electricity prices of Part I with the aim of confirming their possible long-range dependence established by MF-DFA. The third part of the thesis provides an application of the results established in Parts I and II to characterise and classify financial markets. We will pay attention to the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), the NASDAQ Stock Exchange (NASDAQ) and the Toronto Stock Exchange (TSX). The parameters from MF-DFA and those of the short-memory AR(∞) -type models will be employed in this classification. We propose the Fisher discriminant algorithm to find a classifier in the two and three-dimensional spaces of data sets and then provide cross-validation to verify discriminant accuracies. This classification is useful for understanding and predicting the behaviour of different processes within the same market. The fourth part of the thesis investigates the heavy-tailed behaviour of financial processes which may also possess long memory. We consider fractional stochastic differential equations driven by stable noise to model financial processes such as electricity prices. The long memory of electricity prices is represented by a fractional derivative, while the stable noise input models their non-Gaussianity via the tails of their probability density. A method using the empirical densities and MF-DFA will be provided to estimate all the parameters of the model and simulate sample paths of the equation. The method is then applied to analyse daily spot prices for five states of Australia. Comparison with the results obtained from the R/S analysis, periodogram method and MF-DFA are provided. The results from fractional SDEs agree with those from MF-DFA, which are based on multifractal scaling, while those from the periodograms, which are based on the second order, seem to underestimate the long memory dynamics of the process. This highlights the need and usefulness of fractal methods in modelling non-Gaussian financial processes with long memory.

Resolving painful emotional experience during psychodrama

Unresolved painful emotional experiences such as bereavement, trauma and disturbances in core relationships, are common presenting problems for clients of psychodrama or psychotherapy more generally. Emotional pain is experienced as a shattering of the sense of self and disconnection from others and, when unresolved, produces avoidant responses which inhibit the healing process. There is agreement across therapeutic modalities that exposure to emotional experience can increase the efficacy of therapeutic interventions. Moreno proposes that the activation of spontaneity is the primary curative factor in psychodrama and that healing occurs when the protagonist (client) engages with his or her wider social system and develops greater flexibility in response to that system. An extensive case-report literature describes the application of the psychodrama method in healing unresolved painful emotional experiences, but there is limited empirical research to verify the efficacy of the method or to identify the processes that are linked to therapeutic change. The purpose of this current research was to construct a model of protagonist change processes that could extend psychodrama theory, inform practitioners’ therapeutic decisions and contribute to understanding the common factors in therapeutic change. Four studies investigated protagonist processes linked to in-session resolution of painful emotional experiences. Significant therapeutic events were analysed using recordings and transcripts of psychodrama enactments, protagonist and director recall interviews and a range of process and outcome measures. A preliminary study (3 cases) identified four themes that were associated with helpful therapeutic events: enactment, the working alliance with the director and with group members, emotional release or relief and social atom repair. The second study (7 cases) used Comprehensive Process Analysis (CPA) to construct a model of protagonists’ processes linked to in-session resolution. This model was then validated across four more cases in Study 3. Five meta-processes were identified: (i) a readiness to engage in the psychodrama process; (ii) re-experiencing and insight; (iii) activating resourcefulness; (iv) social atom repair with emotional release and (v) integration. Social atom repair with emotional release involved deeply experiencing a wished-for interpersonal experience accompanied by a free flowing release of previously restricted emotion and was most clearly linked to protagonists’ reports of reaching resolution and to post session improvements in interpersonal relationships and sense of self. Acceptance of self in the moment increased protagonists’ capacity to generate new responses within each meta-process and, in resolved cases, there was evidence of spontaneity developing over time. The fourth study tested Greenberg’s allowing and accepting painful emotional experience model as an alternative explanation of protagonist change. The findings of this study suggested that while the process of allowing emotional pain was present in resolved cases, Greenberg’s model was not sufficient to explain the processes that lead to in-session resolution. The protagonist’s readiness to engage and activation of resourcefulness appear to facilitate the transition from problem identification to emotional release. Furthermore, experiencing a reparative relationship was found to be central to the healing process. This research verifies that there can be in-session resolution of painful emotional experience during psychodrama and protagonists’ reports suggest that in-session resolution can heal the damage to the sense of self and the interpersonal disconnection that are associated with unresolved emotional pain. A model of protagonist change processes has been constructed that challenges the view of psychodrama as a primarily cathartic therapy, by locating the therapeutic experience of emotional release within the development of new role relationships. The five meta-processes which are described within the model suggest broad change principles which can assist practitioners to make sense of events as they unfold and guide their clinical decision making in the moment. Each meta-process was linked to specific post-session changes, so that the model can inform the development of therapeutic plans for individual clients and can aid communication for practitioners when a psychodrama intervention is used for a specific therapeutic purpose within a comprehensive program of therapy.

Damage assessment in structures using vibration characteristics

Changes in load characteristics, deterioration with age, environmental influences and random actions may cause local or global damage in structures, especially in bridges, which are designed for long life spans. Continuous health monitoring of structures will enable the early identification of distress and allow appropriate retrofitting in order to avoid failure or collapse of the structures. In recent times, structural health monitoring (SHM) has attracted much attention in both research and development. Local and global methods of damage assessment using the monitored information are an integral part of SHM techniques. In the local case, the assessment of the state of a structure is done either by direct visual inspection or using experimental techniques such as acoustic emission, ultrasonic, magnetic particle inspection, radiography and eddy current. A characteristic of all these techniques is that their application requires a prior localization of the damaged zones. The limitations of the local methodologies can be overcome by using vibration-based methods, which give a global damage assessment. The vibration-based damage detection methods use measured changes in dynamic characteristics to evaluate changes in physical properties that may indicate structural damage or degradation. The basic idea is that modal parameters (notably frequencies, mode shapes, and modal damping) are functions of the physical properties of the structure (mass, damping, and stiffness). Changes in the physical properties will therefore cause changes in the modal properties. Any reduction in structural stiffness and increase in damping in the structure may indicate structural damage. This research uses the variations in vibration parameters to develop a multi-criteria method for damage assessment. It incorporates the changes in natural frequencies, modal flexibility and modal strain energy to locate damage in the main load bearing elements in bridge structures such as beams, slabs and trusses and simple bridges involving these elements. Dynamic computer simulation techniques are used to develop and apply the multi-criteria procedure under different damage scenarios. The effectiveness of the procedure is demonstrated through numerical examples. Results show that the proposed method incorporating modal flexibility and modal strain energy changes is competent in damage assessment in the structures treated herein.

Identification of factors predicting clickthrough in Web searching using neural network analysis

In this research, we aim to identify factors that significantly affect the clickthrough of Web searchers. Our underlying goal is determine more efficient methods to optimize the clickthrough rate. We devise a clickthrough metric for measuring customer satisfaction of search engine results using the number of links visited, number of queries a user submits, and rank of clicked links. We use a neural network to detect the significant influence of searching characteristics on future user clickthrough. Our results show that high occurrences of query reformulation, lengthy searching duration, longer query length, and the higher ranking of prior clicked links correlate positively with future clickthrough. We provide recommendations for leveraging these findings for improving the performance of search engine retrieval and result ranking, along with implications for search engine marketing