Numerical methods for Lévy processes

We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric Lévy model

A Computational Framework for Efficient Reliability Analysis of Complex Networks

With the growing scale and complexity of modern infrastructure networks comes the challenge of developing efficient and dependable methods for analysing their reliability. Special attention must be given to potential network interdependencies as disregarding these can lead to catastrophic failures. Furthermore, it is of paramount importance to properly treat all uncertainties. The survival signature is a recent development built to effectively analyse complex networks that far exceeds standard techniques in several important areas. Its most distinguishing feature is the complete separation of system structure from probabilistic information. Because of this, it is possible to take into account a variety of component failure phenomena such as dependencies, common causes of failure, and imprecise probabilities without reevaluating the network structure. This cumulative dissertation presents several key improvements to the survival signature ecosystem focused on the structural evaluation of the system as well as the modelling of component failures. A new method is presented in which (inter)-dependencies between components and networks are modelled using vine copulas. Furthermore, aleatory and epistemic uncertainties are included by applying probability boxes and imprecise copulas. By leveraging the large number of available copula families it is possible to account for varying dependent effects. The graph-based design of vine copulas synergizes well with the typical descriptions of network topologies. The proposed method is tested on a challenging scenario using the IEEE reliability test system, demonstrating its usefulness and emphasizing the ability to represent complicated scenarios with a range of dependent failure modes. The numerical effort required to analytically compute the survival signature is prohibitive for large complex systems. This work presents two methods for the approximation of the survival signature. In the first approach system configurations of low interest are excluded using percolation theory, while the remaining parts of the signature are estimated by Monte Carlo simulation. The method is able to accurately approximate the survival signature with very small errors while drastically reducing computational demand. Several simple test systems, as well as two real-world situations, are used to show the accuracy and performance. However, with increasing network size and complexity this technique also reaches its limits. A second method is presented where the numerical demand is further reduced. Here, instead of approximating the whole survival signature only a few strategically selected values are computed using Monte Carlo simulation and used to build a surrogate model based on normalized radial basis functions. The uncertainty resulting from the approximation of the data points is then propagated through an interval predictor model which estimates bounds for the remaining survival signature values. This imprecise model provides bounds on the survival signature and therefore the network reliability. Because a few data points are sufficient to build the interval predictor model it allows for even larger systems to be analysed. With the rising complexity of not just the system but also the individual components themselves comes the need for the components to be modelled as subsystems in a system-of-systems approach. A study is presented, where a previously developed framework for resilience decision-making is adapted to multidimensional scenarios in which the subsystems are represented as survival signatures. The survival signature of the subsystems can be computed ahead of the resilience analysis due to the inherent separation of structural information. This enables efficient analysis in which the failure rates of subsystems for various resilience-enhancing endowments are calculated directly from the survival function without reevaluating the system structure. In addition to the advancements in the field of survival signature, this work also presents a new framework for uncertainty quantification developed as a package in the Julia programming language called UncertaintyQuantification.jl. Julia is a modern high-level dynamic programming language that is ideal for applications such as data analysis and scientific computing. UncertaintyQuantification.jl was built from the ground up to be generalised and versatile while remaining simple to use. The framework is in constant development and its goal is to become a toolbox encompassing state-of-the-art algorithms from all fields of uncertainty quantification and to serve as a valuable tool for both research and industry. UncertaintyQuantification.jl currently includes simulation-based reliability analysis utilising a wide range of sampling schemes, local and global sensitivity analysis, and surrogate modelling methodologies

A GENERIC RELIABILITY ANALYSIS AND DESIGN FRAMEWORK WITH RANDOM PARAMETER, FIELD, AND PROCESS VARIABLES

This dissertation aims at developing a generic reliability analysis and design framework that enables reliability prediction and design improvement with random parameter, field, and process variables. The capability of this framework is further improved by predicting and managing reliability even with a dearth of data that can be used to characterize random variables. To accomplish the research goal, three research thrusts are set forth. First, advanced techniques are developed to characterize the random field or process. The fundamental idea of these techniques is to model the random field or process with a set of important field signatures and random variables. These techniques enable the use of random parameter, field, and process variables for reliability analysis and design even with a dearth of data. Second, a generic reliability analysis framework is proposed to accurately assess system reliability in the presence of random parameter, field, and process variables. An advanced probability analysis technique, the Eigenvector Dimension Reduction (EDR) method, is developed by integrating the Dimension Reduction (DR) method with three proposed improvements: 1) an eigenvector sampling approach to obtain statistically independent samples over a random space; 2) a Stepwise Moving Least Square (SMLS) method to accurately approximate system responses over a random space; and 3) a Probability Density Function (PDF) generation method to accurately approximate the PDF of system responses for reliability analysis. Third, a generic Reliability-Based Design Optimization (RBDO) framework is developed to solve engineering design problems with random parameter, field, and process variables. This design framework incorporates the EDR method into RBDO. To illustrate the effectiveness of the developed framework, many numerical and engineering examples are employed to conduct the reliability analysis and RBDO with random parameter, field, and process variables. This dissertation demonstrates that the developed framework is very accurate and efficient for the reliability analysis and RBDO of engineering products and processes

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

Stochastic processes for graphs, extreme values and their causality: inference, asymptotic theory and applications

This thesis provides some theoretical and practical statistical inference tools for multivariate stochastic processes to better understand the behaviours and properties present in the data. In particular, we focus on the modelling of graphs, that is a family of nodes linked together by a collection of edges, and extreme values, that are values above a high threshold to have their own dynamics compared to the typical behaviour of the process. We develop an ensemble of statistical models, statistical inference methods and their asymptotic study to ensure the good behaviour of estimation schemes in a wide variety of settings. We also devote a chapter to the formulation of a methodology based on pre-existing theory to unveil the causal dependency structure behind high-impact events.Open Acces

Multivariate Extremes in Financial Markets: New Statistical Testing Methods and Applications

This thesis studies dependence of extreme events in financial markets. Statistical tests, detecting peculiar properties of tail dependence, are proposed. Statistical properties are derived. Comprehensive simulation experiments illustrate the tests\u27 usefulness. Empirically, unknown tail structures in financial time series are revealed