Methods

Mathematical Methodologies

RiskAnalytica employs a wide range of mathematical methods to produce flexible and dynamical models. A library of standard as well as “problem specific” methods and tools has been assembled to aid in this effort. The methods span a range of theoretical and practical tools used in mathematics, physics and statistics. The methods are used to establish the theoretical foundations of the RiskAnalytica’s risk and rewards framework, determine the dynamic equations (and boundary condition) and provide solutions.

The various observables required to support RiskAnalytica’s risk and rewards framework are computed within a geometrical representation. The equations which are determined to represent the evolution of such geometry are usually found to be non-linear. Whenever possible, analytic methods are employed to obtain exact solution to problem. In general, however, exact solutions cannot be found and numerical approximations must be used as a proxy.

Currently, RiskAnalytica employs a large variety of such method in order to accommodate every possible set of dynamic equations. In order to establish the framework and obtain reasonable solutions, RiskAnalytica adopts tools and methodologies from a diverse range of scientific fields and techniques such as:

• Algebraic Topology – Understanding the basic point set topology of the problem along with the related Homology group structure. In general, the groups are highly dimensional and require clever computational algorithm;
• Differential geometry – Establishing metric properties to provide a geometric representation of the problem. Differential geometry will determine the manner in which coordinate systems change in models involving curved geometry. This allows to model physical reality as a topological manifold upon which the notion of a metric and associated connections gives rise to curvature and torsion;
• Hamiltonian dynamics – Understanding the possible connection between the geometry and the dynamical equations which govern their evolution based on a unique action principle. A non-linear system can be investigated for various dynamical instabilities and chaotic behaviour using Lyapunov exponents;
• Symmetries and group theory - Understanding the symmetries which exist within the problem. The restrictions which such symmetries impose on the equations of motion are represented within the realm of group theory;
• Conformal representations – Understanding the causal structure of the proposed geometry within a set of pictorial representations. In Physics such representations have been proven to be very useful in understanding the singular and boundary properties by introducing various conformal transformations. In General Relativity such pictorial representations are an indispensable part of geometric visualizations of the problem;
• Boundary conditions – The solutions of the dynamical equations which govern the evolution of the geometry are found to satisfy a set of non-linear partial differential equations. In general, the solutions require an understanding of the boundary conditions and the manner in which they effect the evolution of the system;
• Perturbation and Approximation theory – Determining the nature of the exact solutions (to the equation of motion) as a power series expansion. RiskAnalytica employs a wide range of special functions and approximation techniques to provide a series solution whenever possible;
• Numerical methods – In instances in which exact solutions are not possible, numerical solutions are approximated within an acceptable margin of accuracy. Currently, RiskAnalytica has at its disposal a rich variety of numerical recipes to problems in partial and ordinary differential equations, computational topology and probability theory;
• Lattice gauge theory - A powerful method in which solutions can be generated within computer-assisted calculations. In lattice gauge theory, the geometrical space is Wick rotated into Euclidean space and is (discretized) replaced by a lattice with lattice spacing equal to a. The computation of dynamical observables is carried out at every point on the lattice in an effort to reproduce the exact so;ution in the limit of a approaching 0.hep-ph/9911400
• Markov Chain Monte Carlo - A stochastic simulation method which uses state transition probabilities to represent the evolution of a dynamic system. The subsequent states of the system are generated randomly based on an appropriate probability of their occurrence (tested using Kolmogorov-Smirnov tests). This gives rise to a statistical approximation with respect to the exact solution (within the desired accuracy level).
• Sensitivity Analysis - Understanding how sensitive a possible outcome (an effect) is to input and functional assumptions (causes). Various methods which are important in computational physics and bank risk management are employed;
• Decision Tree Analysis - Process maps are used to reflect the conditional, joint and marginal decision dynamics that the decision maker faces, with the incorporation of inherent variability
• Knowledge networking - Organizations form knowledge using several different networks often not reflected by a formal organizational structure. Mapping such networks allows an understanding of how observations are used to form organizational knowledge and action. Knowledge networks are subject to variability in the way observations are gained. This is an important process to understand for sophisticated risk management that requires a low margin of error.
• Data Analysis - Techniques used for analyzing historical or experimental data, allowing determination of the extent that an observed result is connected and/or correlated to input values.
• Forecasting - Techniques used to guide planned activity and responses over time using observation based simulation techniques.
• Linear and Non-Linear Analysis - Techniques used to understand the causal links between the variables, obtain the algorithmic form and to establish a mathematical network representation.