A unified pricing of variable annuity guarantees under the optimal stochastic control framework

In this paper, we review pricing of variable annuity living and death guarantees offered to retail investors in many countries. Investors purchase these products to take advantage of market growth and protect savings. We present pricing of these products via an optimal stochastic control framework, and review the existing numerical methods. For numerical valuation of these contracts, we develop a direct integration method based on Gauss-Hermite quadrature with a one-dimensional cubic spline for calculation of the expected contract value, and a bi-cubic spline interpolation for applying the jump conditions across the contract cashflow event times. This method is very efficient when compared to the partial differential equation methods if the transition density (or its moments) of the risky asset underlying the contract is known in closed form between the event times. We also present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a benchmark for practitioners and researchers developing pricing of variable annuity guarantees.Comment: Keywords: variable annuity, guaranteed living and death benefits, guaranteed minimum accumulation benefit, optimal stochastic control, direct integration metho

Sensor-based Nonlinear and Nonstationary Dynaimc Analysis of Online Structural Health Monitoring

This dissertation focuses on robust online Structural Health Monitoring (SHM) framework for civil engineering structures. The proposed framework improves the diagnostic and prognostic schemes for damage-state awareness and structural life prediction in civil engineering structures. The underlying research achieves three main objectives, namely, (1) sensor placement optimization using partial differential equation modeling and Fisher information matrix, (2) structural damage detection using quasi-recursive correlation dimension (QRCD), and (3) structural damage prediction using online empirical mode decomposition (EMD).The research methodology includes three research tasks: Firstly, to formulate the optimal criteria for the sensor placement optimization damage detection problem based upon a partial differential equation (PDE) analytical model. The PDE model is derived and then validated through experimental results using correlation analysis. Secondly, to develop a novel quasi-recursive correlation dimension method for structural damage detection. The QRCD algorithm is integrated with an attractor analysis and overlapping windowing technique. Thirdly, to design an online structural damage prediction method based on empirical mode decomposition. The proposed SHM prediction scheme consists of two steps: prediction based change point detection using Hilbert instantaneous phase, and damage severity prediction using the energy index of the most representative intrinsic mode function (IMF).Study results show that; (1) the proposed optimal sensor placement method leads to an optimal spatial location for a collection of sensors, which are sensitive to structural damage, (2) the proposed damage detection algorithm can significantly alleviate the complexity of computation for correlation dimension to approximate O(N), making the online monitoring of nonlinear/nonstationary processes more applicable and efficient; and (3) the proposed empirical mode decomposition method for online damage prediction overcomes the boundary effects of the sifting process, and it has significant prediction accuracy improvement (greater than 30%) over other commonly used prediction techniques.Industrial Engineering & Managemen

B-splines for sparse grids : algorithms and application to higher-dimensional optimization

In simulation technology, computationally expensive objective functions are often replaced by cheap surrogates, which can be obtained by interpolation. Full grid interpolation methods suffer from the so-called curse of dimensionality, rendering them infeasible if the parameter domain of the function is higher-dimensional (four or more parameters). Sparse grids constitute a discretization method that drastically eases the curse, while the approximation quality deteriorates only insignificantly. However, conventional basis functions such as piecewise linear functions are not smooth (continuously differentiable). Hence, these basis functions are unsuitable for applications in which gradients are required. One example for such an application is gradient-based optimization, in which the availability of gradients greatly improves the speed of convergence and the accuracy of the results. This thesis demonstrates that hierarchical B-splines on sparse grids are well-suited for obtaining smooth interpolants for higher dimensionalities. The thesis is organized in two main parts: In the first part, we derive new B-spline bases on sparse grids and study their implications on theory and algorithms. In the second part, we consider three real-world applications in optimization: topology optimization, biomechanical continuum-mechanics, and dynamic portfolio choice models in finance. The results reveal that the optimization problems of these applications can be solved accurately and efficiently with hierarchical B-splines on sparse grids.In der Simulationstechnik werden zeitaufwendige Zielfunktionen oft durch einfache Surrogate ersetzt, die durch Interpolation gewonnen werden können. Vollgitter-Interpolationsmethoden leiden unter dem sogenannten Fluch der Dimensionalität, der sie unbrauchbar macht, falls der Parameterbereich der Funktion höherdimensional ist (vier oder mehr Parameter). Dünne Gitter sind eine Diskretisierungsmethode, die den Fluch drastisch lindert und die Approximationsqualität nur leicht verschlechtert. Leider sind konventionelle Basisfunktionen wie die stückweise linearen Funktionen nicht glatt (stetig differenzierbar). Daher sind sie für Anwendungen ungeeignet, in denen Gradienten benötigt werden. Ein Beispiel für eine solche Anwendung ist gradientenbasierte Optimierung, in der die Verfügbarkeit von Gradienten die Konvergenzgeschwindigkeit und die Ergebnisgenauigkeit deutlich verbessert. Diese Dissertation demonstriert, dass hierarchische B-Splines auf dünnen Gittern hervorragend geeignet sind, um glatte Interpolierende für höhere Dimensionalitäten zu erhalten. Die Dissertation ist in zwei Hauptteile gegliedert: Der erste Teil leitet neue B-Spline-Basen auf dünnen Gittern her und untersucht ihre Implikationen bezüglich Theorie und Algorithmen. Der zweite Teil behandelt drei Realwelt-Anwendungen aus der Optimierung: Topologieoptimierung, biomechanische Kontinuumsmechanik und Modelle der dynamischen Portfolio-Wahl in der Finanzmathematik. Die Ergebnisse zeigen, dass die Optimierungsprobleme dieser Anwendungen durch hierarchische B-Splines auf dünnen Gittern genau und effizient gelöst werden können

Reduced-order modelling for high-speed aerial weapon aerodynamics

In this work a high-fidelity low-cost surrogate of a computational fluid dynamics analysis tool was developed. This computational tool is composed of general and physics- based approximation methods by which three dimensional high-speed aerodynamic flow- field predictions are made with high efficiency and an accuracy which is comparable with that of CFD. The tool makes use of reduced-basis methods that are suitable for both linear and non-linear problems, whereby the basis vectors are computed via the proper orthogonal decomposition (POD) of a training dataset or a set of observations. The surrogate model was applied to two flow problems related to high-speed weapon aerodynamics. Comparisons of surrogate model predictions with high-fidelity CFD simulations suggest that POD-based reduced-order modelling together with response surface methods provide a reliable and robust approach for efficient and accurate predictions. In contrast to the many modelling efforts reported in the literature, this surrogate model provides access to information about the whole flow-field. In an attempt to reduce the up-front cost necessary to generate the training dataset from which the surrogate model is subsequently developed, a variable-fidelity POD- based reduced-order modelling method is proposed in this work for the first time. In this model, the scalar coefficients which are obtained by projecting the solution vectors onto the basis vectors, are mapped between spaces of low and high fidelities, to achieve high- fidelity predictions with complete flow-field information. In general, this technique offers an automatic way of fusing variable-fidelity data through interpolation and extrapolation schemes together with reduced-order modelling (ROM). Furthermore, a study was undertaken to investigate the possibility of modelling the transonic flow over an aerofoil using a kernel POD–based reduced-order modelling method. By using this type of ROM it was noticed that the weak non-linear features of the transonic flow are accurately modelled using a small number of basis vectors. The strong non-linear features are only modelled accurately by using a large number of basis vectors