A computationally efficient methodology in pricing a guaranteed minimum accumulation benefit

In this thesis, we consider a framework under which three correlated factors, namely, financial, mortality and lapse risks, are modelled in an integrated way. This modelling framework supports the valuation of a guaranteed minimum accumulation benefit (GMAB). The change-of-measure approach is employed to come up with a compact and implementable valuation expressions. We provide a numerical demonstration to confirm the efficiency and accuracy of our proposed pricing methodology. In particular, our approach on average takes only 0.07% of the computing time entailed by the Monte-Carlo (MC) simulation technique. Furthermore, the standard errors of our approach’s results are lower than those obtained from MC-based computations. When there are no renewal options in a GMAB contract, we get the special case of a guaranteed minimum maturity benefit for which a closed-form pricing solution is derived

Sparse metric learning via smooth optimization

Copyright © 2009 NIPS Foundation23rd Annual Conference on Advances in Neural Information Processing Systems (NIPS 2009), Vancouver, Canada, 7-10 December 2009In this paper we study the problem of learning a low-rank (sparse) distance matrix. We propose a novel metric learning model which can simultaneously conduct dimension reduction and learn a distance matrix. The sparse representation involves a mixed-norm regularization which is non-convex. We then show that it can be equivalently formulated as a convex saddle (min-max) problem. From this saddle representation, we develop an efficient smooth optimization approach [15] for sparse metric learning, although the learning model is based on a non-differentiable loss function. This smooth optimization approach has an optimal convergence rate of O(1=t2) for smooth problems where t is the iteration number. Finally, we run experiments to validate the effectiveness and efficiency of our sparse metric learning model on various datasets

Accelerating Diffusion-based Combinatorial Optimization Solvers by Progressive Distillation

Graph-based diffusion models have shown promising results in terms of generating high-quality solutions to NP-complete (NPC) combinatorial optimization (CO) problems. However, those models are often inefficient in inference, due to the iterative evaluation nature of the denoising diffusion process. This paper proposes to use progressive distillation to speed up the inference by taking fewer steps (e.g., forecasting two steps ahead within a single step) during the denoising process. Our experimental results show that the progressively distilled model can perform inference 16 times faster with only 0.019% degradation in performance on the TSP-50 dataset

Wave Excitation and Dynamics in Disordered Systems

This thesis presents studies of the field and energy excited in disordered systems as well as the dynamics of scattering. Dynamic and steady state aspects of wave propagation are deeply connected in lossless open systems in which the scattering matrix is unitary. There is then an equivalence among the energy excited within the medium through all channels, the Wigner time delay, which is the sum of dwell times in all channels coupled to the medium, and the density of states. But these equivalences fall away in the presence of material loss or gain. In this paper, we use microwave measurements, numerical simulations, and theoretical analysis to discover the changing relationships among the internal field, transmission, transmission time, dwell time, total excited energy, and the density of states in with loss and gain, and their dependence upon dimensionality and spectral overlap. The field, including the contribution of the still coherent incident wave, is a sum over modal partial fractions with amplitudes that are independent of loss and gain. Specifically, we demonstrate in a unitary system that wave statistics within samples of any dimension are independent of the detailed structure of a material and depend only on the net strengths of scattering and reflection between the observation point and each of the boundaries. The total energy excited is proportional to the dwell time. When modes are spectrally isolated, the energy is proportional to the sum of products of the modal contribution to the density of states and the ratio of the linewidth without and with absorption. When modes overlap, however, the total energy is further reduced by loss and enhanced by gain. In 1D, the average transmission time is independent of loss, gain, and scattering strength. In higher dimensions, however, the average transmission time falls with loss, scattering strength and channel number, and increases with gain. It is the sum over Lorentzian functions associated with the zeros as well as the poles of the transmission matrix. This endows transmission zeros with topological characteristics: in unitary media, zeros occur either singly on the real axis or as conjugate pairs in the complex frequency plane. The transmission zeros are moved down or up in the complex plane by an amount equal to the internal rate of decay or growth of the field. In weakly absorbing media, the spectrum of the transmission time of the lowest transmission eigenchannel is the sum of Lorentzians due to transmission zeros plus a background due to far-off-resonance poles. As the scattering strength increases, conjugate pairs of zeros are brought to the real axis and converted to two single zeros which are constrained to move on the real axis until they combine and leave the real axis as a conjugate pair. The average density of transmission zeros in the complex plane is found from the fall of the average transmission time with absorption as zeros are swept into the lower half of the complex frequency plane. As a sample is deformed, two single zeros and a conjugate pair of zeros may interconvert on the real axis with a square root singularity in the sensitivity of the spacing between transmission zeros to structural change. Thus, the disposition of poles and zeros in the complex frequency plane provides a framework for understanding and controlling wave propagation in non-Hermitian systems. Waves propagating through random media experience multiple scattering and give rise to random fluctuations of transmission and reflection. We show using a random-matrix model, microwave experiments, and numerical simulations that the location and reflectivity of an object inside a 1D random medium can be extracted from the transmitted and reflected fields