Nonstandard Finite Difference Schemes with Application to Finance: Option Pricing

2000 Mathematics Subject Classification: 65M06, 65M12.

POSITIVE FILTERED P_N METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM

We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP_N) expansion in the angular variable. The FP_N moment equations are accurate approximations to linear kinetic equations, but they are known to suffer from the occurrence of unphysical, negative particle concentrations. The new positive filtered P_N (FP_N+) closure is developed to address this issue. The FP_N+ closure approximates the kinetic distribution by a spherical harmonic expansion that is non-negative on a finite, predetermined set of quadrature points. With an appropriate numerical PDE solver, the FP_N+ closure generates particle concentrations that are guaranteed to be non-negative. Under an additional, mild regularity assumption, we prove that as the moment order tends to infinity, the FP_N+ approximation converges, in the L2 sense, at the same rate as the FP_N approximation; numerical tests suggest that this assumption may not be necessary. By numerical experiments on the challenging line source benchmark problem, we confirm that the FP_N+ method indeed produces accurate and non-negative solutions. To apply the FP_N+ closure on problems at large temporal-spatial scales, we develop a positive asymptotic preserving (AP) numerical PDE solver. We prove that the propose AP scheme maintains stability and accuracy with standard mesh sizes at large temporal-spatial scales, while, for generic numerical schemes, excessive refinements on temporal-spatial meshes are required. We also show that the proposed scheme preserves positivity of the particle concentration, under some time step restriction. Numerical results confirm that the proposed AP scheme is capable for solving linear transport equations at large temporal-spatial scales, for which a generic scheme could fail. Constrained optimization problems are involved in the formulation of the FP_N+ closure to enforce non-negativity of the FP_N+ approximation on the set of quadrature points. These optimization problems can be written as strictly convex quadratic programs (CQPs) with a large number of inequality constraints. To efficiently solve the CQPs, we propose a constraint-reduced variant of a Mehrotra-predictor-corrector algorithm, with a novel constraint selection rule. We prove that, under appropriate assumptions, the proposed optimization algorithm converges globally to the solution at a locally q-quadratic rate. We test the algorithm on randomly generated problems, and the numerical results indicate that the combination of the proposed algorithm and the constraint selection rule outperforms other compared constraint-reduced algorithms, especially for problems with many more inequality constraints than variables.

Optimized strong stability preserving IMEX Runge-Kutta methods

Esta es la versión no revisada del artículo: Inmaculada Higueras, Natalie Happenhofer, Othmar Koch, and Friedrich Kupka. 2014. Optimized strong stability preserving IMEX Runge-Kutta methods. J. Comput. Appl. Math. 272 (December 2014), 116-140. Se puede consultar la versión final en https://doi.org/10.1016/j.cam.2014.05.011

Secure communication protocol for preserving E-tendering integrity.

This paper presents a secure communication protocol which can be used as the framework for an e-tendering scheme. This protocol is focused on securing the integrity of tendering documents and ensuring that a secure record of document generation is kept. Our protocol provides a mechanism to manage e-tendering contract evidence as a legal record in a unique and effective manner. It is the starting point of reliable record keeping. To a certain extent, it also addresses existing security problems in the traditional tendering processes.

TOPSIG : Topology Preserving Document Signatures

Performance comparisons between File Signatures and Inverted Files for text retrieval have previously shown several significant shortcomings of file signatures relative to inverted files. The inverted file approach underpins most state-of-the-art search engine algorithms, such as Language and Probabilistic models. It has been widely accepted that traditional file signatures are inferior alternatives to inverted files. This paper describes TopSig, a new approach to the construction of file signatures. Many advances in semantic hashing and dimensionality reduction have been made in recent times, but these were not so far linked to general purpose, signature file based, search engines. This paper introduces a different signature file approach that builds upon and extends these recent advances. We are able to demonstrate significant improvements in the performance of signature file based indexing and retrieval, performance that is comparable to that of state of the art inverted file based systems, including Language models and BM25. These findings suggest that file signatures offer a viable alternative to inverted files in suitable settings and positions the file signatures model in the class of Vector Space retrieval models.

A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

The Wright-Fisher model is an Itô stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0,1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0,1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0,1].

Can slide positivity rates predict malaria transmission?

Background: Malaria is a significant threat to population health in the border areas of Yunnan Province, China. How to accurately measure malaria transmission is an important issue. This study aimed to examine the role of slide positivity rates (SPR) in malaria transmission in Mengla County, Yunnan Province, China. Methods: Data on annual malaria cases, SPR and socio-economic factors for the period of 1993 to 2008 were obtained from the Center for Disease Control and Prevention (CDC) and the Bureau of Statistics, Mengla, China. Multiple linear regression models were conducted to evaluate the relationship between socio-ecologic factors and malaria incidence. Results: The results show that SPR was significantly positively associated with the malaria incidence rates. The SPR (beta = 1.244, p = 0.000) alone and combination (SPR, beta = 1.326, p < 0.001) with other predictors can explain about 85% and 95% of variation in malaria transmission, respectively. Every 1% increase in SPR corresponded to an increase of 1.76/100,000 in malaria incidence rates. Conclusion: SPR is a strong predictor of malaria transmission, and can be used to improve the planning and implementation of malaria elimination programmes in Mengla and other similar locations. SPR might also be a useful indicator of malaria early warning systems in China.

Preserving the past in digital memory

The process of researching children’s literature from the past is a growing challenge as resources age and are increasingly treated as rare items, stored away within libraries and other research centres. In Australia, researchers and librarians have collaborated with the bibliographic database AustLit: The Australian Literature Resource to produce the Australian Children’s Literature Digital Resources Project (CLDR). This Project aims to address the growing demand for online access to rare children’s literature resources, and demonstrates the research potential of early Australian children’s literature by supplementing the collection with relevant critical articles. The CLDR project is designed with a specific focus and provides access to full text Australian children’s literature from European settlement to 1945. The collection demonstrates a need and desire to preserve literature treasures to prevent losing such collections in a digital age. The collection covers many themes relevant to the conference including, trauma, survival, memory, survival, hauntings, and histories. The resource provides new and exciting ways with which to research children’s literature from the past and offers a fascinating repository to scholars and professionals of ranging disciplines who are in interested in Australian children’s literature.

Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise

There has been considerable recent work on the development of energy conserving one-step methods that are not symplectic. Here we extend these ideas to stochastic Hamiltonian problems with additive noise and show that there are classes of Runge-Kutta methods that are very effective in preserving the expectation of the Hamiltonian, but care has to be taken in how the Wiener increments are sampled at each timestep. Some numerical simulations illustrate the performance of these methods.

Threshold privacy preserving keyword searches

We consider the following problem: users of an organization wish to outsource the storage of sensitive data to a large database server. It is assumed that the server storing the data is untrusted so the data stored have to be encrypted. We further suppose that the manager of the organization has the right to access all data, but a member of the organization can not access any data alone. The member must collaborate with other members to search for the desired data. In this paper, we investigate the notion of threshold privacy preserving keyword search (TPPKS) and define its security requirements. We construct a TPPKS scheme and show the proof of security under the assumptions of intractability of discrete logarithm, decisional Diffie-Hellman and computational Diffie-Hellman problems.

Non-market use and non-use values for preserving ecosystem services over time : a choice experiment application to coral reef ecosystems in New Caledonia

Non-use values (i.e. economic values assigned by individuals to ecosystem goods and services unrelated to current or future uses) provide one of the most compelling incentives for the preservation of ecosystems and biodiversity. Assessing the non-use values of non-users is relatively straightforward using stated preference methods, but the standard approaches for estimating non-use values of users (stated decomposition) have substantial shortcomings which undermine the robustness of their results. In this paper, we propose a pragmatic interpretation of non-use values to derive estimates that capture their main dimensions, based on the identification of a willingness to pay for ecosystem protection beyond one's expected life. We empirically test our approach using a choice experiment conducted on coral reef ecosystem protection in two coastal areas in New Caledonia with different institutional, cultural, environmental and socio-economic contexts. We compute individual willingness to pay estimates, and derive individual non-use value estimates using our interpretation. We find that, a minima, estimates of non-use values may comprise between 25 and 40% of the mean willingness to pay for ecosystem preservation, less than has been found in most studies.