查看更多>>摘要：We derive the Laplace transforms for the prices and deltas of the powered call and put options, as well as for the price and delta of the capped powered call option under a general framework. These Laplace transforms are expressed in terms of the transform of the underlying asset price at maturity. For any model that can derive the transform of the underlying asset price, we can obtain the Laplace transforms for the prices and deltas of the powered options and the capped powered call option. The prices and deltas of the powered options and the capped powered call option can be computed by numerical inversion of the Laplace transforms. Models to which our method can be applied include the geometric Levy model, the regime-switching model, the Black- Scholes-Vasicek model, and Heston's stochastic volatility model, which are commonly used for pricing of financial derivatives. In this paper, numerical examples are presented for all four models. (c) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper, we propose a recovery-type a posteriori error estimator of the weak Galerkin finite element method for the second order elliptic equation. The reliability and efficiency of the estimator are analyzed by a discrete H-1-norm of the exact error. The estimator is further used in the adaptive weak Galerkin algorithm on the triangular, quadrilateral and other polygonal meshes. Numerical results are provided to demonstrate the effectiveness of the adaptive mesh refinement guided by this estimator. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：We provide a comparative analysis of qualitative features of different numerical methods for the inhomogeneous geometric Brownian motion (IGBM). The limit distribution of the IGBM exists, its conditional and asymptotic mean and variance are known and the process can be characterised according to Feller's boundary classification. We compare the frequently used Euler-Maruyama and Milstein methods, two Lie-Trotter and two Strang splitting schemes and two methods based on the ordinary differential equation (ODE) approach, namely the classical Wong-Zakai approximation and the recently proposed log-ODE scheme. First, we prove that, in contrast to the Euler-Maruyama and Milstein schemes, the splitting and ODE schemes preserve the boundary properties of the process, independently of the choice of the time discretisation step. Second, we prove that the limit distribution of the splitting and ODE methods exists for all stepsize values and parameters. Third, we derive closed-form expressions for the conditional and asymptotic means and variances of all considered schemes and analyse the resulting biases. While the Euler-Maruyama and Milstein schemes are the only methods which may have an asymptotically unbiased mean, the splitting and ODE schemes perform better in terms of variance preservation. The Strang schemes outperform the Lie-Trotter splittings, and the log-ODE scheme the classical ODE method. The mean and variance biases of the log-ODE scheme are very small for many relevant parameter settings. However, in some situations the two derived Strang splittings may be a better alternative, one of them requiring considerably less computational effort than the log-ODE method. The proposed analysis may be carried out in a similar fashion on other numerical methods and stochastic differential equations with comparable features. Crown Copyright (C) 2021 Published by Elsevier B.V.
查看更多>>摘要：This manuscript is devoted to studying approximations of a coupled Klein-Gordon-Zakharov system where different orders of fractional spatial derivatives are utilized. The fractional derivatives involved are in the Riesz sense. It is understood that such a modeling system possesses an energy functional which is conserved throughout the period of time considered, and that its solutions are uniformly bounded. Motivated by these facts, we propose two numerical models to approximate the underlying continuous system. While both approximations remain to be nonlinear, one of them is implicit and the other is explicit. For each of the discretized models, we introduce a proper discrete energy functional to estimate the total energy of the continuous system. We prove that such a discrete energy is conserved in both cases. The existence of solutions of the numerical models is established via fixed-point theorems. Continuing explorations of intrinsic properties of the numerical solutions are carried out. More specifically, we show rigorously that the two schemes constructed are capable of preserving the boundedness of the approximations and that they yield consistent estimates of the true solution. Numerical stability and convergence are likewise proved theoretically. As one of the consequences, the uniqueness of numerical solutions is shown rigorously for both discretized models. Finally, comparisons of the numerical solutions are provided, in order to evaluate the capabilities of these discrete methods to preserve the discrete energy of underlying systems. (c) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In the paper, the authors, (1) by constructing a counterexample and utilizing Minkowski's inequality, demonstrate that there existed errors in the proofs of Theorems 1 and 2 in the paper "Mehrez and Agarwal, (2019)"; (2) with the help of an integral identity and by means of Holder's integral inequality, present a new integral inequality of the Hermite-Hadamard type for convex functions. (c) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper, we present a new parallel accurate algorithm called PAccSumK for computing summation of floating-point numbers. It is based on AccSumK algorithm. In the experiment, for the summation problems with large condition numbers, our algorithm outperforms the PSumK algorithm in terms of accuracy and computing time. The reason is that our algorithm is based on a more accurate algorithm called AccSumK algorithm compared to the SumL algorithm used in PSumK. The proposed parallel algorithm in this paper is designed to compute a result as if computed internally in K-fold the working precision. Numerical results are presented showing the performance and the accuracy of our new parallel algorithm for calculating summation. (c) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper, we present a fourth-order Cartesian grid-based boundary integral method (BIM) for a multiple acoustic scattering problem on closely packed obstacles. We reformulate the exterior Helmholtz boundary value problems (BVPs) as a Fredholm boundary integral equation (BIE) of the second kind for some unknown density function. Unlike the traditional boundary integral method, a distinctive feature of our scheme is that we do not require quadratures and direct evaluations of nearly singular, singular or hyper-singular boundary integrals in the solution of BIEs. Instead, we reinterpret boundary integrals as solutions to equivalent simple interface problems in an extended rectangle domain, which can be solved efficiently by a fourth-order finite difference method coupled with numerical corrections, FFT based solution and interpolations. Extensive numerical experiments show that our method is formally high-order accurate, fast convergent and in particular insensitive to complexity of scatterers. (c) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：This study concerns the Darcy flow problem in a two-dimensional fractured porous domain, in which the fracture is regarded as a one-dimensional interface, interacting with the surrounding media. In this paper, a finite volume element method (FVEM) is first proposed for the multi-dimensional fracture model, and error estimates for the pressure with optimal convergence are discussed. On this basis, a two-grid FVEM is developed for decoupling the multi-domain fracture model by a coarse grid approximation to the interface coupling conditions, and theoretical analysis demonstrates that approximation accuracy does not deteriorate under the two-grid decoupling technique. Finally, numerical experiments for FVEM and two-grid FVEM are presented to confirm the accuracy of theoretical analysis. (c) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：This paper proposes a new efficient operator splitting method for option pricing problem under the Heston model, which is very popular in financial engineering. The key idea of this method is relying on eliminating the cross derivative term in partial differential equation in two dimension by some variable transformation techniques, and then decomposes the original equation in two dimensions into two partial differential equations in one dimension, which can be numerically solved efficiently. Moreover, this method not only keeps the differentiability of model parameters, but also preserves the positivity, monotonicity and convexity of the option prices. Numerical results for a European put option show that this method achieves accuracy of second-order in space and first-order in time, which are coinciding with the theoretical analysis results. Since the algorithm of this paper can be parallelized easily, the option pricing problems in high-dimension can also be dealt with, such as the Basket option written on several assets and etc. Our method can also be applied to pricing American options, Asian options and option pricing problems in stochastic interest-rate models. (c) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：We consider the eigenvalue problem of a class of infinite complex symmetric tridiagonal matrices whose diagonal and off-diagonal elements diverge in modulus, and which have a compact inverse. We regard the matrix as a linear operator mapping a maximal domain in Hilbert space l(2) into l(2). This paper aims to extend the work of Ikebe et al. on a class of eigenvalue problems and for which asymptotic error estimates have been obtained. In this paper we focus on the following points: (1) considering what class of zero finding problems of three-term recurrence relations can be reformulated as eigenvalue problems of the class of infinite tridiagonal matrices stated above; (2) determining a class of matrices for which obtaining good approximate eigenvalues is guaranteed by using those of truncated principal sub-matrices; and (3) determining a class of matrices that permits us the asymptotic error estimates computed as in (2). (C)& nbsp;& nbsp;2021 The Author(s). Published by Elsevier B.V.& nbsp;& nbsp;
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