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Inspired by some works of Kirkby, J. L. [2015. Efficient option pricing by frame duality with the fast Fourier transform. SIAM Journal on Financial Mathematics 6(1), 713-747; 2016. An efficient transform method for Asian option pricing. SIAM Journal on Financial Mathematics 7(1), 845-892], we present a systematic study on effectively computing the Gerber-Shiu function in the Lévy risk model, where the frame duality projection is used for approximation. By introducing an auxiliary function, we provide a smooth extension of the Gerber-Shiu function, which has closed-form Fourier transform and is differentiable over the whole real line under some conditions. The objective function is approximated by its frame duality projection onto a Riesz basis, and the projection coefficients are readily computed by the fast Fourier transform algorithm. Error analysis is made and the effectiveness of our results will be further illustrated in the numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, the compound Poisson risk model is considered. Inspired by Albrecher, Cheung, & Thonhauser. [(2011b). Randomized observation periods for the compound Poisson risk model: dividend. ASTIN Bulletin 41(2), 645-672], it is assumed that the insurer observes its surplus level periodically to decide on dividend payments at the arrival times of an Erlang(n) renewal process. If the observed surplus is larger than the maximum of a threshold b and the last observed (post-dividend) level, then a fraction of the excess is paid as a lump sum dividend. Ruin is declared when the observed surplus is negative. In this proposed periodic threshold-type dividend strategy, the insurer can have a ruin probability of less than one (as opposed to the periodic barrier strategy). The expected discounted dividends before ruin (denoted by V) will be analyzed. For arbitrary claim distribution, the general solution of V is derived. More explicit result for V is presented when claims have rational Laplace transform. Numerical examples are provided to illustrate the effect of randomized observations on V and the optimization of V with respect to b. When claims are exponential, convergence to the traditional threshold strategy is shown as the inter-observation times tend to zero. [ABSTRACT FROM AUTHOR]

The contribution of this article contains two parts: first, we prove a supercloseness result for the partially penalized immersed finite element (PPIFE) methods in (Lin, T. Lin, Y. & Zhang, X. (2015), Partially penalized immersed finite element methods for elliptic interface problems. SIAM J. Numer. Anal. 53, 1121–1144) and then based on the supercloseness result, we show that the gradient recovery method proposed in our previous work (Guo, H. & Yang, X. (2017) Gradient recovery for elliptic interface problem: II. Immersed finite element methods. J. Comput. Phys. 338, 606–619) can be applied to the PPIFE methods and the recovered gradient converges to the exact gradient with a superconvergent rate $$\mathcal{\rm O}(h^{1.5})$$. Hence, the gradient recovery method provides an asymptotically exact a posteriori error estimator for the PPIFE methods. Several numerical examples are presented to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new efficient method for estimating the Gerber-Shiu discounted penalty function in the classical risk model. We develop the Gerber-Shiu function on the Laguerre basis, and then estimate the unknown coefficients based on sample information on claim numbers and individual claim sizes. The convergence rate of the estimate is derived. Some simulation examples are illustrated to show that the estimate performs very well when the sample size is finite. We also show that the proposed estimate outperforms other estimates in the simulation studies. [ABSTRACT FROM AUTHOR]

In this paper, we perform $$L^\infty $$ and $$W^{1,\infty }$$ error estimates for a class of bi- k finite volume schemes on a quadrilateral mesh for elliptic equations, where $$k\ge 2$$ is arbitrary. We show that the errors of the finite volume solution in both the $$L^\infty $$ and $$W^{1,\infty }$$ norms converge to zero with optimal orders, provided the solution $$u\in W^{k+2,\infty }$$ . Our analysis is based mainly on an estimate of the difference between the finite volume and the corresponding finite element bilinear forms, as well as some techniques derived for $$L^\infty $$ and $$W^{1,\infty }$$ estimates of the finite element method. Our theoretical findings are supported by several numerical examples. [ABSTRACT FROM AUTHOR]

In this article, we analyze convergence and supercloseness properties of a class of weak Galerkin (WG) finite element methods for solving second-order elliptic problems. It is shown that the WG solution is superclose to the Lagrange interpolant using Lobatto points. This supercloseness behavior is obtained through some newly designed stabilization terms. A postprocessing technique using polynomial preserving recovery (PPR) is introduced for the WG approximation. Superconvergence analysis is performed for the PPR recovered gradient. Numerical examples are provided to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a spectrally negative Lévy insurance risk process with a barrier-type dividend strategy. In contrast to the traditional barrier strategy in which dividends are payable to the shareholders immedi- ately when the surplus process reaches a fixed level b (as long as ruin has not yet occurred), it is assumed that the insurer only makes dividend decisions at some discrete time points in the spirit of [1]. Under such a dividend strategy with Erlang inter-dividend-decision times, expressions for the Gerber-Shiu ex- pected discounted penalty function proposed in [24] and the moments of total discounted dividends payable until ruin are derived. The results are expressed in terms of the scale functions of a spectrally negative Lévy process and an em- bedded spectrally negative Markov additive process. Our analyses rely on the introduction of a potential measure associated with an Erlang random variable. Numerical illustrations are also given. [ABSTRACT FROM AUTHOR]

In this paper, we consider a compound Poisson model under a mixed dividend strategy. The mixed dividend strategy is a combination of threshold dividend strategy and periodic dividend strategy. Given a positive threshold level b  > 0, whenever the surplus process attains the level b , dividends will be paid off continuously at a rate α  > 0. Furthermore, given a sequence of dividend decision times { Z j } j = 1 ∞ , whenever the observed surplus level at Z j is larger than b , the excess value will also be paid off as dividend. We study the expected discounted dividend payments before ruin and the Gerber–Shiu expected discounted penalty function. Some numerical examples are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate local ultraconvergence properties of the high-order finite element method (FEM) for second order elliptic problems with variable coefficients. Under suitable regularity and mesh conditions, we show that at an interior vertex, which is away from the boundary with a fixed distance, the gradient of the post-precessed kth $$(k\ge 2)$$ order finite element solution converges to the gradient of the exact solution with order $$\mathcal{O}(h^{k+2} (\mathrm{ln} h)^3)$$ . The proof of this ultraconvergence property depends on a new interpolating operator, some new estimates for the discrete Green's function, a symmetry theory derived in [26], and the Richardson extrapolation technique in [20]. Numerical experiments are performed to demonstrate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we propose two systematic strategies to recover the gradient on the boundary of a domain. The recovered gradient has comparable superconvergent property on the boundary as that in the interior of the domain. This superconvergence property has been validated by several numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Markov additive insurance risk process under a randomized dividend strategy in the spirit of Albrecher et al. (). Decisions on whether to pay dividends are only made at a sequence of dividend decision time points whose intervals are Erlang( n) distributed. At a dividend decision time, if the surplus level is larger than a predetermined dividend barrier, then the excess is paid as a dividend as long as ruin has not occurred. In contrast to Albrecher et al. (), it is assumed that the event of ruin is monitored continuously (Avanzi et al. () and Zhang ()), i.e. the surplus process is stopped immediately once it drops below zero. The quantities of our interest include the Gerber-Shiu expected discounted penalty function and the expected present value of dividends paid until ruin. Solutions are derived with the use of Markov renewal equations. Numerical examples are given, and the optimal dividend barrier is identified in some cases. [ABSTRACT FROM AUTHOR]

In the spirit of a publication by Cheung in 2013 we study generalized Gerber-Shiu functions in the compound Poisson risk model perturbed by diffusion. These generalized Gerber-Shiu functions can be used to analyze the moments of the total discounted claim costs until ruin. Integral equations for the generalized Gerber-Shiu functions are derived and a solution procedure is also provided. Some explicit results are given when the claim size density is a combination of exponentials, and some numerical results are also given. [ABSTRACT FROM AUTHOR]

A family of any order finite volume (FV) schemes over quadrilateral meshes is analyzed under the framework of Petrov-Galerkin method. By constructing a special mapping from the trial space to the test space, a unified proof for the inf-sup condition of any order FV schemes is provided under a weak condition that the underlying mesh is an $$h^{1+\gamma },\gamma >0$$ parallelogram mesh. The optimal convergence rate of FV solutions is then obtained with well-known techniques. [ABSTRACT FROM AUTHOR]

In this article, we combine mixed finite element method, multiscale discretization, and Rayleigh quotient iteration to propose a new adaptive algorithm based on residual type a posterior error estimates for the Stokes eigenvalue problem. Both reliability and efficiency of the error indicator are proved. The efficiency of the algorithm is also investigated using Chen's Innovation Finite Element Method (iFEM) package. Numerical results are satisfying.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 31-53, 2015 [ABSTRACT FROM AUTHOR]

Abstract: In this work, we present and analyze a -version finite volume method (FVM) for elliptic problems in the one dimensional setting. Under some regularity assumptions of the exact solution, it is shown that the -version FV solution converges with exponential rates under and -norms. Superconvergence properties at nodal, Lobatto and Gauss points have been also discussed. Numerical results are presented to support our theoretical findings. [Copyright &y& Elsevier]

We consider conforming finite element approximation of fourth-order singularly perturbed problems of reaction diffusion type. We prove superconvergence of standard C1 finite element method of degree p on a modified Shishkin mesh. In particular, a superconvergence error bound of [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a least-squares finite element method for second-order two-point boundary value problems is considered. The problem is recast as a first-order system. Standard and improved optimal error estimates in maximum-norms are established. Superconvergence estimates at interelement, Lobatto, and Gauss points are developed. Numerical experiments are given to illustrate theoretical results. [Copyright &y& Elsevier]

Natural superconvergence of the least-squares finite element method is surveyed for the one-and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method. [ABSTRACT FROM AUTHOR]

In this work, a recovery technique for the intermediate finite element family of the second type is proposed and analysed on a second-order elliptic model problem. It is shown that when the pollution error is properly controlled, the convergence rate of the recovered gradient is two orders higher than the optimal global rate on an interior subdomain where rectangular meshes of regular type are applied. Some numerical results are provided to confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

• We consider the finite time dividend-ruin problems with constant inter-observation times. • Recursive methods for computing the finite time expected discounted penalty function and the present value of dividend payments are proposed. • Error analysis are made for our algorithm. • Numerical results are provided to show effectiveness of our method. In this paper, we use a Lévy process to model the surplus flow of an insurance company. It is assumed that the surplus level is observed at a sequence of fixed times and dividend decisions are made at each observation time. If the observed surplus level is larger than a given barrier, then the excess amount would be paid off as a lump sum of dividends. Further, we assume that ruin is declared as soon as the observed surplus level is negative. Using the Fourier cosine series expansion method, we propose some numerical methods for computing the finite-time expected discounted dividend payments before ruin and the finite-time expected discounted penalty function. Both error analysis and numerical examples are given to show accuracy and efficiency of our method. [ABSTRACT FROM AUTHOR]

In this paper, we present an hp analysis of the mortar spectral element method for the Schrödinger eigenvalue problem (− Δ + c 2 ‖ x ‖ 2 ) u = λ u , and thereby justify the numerical findings in [30] , where the method was demonstrated to be efficient to handle the singularities arising from both the inverse square potential and the reentrant/obtuse corners with exponential order of convergence. Non-uniformly weighted Sobolev spaces are introduced to accommodate singularities and to measure the regularity of the eigenfunctions. Optimal error estimates for the mortar spectral element method and the lifting theorem for the eigenfunctions are established. [ABSTRACT FROM AUTHOR]

We present an efficient valuation approach for guaranteed minimum maturity benefits (GMMBs) embedded in variable annuity (VA) contracts in a regime-switching jump diffusion model. We allow early surrender of the VA contract and impose surrender charges, which are important in practice to discourage early termination/lapse of the contract. We consider both continuously-monitored and discretely-monitored surrender behaviors before maturity, and utilize an intensity-based framework. Based on the continuous-time Markov chain (CTMC) approximation combined with the Fourier cosine series expansion method, we find that the valuation problem can be solved under a regime-switching jump diffusion framework. Both error analysis and numerical experiments demonstrate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider discontinuous Galerkin approximations to the solution of Naghdi arches and show how to post-process them in an element-by-element fashion to obtain a far better approximation. Indeed, we prove that, if polynomials of degree k are used, the postprocessed approximation converges with order 2k+1 in the L2-norm throughout the domain. This has to be contrasted with the fact that before post-processing, the approximation converges with order k + 1 only. Moreover, we show that this superconvergence property does not deteriorate as the thickness of the arch becomes extremely small. Numerical experiments verifying the abovementioned theoretical results are displayed. [ABSTRACT FROM AUTHOR]