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Abstract: Abstract In this paper, we study the valuation of a variable annuity embedding an early surrender option in which the underlying equity price follows the multiscale stochastic volatility model. Utilizing singular and regular perturbation techniques by Fouque et al. (in Multiscale Stochastic volatility for equity, interest rate, and credit derivatives. Cambridge University Press: Cambridge, 2011), we provide a closed-form solution of the valuation in an asymptotical sense. Comparison with another simulation-based method is presented to confirm the accuracy of our valuation methodology. We show that the fair insurance fee of the variable annuity with the surrender option far more decreases in an increase of the underlying equity price than the fair insurance fee of the annuity without the option does. The multiscale model’s conventional feature (i.e., the fast and slow time scale volatilities are influential in the short-term and long-term products, respectively) is observed in the fair insurance fee of the annuity without the surrender option. When the annuity contract embeds the surrender option, however, the effects of the fast scale volatility on the fair insurance fee becomes more remarkable. PubDate: 2022-05-02
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Abstract: Abstract In this paper, we investigate application of mathematical optimization to construction of a cubature formula on Wiener space, which is a weak approximation method of stochastic differential equations introduced by Lyons and Victoir (Proc R Soc Lond A 460:169–198, 2004). After giving a brief review on the cubature theory on Wiener space, we show that a cubature formula of general dimension and degree can be obtained through a Monte Carlo sampling and linear programming. This paper also includes an extension of stochastic Tchakaloff’s theorem, which technically yields the proof of our primary result. PubDate: 2022-05-01
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Abstract: Abstract Saddle point linear systems arise in many applications in computational sciences and engineering such as finite element approximations to Stokes problems, image reconstructions, tomography, genetics, statistics, and model order reductions for dynamical systems. In this paper, we present a least-squares approach to solve saddle point linear systems. The basic idea is to construct a projection matrix and transform a given saddle point linear system to a least-squares problem and then solve the least-squares problem by an iterative method such as LSMR: an iterative method for sparse least-squares problems. The proposed method rivals LSMR applied to the original problem in simplicity and ease to use. Numerical experiments demonstrate that the new iterative method is efficient and converges fast PubDate: 2022-04-10
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Abstract: Abstract We are concerned with the validity of the mean value theorem for the noncausal stochastic integral \(\int _s^t f(X_r)d_*W_r\) with respect to Brownian motion \(W_t(\omega )\) , where \(X_t\) is an Itô process. We establish first a mean value theorem for the noncausal stochastic integral \(\int _s^t f(X_r)dX_r\) and based on the result we show the corresponding formulae for the noncausal integral \(\int _s^t f(X_r)d_*W_r\) or for Itô integral \(\int _s^t f(X_r)d_0W_r\) as well. We also study the problem for such a genuin noncausal case where the process \(X_t\) is noncausal, that is, not adapted to the natural filtration associated to Brownian motion. The discussions are developed in the framework of the noncausal calculus. Hence some materials and basic facts in the theory of noncausal stochastic calculus are briefly reviewed as preliminary. PubDate: 2022-04-10
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Abstract: Abstract When solving the American options with or without dividends, numerical methods often obtain lower convergence rates if further treatment is not implemented even using high-order schemes. In this article, we present a fast and explicit fourth-order compact scheme for solving the free boundary options. In particular, the early exercise features with the asset option and option sensitivity are computed based on a coupled of nonlinear PDEs with fixed boundaries for which a high order analytical approximation is obtained. Furthermore, we implement a new treatment at the left boundary by introducing a third-order Robin boundary condition. Rather than computing the optimal exercise boundary from the analytical approximation, we simply obtain it from the asset option based on the linear relationship at the left boundary. As such, a high order convergence rate can be achieved. We validate by examples that the improvement at the left boundary yields a fourth-order convergence rate without further implementation of mesh refinement, Rannacher time-stepping, and/or smoothing of the initial condition. Furthermore, we extensively compare the performance of our present method with several 5(4) Runge–Kutta pairs and observe that Dormand and Prince and Bogacki and Shampine 5(4) pairs are faster and provide more accurate numerical solutions. Based on numerical results and comparison with other existing methods, we can validate that the present method is very fast and provides more accurate solutions with very coarse grids. PubDate: 2022-03-29
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Abstract: Abstract We propose a verification method for specification of homoclinic orbits as application of our previous work for constructing local Lyapunov functions by verified numerics. Our goal is to specify parameters appeared in the given systems of ordinary differential equations (ODEs) which admit homoclinic orbits to equilibria. Here we restrict ourselves to cases that each equilibrium is independent of parameters. The feature of our methods consists of Lyapunov functions, integration of ODEs by verified numerics, and Brouwer’s coincidence theorem on continuous mappings. Several techniques for constructing continuous mappings from a domain of parameter vectors to a region of the phase space are shown. We present numerical examples for problems in 3 and 4-dimensional cases. PubDate: 2022-03-28
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Abstract: Abstract Consider using the right-preconditioned GMRES (AB-GMRES) for obtaining the minimum-norm solution of inconsistent underdetermined systems of linear equations. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and ill-conditioned problems, the iterates may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very ill-conditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized GMRES based on solving the normal equations corresponding to the above triangular system using the standard Cholesky decomposition. This has the effect of shifting upwards the tiny singular values of the Hessenberg matrix which lead to an inaccurate solution. We analyze why the method works. Numerical experiments show that the proposed method is robust and efficient, not only for applying AB-GMRES to underdetermined systems, but also for applying GMRES to severely ill-conditioned range-symmetric systems of linear equations. PubDate: 2022-03-28
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Abstract: Abstract The soliton solution of the ultradiscrete BKP equation is obtained, via ultradiscretization of the generalized discrete BKP equation. It is shown that these solutions consist of three different kinds of soliton solutions for the ultradiscrete BKP equation. We also derive the soliton solution of the B-type box and ball system by taking the reduction of the ultradiscrete BKP equation and clarify the origin of the two types of soliton solutions of the B-type box and ball system. PubDate: 2022-03-18
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Abstract: Abstract For positive integers m, n, let \(\mathcal {I}_1=\{I_1,\ldots ,I_m\}\) be a given collection of face ideals in \(\mathbb {Z}[X_1,\ldots ,X_n]\) . This paper presents a progressive multi-branch recursive algorithm for finding the intersection of \(\mathcal {I}_1\) . Based on a constructive method, confluent complement ( \(\textsf {c-}\) )tuples are the new entities from which the monomials in the minimal system of generators of the intersection are formed. Solid superiority of a series of computer codes called ConFuL that supports the underlying theory in comparison with a leading computer algebra system is shown. Empirically, in a series of large scale examples, it is shown that ConFuL on average is at least 50 times faster. ConFuL successfully calculated two largest ever minimal generating sets of face ideals that are 5, 789, 900 and 6, 239, 350 in record time. The time complexity for core parts of the algorithms is shown to be linear polynomial-time in m. PubDate: 2022-03-02 DOI: 10.1007/s13160-022-00504-3
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Abstract: Abstract We propose an immersed hybrid difference (IHD) method for elliptic interface problems. An essential feature of the IHD method lies in the VR(virtual to real) transformation, which makes it possible to derive accurate finite difference approximations with functions of low regularity on interface cells. The VR transformation is consisting of the interface conditions in addition to the consistency equations, which are derived from the governing equation. The method is easy to be implemented and high order methods are conveniently derived. Numerical tests on several types of interfaces with low and high order methods are presented, which demonstrates efficiency of the IHD method. Numerical analysis for the one dimensional case is provided. PubDate: 2022-02-22 DOI: 10.1007/s13160-022-00503-4
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Abstract: Abstract A pursuit-evasion differential game of countably many pursuers and one evader is investigated. Integral constraints are imposed on the control functions of the players. Duration of the game is fixed, and the payoff functional is the greater lower bound of distances between the pursuers and evader when the game is completed. The pursuers want to minimize, and the evader to maximize the payoff. In this paper, we find the value of the game and construct optimal strategies for the players. PubDate: 2022-02-10 DOI: 10.1007/s13160-022-00501-6
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Abstract: Abstract In this paper, we propose a modified formal Lagrangian formulation by introducing dummy dependent variables and prove the existence of such a formulation for any system of differential equations. The corresponding Euler–Lagrange equations, consisting of the original system and its adjoint system about the dummy variables, reduce to the original system via a simple substitution for the dummy variables. The formulation is applied to study conservation laws of differential equations through Noether’s Theorem and in particular, a nontrivial conservation law of the Fornberg–Whitham equation is obtained by using its Lie point symmetries. Finally, a correspondence between conservation laws of the incompressible Euler equations and variational symmetries of the relevant modified formal Lagrangian is shown. PubDate: 2022-02-03 DOI: 10.1007/s13160-022-00500-7
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Abstract: Abstract Discrete Fenchel duality is one of the central issues in discrete convex analysis. The Fenchel-type min–max theorem for a pair of integer-valued M \(^{\natural }\) -convex functions generalizes the min–max formulas for polymatroid intersection and valuated matroid intersection. In this paper we establish a Fenchel-type min–max formula for a pair of integer-valued integrally convex and separable convex functions. Integrally convex functions constitute a fundamental function class in discrete convex analysis, including both M \(^{\natural }\) -convex functions and L \(^{\natural }\) -convex functions, whereas separable convex functions are characterized as those functions which are both M \(^{\natural }\) -convex and L \(^{\natural }\) -convex. The theorem is proved by revealing a kind of box integrality of subgradients of an integer-valued integrally convex function. The proof is based on the Fourier–Motzkin elimination. PubDate: 2022-02-02 DOI: 10.1007/s13160-022-00499-x
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Abstract: Abstract Theoretical analysis using mathematical models is often used to understand a mechanism of collective motion in a self-propelled system. In the experimental system using camphor disks, several kinds of characteristic motions have been observed due to the interaction of two camphor disks. In this paper, we understand the emergence mechanism of the motions caused by the interaction of two self-propelled bodies by analyzing the global bifurcation structure using the numerical bifurcation method for a mathematical model. Finally, it is also shown that the irregular motion, which is one of the characteristic motions, is chaotic motion and that it arises from periodic bifurcation phenomena and quasi-periodic motions due to torus bifurcation. PubDate: 2022-01-17 DOI: 10.1007/s13160-021-00498-4
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Abstract: Abstract This paper is concerned with the dynamic behaviour of a three species competitive-cooperative system with nonlocal dispersal, which describes two species are cooperating with each other and competing with the third species together. Firstly, by using the theory of monotone semiflows, we obtain the existence of bistable traveling wavefronts, which reflects that two cooperative species are invading the third one along the x-axis. We then investigate the global asymptotic stability of these wavefronts by applying a dynamical systems approach and constructing some suitable super-sub solutions. And we also obtain that such wavefronts are unique up to translation with the unique wave speed. In the end, we give the exact min-max representation of the unique wave speed by strict analysis. PubDate: 2022-01-07 DOI: 10.1007/s13160-021-00497-5
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Abstract: Abstract In this paper, we extend the Dashnic-Zusmanovic inclusion sets of matrices to the tensor case, which is proved to be always tighter than the Brauer-type inclusion sets of tensors introduced by Bu et al. (Linear Algebra Appl 512:234–248, 2017). As applications, we presented that, a DZ-type Z-tensor with nonnegative diagonal entries a strong M-tensor, and an even order DZ-type symmetric tensor with positive diagonal entries is positive definite. PubDate: 2022-01-01 DOI: 10.1007/s13160-021-00482-y
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Abstract: Abstract Let \(\mathcal {G}=(\mathcal {V}_{\mathcal {G}},\mathcal {E}_{\mathcal {G}})\) be a connected graph with n vertices. The eccentricity \(ec_{\mathcal {G}}(w)\) of a vertex w in \(\mathcal {G}\) is the maximum distance between w and any other vertex of \(\mathcal {G}\) . The total eccentricity index \(\tau (\mathcal {G})\) of \(\mathcal {G}\) is defined as \(\tau (\mathcal {G})=\sum \nolimits _{w\in \mathcal {V}_{\mathcal {G}}} ec_{\mathcal {G}}(w)\) . In this paper, we derive the trees with minimum and maximum total eccentricity index among the class of n-vertex trees with p pendent vertices. We also determine the trees with minimum and maximum total eccentricity index among the class of n-vertex trees with a given diameter. PubDate: 2021-11-23 DOI: 10.1007/s13160-021-00494-8
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Abstract: Abstract The paper studies asymptotic infinite-time ruin probabilities for a bidimensional time-dependent risk model, in which two insurance companies divide between them both the premium income and the aggregate claims in different positive proportions (modeling an insurer–reinsurer scenario, where the reinsurer takes over a proportion of the insurer’s losses). In the model, the claim sizes and the inter-arrival times correspondingly form a sequence of independent and identically distributed random vectors, where each pair of the vectors follows the time-dependence structure. Under the assumption that the claim sizes have consistently varying tails, asymptotic formulas for two kinds of infinite-time ruin probabilities are derived. PubDate: 2021-10-07 DOI: 10.1007/s13160-021-00487-7
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Abstract: Abstract In this paper, improved algorithms are proposed for preconditioned bi-Lanczos-type methods with residual norm minimization for the stable solution of systems of linear equations. In particular, preconditioned algorithms pertaining to the bi-conjugate gradient stabilized method (BiCGStab) and the generalized product-type method based on the BiCG (GPBiCG) have been improved. These algorithms are more stable compared to conventional alternatives. Further, a stopping criterion changeover is proposed for use with these improved algorithms. This results in higher accuracy (lower true relative error) compared to the case where no changeover is done. Numerical results confirm the improvements with respect to the preconditioned BiCGStab, the preconditioned GPBiCG, and stopping criterion changeover. These improvements could potentially be applied to other preconditioned algorithms based on bi-Lanczos-type methods. PubDate: 2021-09-02 DOI: 10.1007/s13160-021-00480-0
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