A closed-form analytic correction to the Black–Scholes–Merton price for perpetual American options

This is a complementary study of a recent work by Yoon et al. (2013) [1] [J.-H. Yoon, J.-H. Kim, S.-Y. Choi, Multiscale analysis of a perpetual American option with the stochastic elasticity of variance, Appl. Math. Lett. 26 (7) (2013)] which excludes a certain level of the elasticity of variance. A second-order correction to the Black–Scholes option price and optimal exercise boundary for a perpetual American put option is made under the stochastic elasticity of variance of a risky asset. Contrary to the case of Yoon et al. (2013) [1], it is given by an explicit closed-form analytic expression so that one can access clearly the sensitivity of the option price and the optimal exercise boundary to changes in model parameters as well as the impact of the presence of a stochastic elasticity term on the option price and the optimal time to exercise.     

Basket options valuation for a local volatility jump–diffusion model with the asymptotic expansion method

In this paper we discuss the basket options valuation for a jump–diffusion model. The underlying asset prices follow some correlated local volatility diffusion processes with systematic jumps. We derive a forward partial integral differential equation (PIDE) for general stochastic processes and use the asymptotic expansion method to approximate the conditional expectation of the stochastic variance associated with the basket value process. The numerical tests show that the suggested method is fast and accurate in comparison with the Monte Carlo and other methods in most cases.     

Mapped WENO schemes based on a new smoothness indicator for Hamilton–Jacobi equations

In this paper, we introduce an improved version of mapped weighted essentially non-oscillatory (WENO) schemes for solving Hamilton–Jacobi equations. To this end, we first discuss new smoothness indicators for WENO construction. Then the new smoothness indicators are combined with the mapping function developed by Henrick et al. (2005) [31]. The proposed scheme yields fifth-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives. Numerical experiments are provided to demonstrate the performance of the proposed schemes on a variety of one-dimensional and two-dimensional problems.     

Exact and approximation algorithms for the min–max k-traveling salesmen problem on a tree

Given k identical salesmen, where k ? 2 is a constant independent of the input size, the min–max k-traveling salesmen problem on a tree is to determine a set of k tours for the salesmen to serve all customers that are located on a tree-shaped network, so that each tour starts from and returns to the root of the tree with the maximum total edge weight of the tours minimized. The problem is known to be NP-hard even when k = 2. In this paper, we have developed a pseudo-polynomial time exact algorithm for this problem with any constant k ? 2, closing a question that has remained open for a decade. Along with this, we have further developed a (1 + ?)-approximation algorithm for any ? > 0.     

An Oseen scheme for the conduction–convection equations based on a stabilized nonconforming method

An Oseen iterative scheme for the stationary conduction–convection equations based on a stabilized nonconforming finite element method is given. The stability and error estimates are analyzed, which show that the presented method is stable and has good precision. Numerical results are shown to support the developed theory analysis and demonstrate the good effectiveness of the given method.     

Exact solutions of a reaction–diffusion system for Proteus mirabilis bacterial colonies

A mathematical model for Proteus mirabilis colonies is considered in the framework of transformation groups. New solutions via classical and non-classical symmetries are obtained.     

Concept and prototype of a web tool for public–private project contracting based on a system dynamics model

Central European Journal of Operations Research - Contracting has a significant impact on the efficiency of acquisition processes, especially in the context of so-called public–private...     

Fast calculation based on a spatial two-grid finite element algorithm for a nonlinear space–time fractional diffusion model

In this article, a spatial two-grid finite element (TGFE) algorithm is used to solve a two-dimensional nonlinear space–time fractional diffusion model and improve the computational efficiency. First, the second-order backward difference scheme is used to formulate the time approximation, where the time-fractional derivative is approximated by the weighted and shifted Grünwald difference operator. In order to reduce the computation time of the standard FE method, a TGFE algorithm is developed. The specific algorithm is to iteratively solve a nonlinear system on the coarse grid and then to solve a linear system on the fine grid. We prove the scheme stability of the TGFE algorithm and derive a priori error estimate with the convergence result O(Δt² + h^{r + 1 − η} + H^{2r + 2 − 2η}) . Finally, through a two-dimensional numerical calculation, we improve the computational efficiency and reduce the computation time by the TGFE algorithm.     

An analytic algorithm for time fractional nonlinear reaction–diffusion equation based on a new iterative method

A new analytic algorithm for highly nonlinear time fractional reaction–diffusion equations is proposed in this paper. The proposed method is an amalgamation of variational iteration method (VIM), Adomian decomposition method (ADM) and further refined by introducing a new correction functional. This new correction functional is obtained from the standard correction functional of VIM by introducing an auxiliary parameter γ and an auxiliary function H(x) in it. Further, a sequence G_n(x, t), with suitably chosen support, is also introduced in the new correction functional. The algorithm is easy to implement and only four to six iterations are sufficient for fairly accurate solutions. The algorithm is tested on Fitzhugh – Nagumo and generalized Fisher equations with nonlinearity ranging from 2 to 5.     

Numerical methods based on the Floater–Hormann interpolants for stiff VIEs

Numerical Algorithms - In this paper, a normal S-iterative algorithm is studied and analyzed for solving a general class of variational inequalities involving a set of fixed points of nonexpansive...     

Levenberg–Marquardt method based on probabilistic Jacobian models for nonlinear equations

In this paper, we propose a Levenberg–Marquardt method based on probabilistic models for nonlinear equations for which the Jacobian cannot be computed accurately or the computation is very expensive. We introduce the definition of the first-order accurate probabilistic Jacobian model, and show how to construct such a model with sample points generated by standard Gaussian distribution. Under certain conditions, we prove that the proposed method converges to a first order stationary point with probability one. Numerical results show the efficiency of the method.

     

Exact solutions for a variable coefficient and Wick-type stochastic long–short wave resonance equation

A Wick-type stochastic long–short wave resonance equation is studied. Using white noise analysis, Hermite transform, we obtained a number of Wick versions of period-like and soliton-like solutions.     

Dynamic maintenance decision-making for series–parallel manufacturing system based on MAM–MTW methodology

Proper maintenance schedule is required to improve manufacturing systems’ profitability and productivity. A novel dynamic maintenance strategy is thus developed to incorporate both the single-machine optimization and the whole-system schedule for series–parallel system. Firstly, multiple attribute value theory and maintenance effects are considered in the single-machine optimization. A developed multi-attribute model (MAM) is used to determine the optimal maintenance intervals. Then, a series–parallel structure of the system is investigated in terms of the whole-system schedule. Maintenance time window (MTW) programming is presented to make a cost-effective system schedule by dynamically utilizing maintenance opportunities. The maintenance scheme achieved by using the proposed MAM–MTW methodology is demonstrated through a case study in a hydraulic steering factory. It is concluded that proper consideration of maintenance effects and time window leads to a significant cost reduction.     

A polynomial algorithm for a two parameter extension of Wythoff NIM based on the Perron–Frobenius theory

For any positive integer parameters a and b, Gurvich recently introduced a generalization mex _b of the standard minimum excludant mex = mex₁, along with a game NIM(a, b) that extends further Fraenkel’s NIM = NIM(a, 1), which in its turn is a generalization of the classical Wythoff NIM = NIM(1, 1). It was shown that P-positions (the kernel) of NIM(a, b) are given by the following recursion: $$x_n = {\rm mex}_b(\{x_i, y_i \;|\; 0 \leq i < n\}), \;\; y_n = x_n + an; \;\; n \geq 0,$$ and conjectured that for all a, b the limits ?(a, b) = x _n (a, b)/n exist and are irrational algebraic numbers. In this paper we prove that showing that ${\ell(a,b) = \frac{a}{r-1}}$ , where r > 1 is the Perron root of the polynomial $$P(z) = z^{b+1} - z - 1 - \sum_{i=1}^{a-1} z^{\lceil ib/a \rceil},$$ whenever a and b are coprime; furthermore, it is known that ?(ka, kb) = k?(a, b). In particular, ${\ell(a, 1) = \alpha_a = \frac{1}{2} (2 - a + \sqrt{a^2 + 4})}$ . In 1982, Fraenkel introduced the game NIM(a) = NIM(a, 1), obtained the above recursion and solved it explicitly getting ${x_n = \lfloor \alpha_a n \rfloor, \; y_n = x_n + an = \lfloor (\alpha_a + a) n \rfloor}$ . Here we provide a polynomial time algorithm based on the Perron–Frobenius theory solving game NIM(a, b), although we have no explicit formula for its kernel.     

Optimal error estimates of a decoupled scheme based on two-grid finite element for mixed Stokes–Darcy model

Although the numerical results suggest the optimal convergence order of the two-grid finite element decoupled scheme for mixed Stokes–Darcy model with Beavers–Joseph–Saffman interface condition in literatures, the numerical analysis only gets the optimal error order for porous media flow and a non-optimal error order that is half order lower than the optimal one in fluid flow. The purpose of this paper is to fill in the gap between the numerical results and the theoretical analysis.     

Development of an analytical–numerical solution for a steady and axisymmetric turbulent jet diffusion flame for the hydrogen based on a reduced kinetic mechanism

The hydrogen fuel is considered to be an ideal source of energy, because its complete combustion generates no pollutants, only water vapor. Therefore, the hydrogen has been suggested as a clean fuel. A detailed kinetic mechanism for the combustion of hydrogen that comprises eight species (_H2,_O2,O,OH,_H2O,H,_HO2$H_2\text{,}{O}_2\text{,}O\text{,}\mathit{OH}\text{,}{H}_{2}O\text{,}H\text{,}{\mathit{HO}}_2$ and _H2O2$H_2{O}_2$) and 20 elementary reactions, was reduced to two-step mechanism for nonpremixed flames involving four reactive species (_H2,_O2,H,_H2O$H_2\text{,}{O}_2\text{,}H\text{,}{H}_{2}O$). We performed, for this mechanism, a numerical analysis of the equations, including the velocity, mixture fraction, mass fractions and temperature. To quantify the components of intermediary reactions, the mixture fraction is decomposed into three parts, each part directly related to the mass fraction of each species. The results compare favorably with data in the literature for a jet diffusion flame of 50/50% in volume of _H2-_N2$H_2-N_2$. The main advantage of the strategy employed here is to decrease the work needed to solve the system of equations of the reactive flow.