Closed-form Formula Research Articles

A closed form formula for equity valuation model based on differential equation

A closed form formula for equity valuation model based on differential equation

Secure communication with a multiple-antennas untrusted relay in presence of randomly distributed adversary jammers

In this article, we examine the achievable secrecy rate of a cooperative network including one source, one destination, and one untrusted amplify-and-forward relay, where randomly located adversary jammers attempt to disrupt the wireless communications. The relay has multiple antennas, while the other nodes have a single antenna. The adversary jammers are placed following a Poisson Point Process (PPP) and aim to interfere with the untrusted relay's channel. The relay employs maximal-ratio combining (MRC) to mitigate the jammers' impact and retransmits the signal to the destination using maximum ratio transmission (MRT). To prevent the relay from capturing the message, the destination injects pre-known artificial noise, known as destination-assisted cooperative jamming. We present a closed-form solution for the Ergodic secrecy rate (ESR) of the system with Rayleigh fading channels. An optimization problem for maximizing the secrecy rate is also designed, resulting in a closed-form formula for optimal power allocation (OPA) between the source and destination.

Optimal designs using generalized estimating equations in cluster randomized crossover and stepped wedge trials.

Cluster randomized crossover and stepped wedge cluster randomized trials are two types of longitudinal cluster randomized trials that leverage both the within- and between-cluster comparisons to estimate the treatment effect and are increasingly used in healthcare delivery and implementation science research. While the variance expressions of estimated treatment effect have been previously developed from the method of generalized estimating equations for analyzing cluster randomized crossover trials and stepped wedge cluster randomized trials, little guidance has been provided for optimal designs to ensure maximum efficiency. Here, an optimal design refers to the combination of optimal cluster-period size and optimal number of clusters that provide the smallest variance of the treatment effect estimator or maximum efficiency under a fixed total budget. In this work, we develop optimal designs for multiple-period cluster randomized crossover trials and stepped wedge cluster randomized trials with continuous outcomes, including both closed-cohort and repeated cross-sectional sampling schemes. Local optimal design algorithms are proposed when the correlation parameters in the working correlation structure are known. MaxiMin optimal design algorithms are proposed when the exact values are unavailable, but investigators may specify a range of correlation values. The closed-form formulae of local optimal design and MaxiMin optimal design are derived for multiple-period cluster randomized crossover trials, where the cluster-period size and number of clusters are decimal. The decimal estimates from closed-form formulae can then be used to investigate the performances of integer estimates from local optimal design and MaxiMin optimal design algorithms. One unique contribution from this work, compared to the previous optimal design research, is that we adopt constrained optimization techniques to obtain integer estimates under the MaxiMin optimal design. To assist practical implementation, we also develop four SAS macros to find local optimal designs and MaxiMin optimal designs.

The revised Riemann–Hilbert approach to the Kaup–Newell equation with a non-vanishing boundary condition: Simple poles and higher-order poles

With a non-vanishing boundary condition, we study the Kaup–Newell (KN) equation (or the derivative nonlinear Schrödinger equation) using the Riemann–Hilbert approach. Our study yields four types of Nth order solutions of the KN equation that corresponding to simple poles on or not on the ρ circle (ρ related to the non-vanishing boundary condition), and higher-order poles on or not on the ρ circle of the Riemann–Hilbert problem (RHP). We make revisions to the usual RHP by introducing an integral factor that ensures the RHP satisfies the normalization condition. This is important because the Jost solutions go to an integral factor rather than the unit matrix when the spectral parameter goes to infinity. To consider the cases of higher-order poles, we study the parallelization conditions between the Jost solutions without assuming that the potential has compact support, and present the generalizations of residue conditions of the RHP, which play crucial roles in solving the RHP with higher-order poles. We provide explicit closed-form formulae for four types of Nth order solutions, display the explicit first-order and double-pole solitons as examples and study their properties in more detail, including amplitude, width, and exciting collisions.

A new defuzzification method and ranking method for type-2 fuzzy numbers

The type-2 fuzzy number (T2 FN) is a special type-2 fuzzy set whose all secondary membership functions are fuzzy numbers. This paper proposes a new defuzzification method and ranking method for T2 FNs. The defuzzification method can be expressed in a closed-form formula or can be divided into two computational steps. The first step computes the centroid of each secondary membership function to obtain a type-reduced set using the method derived from generalizing the existing alpha-cut defuzzification methods by introducing a parameter that controls the importance of secondary membership degrees, and the second step calculates the centroid of the type-reduced set. The defuzzified values are used to provide position information of T2 FNs. Secondly, we define a function to calculate the area enclosed by the image of the type-reduced set and the coordinate axes, and prove that this function is continuous. The values of this function are used to provide shape information of T2 FNs. Finally, to rank T2 FNs, we define a family of quasi-orders based on their position and shape information, and illustrate the validity and rationality of these quasi-orders through examples.

European Call Option under Stochastic Interest Rate in a Fractional Brownian Motion with Transaction Cost

This paper deals on the valuation of European call option price in a stochastic environment by employing three factors which are the stochastic model of the asset value, the stochastic interest rate and the transaction cost. We specify that our underlying asset and the stochastic interest rate, particularly Hull-White model, follows a fractional Brownian Motion governed by Hurst parameter H. We used the hedging and replicating technique to established the zero-coupon bond on the European option. Finally, we give a closed-form formula of the European call option price.

Vibrations in piezothermoelastic micro-/nanobeam with voids utilizing modified couple stress theory

The present study examines the vibrations in micro- and nano-piezothermoelastic beams with voids. In this work, the modified couple stress theory is employed to generate closed-form formulas for deflection, volume fraction, electric potential and temperature distribution. The effects of voids, modified couple stress, beam dimensions and electric potential on energy dissipation (thermoelastic damping) are examined for beams under various boundary conditions. We have established analytical formulas for frequency shift, attenuation and thermoelastic damping. MATLAB software is used to get numerical results and exhibit them visually.

Valuing American options using multi-step rebate options

The determination of optimal exercise boundaries is a critical aspect of pricing American options, which often requires costly numerical methods. This paper proposes a new approach that employs multi-step rebate options to approximate American option prices. Since the rebate options offer payoffs when the multi-step boundaries are touched, the prices of American options are estimated by maximizing the multi-step rebate option prices, and the optimal multi-step barriers replace the true optimal exercise boundaries. To this end, the closed-form pricing formulas for multi-step rebate options are derived and utilized to approximate several American option prices. Through extensive numerical experiments, we demonstrate the validity and performance of our approach.

Inexact Fixed-Point Proximity Algorithm for the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} Sparse Regularization Problem

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} norm. Specifically, the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} norm regularization term. Such an ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} models found by the proposed inexact algorithm outperform global minimizers of the corresponding ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document} models, in terms of approximation accuracy and sparsity of the solutions.

Second-order reliability analysis of an energy pile with CPT data

As pile foundations and ground source heat producers, energy piles are becoming more and more popular. Even after much research, engineers find their design to be challenging. An effective way for using the Second-Order Reliability method (SORM) in foundation engineering scenarios including explicit limit state functions is shown in this article. The suggested SORM process is simple to implement in a spreadsheet and is based on an approximate paraboloid fitted to the limit state surface close to the design point. Using well-established closed-form SORM formulae, the failure probability is automatically determined using the primary curvatures of the limit state surface and the reliability index. An energy pile foundation engineering design examples are analyzed using the proposed method and discussed. Comparison with FORM, SORM and Genetic Programming (GP)-based FORM and SORM are made. In the case of determination of group capacity of pile and thermal load the input random variables are qc0, qc1, qc2, fs and E. The reliability index and probability of failure value are found to be 9.34 for FORM, 9.55 for SORM, 9.24 for GP-based FORM, and 9.45 for GP-based SORM. The probability of failure value for FORM 4.60E-21, SORM 3.24E-21, GP-based FORM 1.20E-20, and GP-based SORM is 8.50E-21. The relative percentage error for the average second-order approximation corresponding to FORM is also calculated, and found that Pf2 is 29.48 % in the case of simple SORM and 29.15 % in the case of GP-based SORM. The Breitung provides the most accurate estimation, and the Cai_Elishakoff provides the least accurate estimate.

Pricing airbag option via first passage time approach

Airbag option is a new structured product that has been highly favored by investors in the over-the-counter derivatives market. It provides protections against the downside loss to investors like the ‘airbag’ in a car crash. In contrast with its popularity, there is little literature on the pricing of airbag options, especially from the mathematical finance point of view. In this paper, we study the pricing of airbag options using the first passage time approach. We propose two innovative new types of airbag options, namely airbag-TB and airbag-ER, which enhance the downside loss protections by integrating a time-varying barrier and early redemption features respectively in the payoffs. Explicit and closed-form pricing formulas for the original and the two new types of airbag options are derived under the mixed-exponential jump diffusion (MEJD) model, double exponential jump diffusion (DEJD) model and Black-Scholes (BS) model. In doing so, we significantly improve the Hybrid model proposed by Jaimungal and Sigloch (Incorporating risk and ambiguity aversion into a hybrid model of default. Math. Finance, 2012, 22, 57–81) by incorporating a stochastic redemption intensity model and more general underlying asset dynamics. We further demonstrate the performance of our pricing formulas and the two new types of airbag options with numerical examples based on Monte Carlo simulation, which verifies the accuracy and efficiency of our pricing formulas.

Microstructural and transport characteristics of triply periodic bicontinuous materials

Three-dimensional (3D) bicontinuous two-phase materials are increasingly gaining interest because of their unique multifunctional characteristics and advancements in techniques to fabricate them. Because of their complex topological and structural properties, it still has been nontrivial to develop explicit microstructure-dependent formulas to predict accurately their physical properties. A primary goal of the present paper is to ascertain various microstructural and transport characteristics of five different models of triply periodic bicontinuous porous materials at a porosity ϕ1=1/2: those in which the two-phase interfaces are the Schwarz P, Schwarz D and Schoen G minimal surfaces as well as two different pore-channel structures. We ascertain their spectral densities, pore-size distribution functions, local volume-fraction variances, and hyperuniformity order metrics and then use this information to estimate certain effective steady-state as well as time-dependent transport properties via closed-form microstructure–property formulas. Specifically, the recently introduced time-dependent diffusion spreadability is determined exactly from the spectral density. Moreover, we accurately estimate the fluid permeability of such porous materials from a closed-form formula that depends on the second moment of the pore-size function and the formation factor, a measure of the tortuosity of the pore space, which is exactly obtained for the three minimal-surface structures. We also rigorously bound the permeability from above using the spectral density. For the five models with identical cubic unit cells, we find that the permeability, inverse of the specific surface, hyperuniformity order metric, pore-size second moment and long-time spreadability behavior are all positively correlated and rank order the structures in exactly the same way. We also conjecture what structures maximize the fluid permeability for arbitrary porosities and show that this conjecture must be true in the extreme porosity limits by identifying the corresponding optimal structures.

Sedimentation of a Charged Soft Sphere within a Charged Spherical Cavity.

The sedimentation of a soft particle composed of an uncharged hard sphere core and a charged porous surface layer inside a concentric charged spherical cavity full of a symmetric electrolyte solution is analyzed in a quasi-steady state. By using a regular perturbation method with small fixed charge densities of the soft sphere and cavity wall, a set of linearized electrokinetic equations relevant to the fluid velocity field, electrical potential profile, and ionic electrochemical potential energy distributions are solved. A closed-form formula for the sedimentation velocity of the soft sphere is obtained as a function of the ratios of core-to-particle radii, particle-to-cavity radii, particle radius-to-Debye screening length, and particle radius-to-porous layer permeation length. The existence of the surface charge on the cavity wall increases the settling velocity of the charged soft sphere, principally because of the electroosmotic enhancement of fluid recirculation within the cavity induced by the sedimentation potential gradient. When the porous layer space charge and cavity wall surface charge have the same sign, the particle velocity is generally enhanced by the presence of the cavity. When these fixed charges have opposite signs, the particle velocity will be enhanced/reduced by the presence of the cavity if the wall surface charge density is sufficiently large/small relative to the porous layer space charge density in magnitude. The effect of the wall surface charge on the sedimentation of the soft sphere increases with decreases in the ratios of core-to-particle radii, particle-to-cavity radii, and particle radius-to-porous layer permeation length but is not a monotonic function of the ratio of particle radius-to-Debye length.

Modeling the measurement accuracy of one-dimensional boundary subsets in digital image correlation

Boundary subsets, which contain unwanted pixels from background image or other regions, generally occur at areas of vital mechanical importance. In this work, we establish a theoretical model characterizing the measurement accuracy of 1-dimensional boundary subsets, which explicitly expresses the retrieved displacement in terms of subset size, invalid pixels, shape function order, and underlying deformation field. Particularly, the transfer function of digital image correlation at boundary subsets is derived and verified, indicating possible amplification of amplitude and the existence of phase shift, which are quite different from standard full square subsets. Finally, the retrieved deformation near boundary is theoretically analyzed, and closed-form formulae are deduced for polynomial underlying displacements in consideration of the Weierstrass approximation theorem. To the author's knowledge, this is the first general mathematical model for boundary subsets, and paves the way for enhanced measurement accuracy.

Secrecy performance analysis of a IRS-assisted underwater optical wireless communication system

In response to the dearth of radio frequency (RF) equivalents, there has been a recent surge in interest in optical wireless communication in underwater environments. To ensure a strong line-of-sight (LOS) connection, the intelligent reflecting surface (IRS) is installed to create a virtual LOS. Then, the first part of this study looks into the security of underwater optical wireless communication (UOWC) in relation to a number of real-world phenomena, including oceanic propagation loss, oceanic turbulence, and IRS-induced geometric loss. Then, a wiretap channel with three authorized users - a reputable broadcaster named Alice (the submarine), a law-abiding user named Bob, and an eavesdropper named Eve - is examined over turbulent channels that exhibit the Log-normal distribution. Furthermore, our study derives the closed-form formulas for the secrecy performance measures, secrecy outage probability, and secrecy throughput. Finally, the numerical results show how the impact of oceanic turbulence-induced fading and distance between Bob’s and Eve’s positions on the secrecy system performance.

Critical velocities of a three-layer composite tube incorporating the rotary inertia and material anisotropy

Critical velocities of a three-layer composite tube subjected to a uniform internal pressure moving at a constant velocity are obtained in closed-form expressions. A Love–Kirchhoff thin shell model including the rotary inertia and material anisotropy effects is used in the formulation. The composite tube is made of three perfectly bonded cylindrical layers of dissimilar materials, each of which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form formulas for the critical velocities are first derived for the general case by incorporating the effects of material anisotropy, rotary inertia and radial stress. Specific formulas are then obtained for composite tubes without the rotary inertia effect and/or the radial stress effect and with various types of material symmetry for each layer as special cases. It is also shown that the current model for three-layer tubes can be reduced to those for single- and two-layer tubes. To illustrate the newly derived formulas, an example is provided for a composite tube consisting of an isotropic inner layer, an orthotropic core, and an isotropic outer layer. All four critical velocities of the composite tube are computed using the new closed-form formulas. Three values of the lowest critical velocity of the three-layer composite tube are analytically obtained from three sets of the new formulas, which agree well with the value computationally determined by others.

Diffusion LMS algorithm in the presence of second order nonlinearities with theoretical bounds

In this paper, we propose an algorithm for the estimation and compensation of second-order nonlinearity in Wireless Sensor Networks (WSNs) within a distributed estimation framework. First, we investigate the impact of second-order nonlinearity on the performance of the Diffusion Least Mean Square (DLMS) algorithm and derive an upper bound for the l2-norm of the error caused by nonlinearity. Next, we suggest a distributed algorithm incorporating additional nonlinearity estimation and compensation units. Furthermore, considering second-order nonlinearity, we calculate the Cramer-Rao Bound for estimating both the unknown vector and the nonlinearity coefficient vector, with the Fisher information matrix derived in a closed-form formula. Simulation results demonstrate the effectiveness of the proposed algorithm in enhancing the performance of distributed estimation in the presence of nonlinear sensors in a WSN.

Generalized Moment Estimators Based on Stein Identities

For parameter estimation of continuous and discrete distributions, we propose a generalization of the method of moments (MM), where Stein identities are utilized for improved estimation performance. The construction of these Stein-type MM-estimators makes use of a weight function as implied by an appropriate form of the Stein identity. Our general approach as well as potential benefits thereof are first illustrated by the simple example of the exponential distribution. Afterward, we investigate the more sophisticated two-parameter inverse Gaussian distribution and the two-parameter negative-binomial distribution in great detail, together with illustrative real-world data examples. Given an appropriate choice of the respective weight functions, their Stein-MM estimators, which are defined by simple closed-form formulas and allow for closed-form asymptotic computations, exhibit a better performance regarding bias and mean squared error than competing estimators.

On Pólya’s random walk constants

A celebrated result in probability theory is that a simple symmetric random walk on the d d -dimensional lattice Z d \mathbb {Z}^d is recurrent for d = 1 , 2 d=1,2 and transient for d ≥ 3 d\geq 3 . In this note, we derive a closed-form expression, in terms of the Lauricella function F C F_C , for the return probability for all d ≥ 3 d\geq 3 . Previously, a closed-form formula had only been available for d = 3 d=3 .

Steady-state flow under surface recharge through unconfined aquifer with depth-decaying hydraulic conductivity: Analytical results

This paper presents some analytical results related to groundwater flow through an unconfined horizontal aquifer induced by areal recharge on the top and a decaying hydraulic conductivity. Under steady-state conditions, and in the context of Boussinesq equation, exact analytical solutions describing drainage into one or two penetrating channels at the aquifer boundaries have been derived in closed forms. The analytical expressions have been used to investigate the existence of water divide inside the unconfined aquifer. Based on the derived analytical solutions, a closed-form analytical formula is developed for calculating groundwater transit time within Dupuit-type flow system. The analytical transit time expression is simply expressed in terms of common mathematical functions. Analytical results are verified against a numerical model and show excellent agreements. The analytical transit time is validated against a numerical one obtained from a two-dimensional model representing a rectangular unconfined aquifer. Based on the explicit closed-form solution, a fully analytical inverse problem to estimate the decay parameter is proposed using a least-square objective function. The solution of the inverse problem simply corresponds to the solution of a nonlinear algebraic equation obtained from the definition of the objective function and the form of the analytical solution.