Closed-form Solution Research Articles

Deep learning of optimal exercise boundaries for American options

Efficiently determining a price and optimal exercise boundary for an American option is a critical subject in the financial sector. This study introduces a novel application of long short-term memory neural networks to solve a relevant Volterra equation, enhancing the accuracy and efficiency of American option pricing. The proposed approach outperforms traditional numerical techniques, including finite difference methods, binomial trees, and Monte Carlo methods, delivering an impressive speed improvement by a factor of thousands while maintaining industry-accepted accuracy levels. It exhibits computational speeds up to about a hundred times faster than the state-of-the-art method used by Andersen et al. [High-performance American option pricing, J. Comput. Finance 20(1) (2016), pp. 39–87] when evaluating numerous options. The proposed network, trained on a reasonable range of parameters related to the Black–Scholes model, swiftly determines option prices and exercise boundaries, serving as a practical closed-form solution.

BIEM_CH: An Efficient Algorithm for Dynamic Rupture Simulation of Complex Fault Systems with Unstructured Meshes and Half-Space Green’s Function

Abstract In this study, a fast 3D dynamic rupture simulation algorithm, named BIEM_CH (Boundary Integral Equation Method for Complex fault systems in Half-space), is presented. This algorithm, based on exact half-space Green’s functions, supports both structured and unstructured discretization schemes, allowing for the effective handling of a wide range of fault geometries, from simple to complex. Because of the semianalytical nature of the boundary integral equation method, the integral kernel (surface integral of the spatial derivatives of Green’s functions) and rupture processes can be computed separately, making BIEM_CH particularly suitable for applications requiring numerous forward simulations, such as dynamic source inversion, for which only the rupture process needs to be recalculated once the integral kernel is obtained. The performance of the algorithm has been significantly enhanced, achieving up to a hundredfold speed increase through the use of exact closed-form solutions for the time-domain half-space Green’s function and matrix operations leveraging graphical processing unit acceleration, resulting in dynamic rupture simulations that can be completed in a matter of seconds. Moreover, BIEM_CH maintains excellent stability when the mesh dimension does not exceed 375 m, irrespective of whether structured or unstructured discretization schemes are used. In addition, this algorithm demonstrates good agreement with other methods in benchmark exercises conducted by the Southern California Earthquake Center and the U.S. Geological Survey’s dynamic rupture code verification project.

Single machine minsum and minmax common due-date assignment and scheduling problems involving general position-dependent workloads

This study addresses the well-known scheduling and assignment problem with a common due date. Four cost components are assumed, i.e., earliness, tardiness, the cost of delaying the due date, and extended total completion time. The minsum and the minmax versions of the fundamental problem are solved. For each of the major versions of the classic scheduling theory assuming jobs with position-independent processing times, a detailed analysis is provided, and, consequently, the properties of an optimal solution and a closed-form solution. These elementary results lay the foundation for two extensions. The first extension is for jobs with variable processing times and focuses on general position-dependent processing times. The second extension considers the recently introduced convex resource allocation method with general position-dependent workloads and continuous non-renewable resource. All studied problems are shown to be solved in polynomial time, such that the computational complexity of the minsum and minmax position-independent processing time variants are O(n log⁡n ) and O(n), respectively, and the computational complexity of the variants involving variable processing times is O(n^3 ).

Quantum Kernel Machine Learning With Continuous Variables

The popular qubit framework has dominated recent work on quantum kernel machine learning, with results characterising expressivity, learnability and generalisation. As yet, there is no comparative framework to understand these concepts for continuous variable (CV) quantum computing platforms. In this paper we represent CV quantum kernels as closed form functions and use this representation to provide several important theoretical insights. We derive a general closed form solution for all CV quantum kernels and show every such kernel can be expressed as the product of a Gaussian and an algebraic function of the parameters of the feature map. Furthermore, in the multi-mode case, we present quantification of a quantum-classical separation for all quantum kernels via a hierarchical notion of the “stellar rank" of the quantum kernel feature map. We then prove kernels defined by feature maps of infinite stellar rank, such as GKP-state encodings, can be approximated arbitrarily well by kernels defined by feature maps of finite stellar rank. Finally, we simulate learning with a single-mode displaced Fock state encoding and show that (i) accuracy on our specific task (an annular data set) increases with stellar rank, (ii) for underfit models, accuracy can be improved by increasing a bandwidth hyperparameter, and (iii) for noisy data that is overfit, decreasing the bandwidth will improve generalisation but does so at the cost of effective stellar rank.

Further study on the RCB-like steering vector estimation method: closed-form solution derivation via accurate Lagrange multiplier calculation

Further study on the RCB-like steering vector estimation method: closed-form solution derivation via accurate Lagrange multiplier calculation

Two-Dimensional Numerical Method for Predicting the Resistance of Ships in Pack Ice: Development and Validation

This study presents a 2D numerical simulation method for predicting the resistance of ships navigating in pack ice. The key contribution of this study lies in the derivation of analytical closed-form solutions for calculating the flexural deformation and stress distribution in an elastic plate using Symplectic Mechanics and Hooke’s laws. These solutions are used to determine the failure mode of ice floes. Linear Elastic Fracture Mechanics (LEFMs) and the weight function method are utilized to analyze crack initiation, propagation, and fracture. Ice is broken when a crack propagates to 14.5% of the ice length. The compressive strength of ice and the contact area are used to calculate the ice load. A collision method was developed based on the Sweep and Prune (SAP) and Gilbert–Johnson–Keerthi (GJK) algorithms. A program for predicting the resistance of ships navigating in pack ice was developed based on MATLAB and the aforementioned theories. The navigation resistance of RV Xuelong at different ice concentrations and speeds was simulated and compared with the model test results from an ice tank. The comparison shows that the simulation results are consistent with the test results, with an average error of 9.05%, indicating the effectiveness and reliability of this numerical method. This study lays a solid foundation for future research on autonomous ship navigation in pack ice.

Electrical contact with dielectric breakdown of interfacial gap

Abstract Electrical contact is fundamental to almost every aspect of modern industry including the fast-growing electric vehicle industry. In metallic contacts in atmospheric conditions, most of the electrical current passes via the micro-junctions formed between two electrodes. The classic electrical contact theory predicts an infinite current density at the circular contact periphery. In the present work, we explore the influence of the dielectric breakdown of air outside the contact area on the electrical contact interface. Incorporating the discharging boundary condition governed by the modified Paschen law, we develop the numerical model as well as two sets of closed-form solutions for low applied voltage cases where two electrodes are in solid-solid contact and complete separation, respectively. For Hertzian contact, the present work theoretically proves that the ignorance of discharge can lead to a singular current density at the contact periphery and an overestimation of the electrical contact resistance. The current density monotonically increases along the radial direction to a finite value at the contact area periphery and followed by a monotonic drop within the discharge zone. The present study serves as a foundation for the modeling of discharging rough surface electrical contact and sheds light on the machine element surface damages caused by the electrical discharge machining.

Model predictive control based on single-phase shift modulation for triple active bridge DC-DC converter

The triple-active bridge (TAB) converter is widely used in various applications due to its high efficiency and power density. However, the high-frequency (HF) transformer coupling between the ports presents challenges for controller design. This article presents a model predictive control (MPC) approach based on single-phase shift modulation for the TAB converter. The developed MPC offers improved transient performance, control flexibility, and precision, ensuring compliance with DC voltage regulations and achieving optimal solutions for port decoupling. The MPC utilizes a cost function to provide robust voltage regulation, and an algorithm based on Karush-Kuhn-Tucker (KKT) conditions is developed to derive closed-form solutions for optimal control parameters. To validate the performance of the TAB converter with the proposed MPC control, Typhoon 602 hardware-in-loop (HIL) experimental case study is conducted. Additionally, a comparison with previous works is carried out to confirm the effectiveness of the proposed method. The results of the HIL experimental setup and the comparative analysis demonstrate that the developed method is effective, providing faster dynamic characteristics and port power decoupling operation capability.

An optimization-based equilibrium measure describing fixed points of non-equilibrium dynamics: application to the edge of chaos

Understanding neural dynamics is a central topic in machine learning, non-linear physics, and neuroscience. However, the dynamics are non-linear, stochastic and particularly non-gradient, i.e., the driving force cannot be written as the gradient of a potential. These features make analytic studies very challenging. The common tool is the path integral approach or dynamical mean-field theory. Still, the drawback is that one has to solve the integro-differential or dynamical mean-field equations, which is computationally expensive and has no closed-form solutions in general. From the associated Fokker–Planck equation, the steady-state solution is generally unknown. Here, we treat searching for the fixed points as an optimization problem, and construct an approximate potential related to the speed of the dynamics, and find that searching for the ground state of this potential is equivalent to running approximate stochastic gradient dynamics or Langevin dynamics. Only in the zero temperature limit, can the distribution of the original fixed points be achieved. The resultant stationary state of the dynamics exactly follows the canonical Boltzmann measure. Within this framework, the quenched disorder intrinsic in the neural networks can be averaged out by applying the replica method, which leads naturally to order parameters for the non-equilibrium steady states. Our theory reproduces the well-known result of edge-of-chaos. Furthermore, the order parameters characterizing the continuous transition are derived, and the order parameters are explained as fluctuations and responses of the steady states. Our method thus opens the door to analytically studying the fixed-point landscape of the deterministic or stochastic high dimensional dynamics.

Dynamic Response to Moving Load of Prestressed Damped Shear Beam Resting on Bi-Parametric Elastic Foundation

The dynamic behaviour of damped shear beam resting on bi-parametric elastic foundation when traversed by moving load travelling at constant velocity is investigated in this present study. The beam-type structure has a uniform cross-sectional area and it is assumed to be simply supported. The governing equations are coupled second order partial differential equations. The method of integral transformation called Finite Fourier series was first used to reduce the sets of coupled second order partial differential equations governing the motion of this class of dynamical problem to sequence of coupled second order ordinary differential equations. Thereafter, the simplified coupled equations describing the motion of the beam-load system were then solved by Laplace transformation in conjunction with convolution theory to obtain the solution. The closed form solution obtained was analyzed to obtain the conditions under which the beam-load system will experience resonance phenomenon and speeds at which this may occur are also established. The effects of pertinent structural parameters on the response of a damped shear beam when under the action of the moving load were presented in plotted curves. From the graphs, it is interestingly found that increase in the values of vital structural parameters, such as axial force Nf, circular frequency CF, foundation stiffness K and shear modulus G, reduces the transverse displacement of the damped shear beam when under the action of the moving load. Practically speaking, increase in the values of these structural parameters significantly enhances the stability of the beam and increases the critical speed of the dynamical system. Consequently, the resonance risk of the vibrating system is reduced and thus the safety of the occupant of this structural member is guaranteed.

Analysis of the dynamic responses of a piezothermoelastic microbeam resonator under dual-phase-lag heat conduction

Thermoelastic vibration responses of a piezothermoelastic (PTE) microbeam resonator have been investigated in the present work under simply supported boundary conditions. The basic governing equations for a transversely isotropic piezothermoelastic solid are formulated by assuming the Euler–Bernoulli beam theory with hyperbolic dual-phase-lag (DPL) heat conduction model. Later on, non-dimensionalization of the variables have been carried out for simplicity to obtain the set of coupled governing equations for thermal deflection, electric field, and thermal moment. Initial and boundary conditions are defined to solve for the non-dimensional thermal deflection and thermal moment by using Laplace transform technique along with the finite Fourier sine transform. Closed-form solutions of the problem are obtained. Moreover, an attempt is made to illustrate the analytical results through numerical simulations by taking a PTE material of Lead Zirconate Titanate (PZT-5A). The influence of the quadratic term of the phase-lag parameter of heal flux in the DPL model is investigated to compare the effects of phase-lags under parabolic and hyperbolic DPL models. In addition, the long-term and short-term vibration responses of thermal moment and thermal deflection are analyzed in detail to observe the impact of different factors which characterize the piezoelectric phenomena and dual-phase-lag effects. Comparative analysis of different well-known models for responses of thermal moment and thermal deflection is also presented for short- and long-time ranges. Some important observations about how piezoelectric parameters, dual-phase-lag parameters, and beam parameters affect the behavior of piezothermoelastic vibrations have been highlighted.

Modeling and Analysis of Multiconductor Transmission Lines by Distributed Transfer Function Method

Multiconductor transmission lines (MCTLs) have various applications in electrical engineering. In modeling and analysis of MCTLs, which are described by a set of coupled partial differential equations, most existing methods rely on approximation or discretization. For high-speed high-frequency applications, accurate analytical methods are always desirable. Such a method, however, is not currently available for complex MCTLs. This paper presents an innovative analytical method for modeling and analysis of MCTLs with various configurations. The proposed method, which is called the Distributed Transfer Function Method, is capable of delivering closed-form analytical solutions for complex MCTLs, in both the frequency domain and the time domain. One highlight of the proposed method is that it gives exact and closed-form solutions for branched transmission lines for the first time. The accuracy, efficiency, and high-frequency utility of the method is demonstrated in numerical examples.

Contingent convertible Ijara pricing model

Purpose This study aims to conceive and develop a pricing model for the Ijara contingent convertible contract (ICCC, hereafter), considering the possibility that the lessee may default. The ICCC model grants the lessor the option of converting the unpaid amount into equity or recovering the leased equipment and selling it at market price in case of financial distress. Design/methodology/approach The ICCC is consistent with the profit-sharing approach and the new risk management techniques, which are compatible with Islamic philosophy. Relying on real options theory and the contingent claim approach, a closed-form solution of the firm’s assets is developed in a dynamic environment, where the rate of return is generated by a Cox-Ingersoll-Ross stochastic process. Findings Examining the numerical analysis reveals the impact of the firm value, the conversion or sell decision and the conversion ratio and volatility on the ICCC value. The value of the ICCC can increase substantially as the value of the firm approaches the conversion threshold. The conversion ratio as well as the asset market price play equally an important role in the decision to convert or sell. Originality/value This paper develops a pricing model for a contingent Ijara contract, which incorporates a conversion option to mitigate the lessee’s credit risk during periods of economic instability. The ICCC is a cooperative strategy that would be advantageous to all parties, including the lessor and lessee. In the event of a conversion, businesses may be able to continue operating thanks to this financial innovation, and the lessor may profit from the company’s recovery by freeing up more resources for the use of more profitable ventures.

Closed-form solution of the deflection of a prestressed multilayer cantilever

Abstract In this work, a closed-form solution for the deflection of a multilayer cantilever with initial residual stress in each layer is presented. In particular, this model aims to take into account the relaxation of stresses due to the curvature of the cantilever and to calculate the curvature radius and the bending moment due to stresses. Stress-induced deflections and force & stress-induced deflections of cantilever with different shapes (rectangular, triangular, truncated triangular, V-shaped and truncated V-shaped cantilevers) are obtained from the Euler-Bernoulli beam theory. Taking a SiN/SiO2/Si multilayer as an example, the calculated stress-induced profiles do not depend on the shape of the beam and are in good agreement with finite element results. A comparison with results obtained from a model under generalized plane strain condition is discussed. Stress \& Force-induced deflections are also successfully compared to finite element results. Finally, a new formula is proposed to accurately determine the stress in a thin film material from the study of the curvature of a multilayer cantilever. As illustrated by the example of a virtual SiO2/Si bi-layer rectangular cantilever, the initial stress value which is an input parameter of the finite element analysis is calculated back from the simulated curvature with an accuracy of 0.2%.

Dynamic Asset Allocation and Retirement Decision with Consumption Ratcheting and Effort Choice

This study examines the problem of choosing optimal consumption, investment, early retirement, and effort levels for an economic agent who is unwilling to reduce their consumption over time. The agent’s income depends on a fixed wage rate, and varies proportionally with their level of effort, which incurs disutility due to labor. By applying the dual-martingale method, we reformulate the problem into its dual form, dividing it into two distinct sub-problems: one focuses on selecting a consumption path that does not decrease, while the other determines the optimal timing for early retirement. Using this method, we find optimal strategies in the form of explicit closed-form solutions.

Advanced A-B fractional modeling of nonlocalized viscoelastic polymer micro-rod caused by mobile heat source including fractional strain

Current investigation is aimed to develop the concept of fractional derivatives proposed by Atangana–Baleanu that captures the memory effects of heat transfer and stress–strain relation inside a nonlocal viscoelastic polymer micro rod affected from a mobile heat input. Laplace Transform algorithm is bestowed to stimulate the closed-form solutions of temperature, displacement, thermal stress. Structured mechanism of numerical inversion of Laplace Transform is carried out for obtaining the quantitative outcomes in physical domain. Impacts of the fractional parameter, fractional order strain quantity, nonlocal parameter, and heat source velocity are analyzed. Results under fractional derivatives reveal stable characteristics of thermal waves.

Attention-Guided Low-Rank Tensor Completion.

Low-rank tensor completion (LRTC) aims to recover missing data of high-dimensional structures from a limited set of observed entries. Despite recent significant successes, the original structures of data tensors are still not effectively preserved in LRTC algorithms, yielding less accurate restoration results. Moreover, LRTC algorithms often incur high computational costs, which hinder their applicability. In this work, we propose an attention-guided low-rank tensor completion (AGTC) algorithm, which can faithfully restore the original structures of data tensors using deep unfolding attention-guided tensor factorization. First, we formulate the LRTC task as a robust factorization problem based on low-rank and sparse error assumptions. Low-rank tensor recovery is guided by an attention mechanism to better preserve the structures of the original data. We also develop implicit regularizers to compensate for modeling inaccuracies. Then, we solve the optimization problem by employing an iterative technique. Finally, we design a multistage deep network by unfolding the iterative algorithm, where each stage corresponds to an iteration of the algorithm; at each stage, the optimization variables and regularizers are updated by closed-form solutions and learned deep networks, respectively. Experimental results for high dynamic range imaging and hyperspectral image restoration show that the proposed algorithm outperforms state-of-the-art algorithms.

Seepage and stability analysis of hydraulically anisotropic unsaturated infinite slopes under steady infiltration

An analytical model is derived for predicting the flow field and stability of an unsaturated infinite slope subjected to steady infiltration. The proposed model is novel because it accounts for the hydraulic anisotropy of unsaturated soil. The governing equation for steady-state seepage in an infinite slope is established in terms of matric suction under a constant surface flux boundary condition. On the basis of the available experimental findings on the hydraulic anisotropy behavior of unsaturated soils, the relative hydraulic conductivity for a soil under unsaturated conditions with respect to the soil at saturation is postulated to be a direction-independent scalar. This postulation simplifies the governing equation to a form that is directly solvable via the relative hydraulic conductivity and the saturated hydraulic conductivity tensor. To enable sophisticated applications, an exponential law and a power law that are well established in the unsaturated soil literature are used to relate the relative hydraulic conductivity to the matric suction and the effective degree of saturation, respectively. Closed-form solutions are derived for the matric suction, the flow net (potential function and stream function), and the effective degree of saturation. Analytical solutions are also derived for the soil unit weight and overburden stress. These solutions are incorporated into the unsaturated infinite slope stability formula constructed on a suction stress-based effective stress failure criterion. Hydraulic anisotropy has been shown to directly affect the flow field and the change in matric suction, which, in turn, drastically affects the slope safety factor against shallow landslides. This finding demonstrates that neglecting hydraulic anisotropy can cause a considerable overestimation of the safety factor, resulting in an unsafe slope stability prediction. The proposed model is useful for preliminary evaluation of the long-term stability of unsaturated slopes during wet periods and the antecedent slope conditions for shallow landslide initiation under transient infiltration.

Joint sparse subspace clustering via fast [formula omitted]-norm constrained optimization

Subspace clustering gains popularity in unsupervised machine learning due to its excellent dimensionality reduction capability and interpretability. Although existing research has made significant progress in clustering performance, it still faces two limitations. The first is the challenge of poor feature representation, which makes it hard to explore global structural sparsity. The second is the issue of long runtime, which makes it computationally inefficient. In this paper, we propose a joint sparse subspace clustering (JSSC) method to capture the structural features in the representation matrix via the ℓ2,0-norm constraint, which encourages sparsity across rows and leads to enhance clustering performance. In algorithms, an optimization scheme based on proximal alternating minimization (PAM) is developed, in which each subproblem has a closed-form solution, thus ensuring the effectiveness. Moreover, the convergence guarantee is rigorously proved in theory. Numerical experiments on image and hyperspectral datasets validate its excellent performance in terms of high accuracy with a high computational speed. In particular, the proposed method improves the clustering accuracy by at least 6.1% and the runtime by at least 14 times compared with competitors on the ORL dataset. The code of our proposed JSSC is available at https://github.com/zhudafa/JSSC.

Analytical solutions for autonomous differential equations with weighted derivatives

In this work, we introduce a new definition of weighted derivatives along with corresponding integral operators, which aim to facilitate the solution of both linear and non-linear differential equations. A significant finding is that the fractional derivative of Caputo–Fabrizio type is a special case within this framework, allowing us to build upon existing research in this area. Additionally, we provide closed-form analytical solutions for autonomous and logistic equations using our newly defined derivatives and integrals. We thoroughly explore the properties associated with these weighted derivatives and integrals. To demonstrate the reliability and practical applicability of our results, we include several examples and applications that highlight the effectiveness of our approach.