Search Results (29)

This study investigates the theoretical foundations and practical advantages of fractional-order optimization in computational machine learning, with a particular focus on stock price forecasting using long short-term memory (LSTM) networks. We extend several widely used optimization algorithms—including Adam, RMSprop, SGD, Adadelta, FTRL, Adamax, and Adagrad—by incorporating fractional derivatives into their update rules. This novel approach leverages the memory-retentive properties of fractional calculus to improve convergence behavior and model efficiency. Our experimental analysis evaluates the performance of fractional-order optimizers on LSTM networks tasked with forecasting stock prices for major companies such as AAPL, MSFT, GOOGL, AMZN, META, NVDA, JPM, V, and UNH. Considering four metrics (Sharpe ratio, directional accuracy, cumulative return, and MSE), the results show that fractional orders can significantly enhance prediction accuracy for moderately volatile stocks, especially among lower-cap assets. However, for highly volatile stocks, performance tends to degrade with higher fractional orders, leading to erratic and inconsistent forecasts. In addition, fractional optimizers with short-memory truncation offer a favorable trade-off between computational efficiency and modeling accuracy in medium-frequency financial applications. Their enhanced capacity to capture long-range dependencies and robust performance in noisy environments further justify their adoption in such contexts. These results suggest that fractional-order optimization holds significant promise for improving financial forecasting models—provided that the fractional parameters are carefully tuned to balance memory effects with system stability. Full article

Deviation of an asset price from its fundamental value, commonly referred to as a price bubble, is a well-known phenomenon in financial markets. Mathematically, a bubble arises when the deflated price process transitions from a martingale to a strict local martingale. This paper explores price bubbles using the framework of optional semimartingale calculus within nonstandard probability spaces, where the underlying filtration is not necessarily right-continuous or complete. We present two formulations for financial markets with bubbles: one in which asset prices are modeled as càdlàg semimartingales and another where they are modeled as làdlàg semimartingales. In both models, we demonstrate that the formation and re-emergence of price bubbles are intrinsically tied to the lack of right continuity in the underlying filtration. These theoretical findings are illustrated with practical examples, offering novel insights into bubble dynamics that hold significance for both academics and practitioners in the field of mathematical finance. Full article

Our aim in this paper is to analytically compute the at-the-money second derivative of the Bachelier implied volatility curve as a function of the strike price for correlated stochastic volatility models. We also obtain an expression for the short-term limit of this second derivative in terms of the first and second Malliavin derivatives of the volatility process and the correlation parameter. Our analysis does not need the volatility to be Markovian and can be applied to the case of fractional volatility models, both with $H<1/2$ and $H>1/2.$ More precisely, we start our analysis with an adequate decomposition formula of the curvature as the curvature in the uncorrelated case (where the Brownian motions describing asset price and volatility dynamics are uncorrelated) plus a term due to the correlation. Then, we compute the curvature in the uncorrelated case via Malliavin calculus. Finally, we add the corresponding correlation correction and we take limits as the time to maturity tends to zero. The presented results can be an interesting tool in financial modeling and in the computation of the corresponding Greeks. Moreover, they allow us to obtain general formulas that can be applied to a wide class of models. Finally, they provide us with a precise interpretation of the impact of the Hurst parameter H on this curvature. Full article

This study addresses the challenge of capturing both short-run volatility and long-run dependencies in global stock markets by introducing fractional transfer entropy (FTE), a new framework that embeds fractional calculus into transfer entropy. FTE allows analysts to tune memory parameters and thus observe how different temporal emphases reshape the network of directional information flows among major financial indices. Empirical evidence reveals that when short-memory effects dominate, markets swiftly incorporate recent news, creating networks that adapt quickly but remain vulnerable to transient shocks. In contrast, balanced memory parameters yield a more stable equilibrium, blending immediate reactions with persistent structural ties. Under long-memory configurations, historically entrenched relationships prevail, enabling established market leaders to remain central despite ongoing fluctuations. These findings demonstrate that FTE uncovers nuanced dynamics overlooked by methods focusing solely on either current events or deep-rooted patterns. Although the method relies on price returns and does not differentiate specific shock types, it offers a versatile tool for investors, policymakers, and researchers to gauge financial stability, evaluate contagion risk, and better understand how ephemeral signals and historical legacies jointly govern global market connectivity. Full article

Electricity is one of the most widely used energy sources. The climate crisis, public pressure to invest in renewable and low-carbon energy sources, and the reduction in industrial electricity consumption caused by the COVID-19 pandemic have a significant impact on the energy sector. In addition, military action in Europe is affecting energy generation capacity and availability, which raises the question of economic calculus, particularly regarding the cost of generation and supply. These factors affect the cost structure of those responsible for supplying energy and, in extreme cases, can lead to energy exclusion. The article aimed to identify differences in the presentation and interpretation of operating cost data from the individual and consolidated financial statements of Polish energy groups, which is of key importance for investors, analysts and decision-makers in the energy sector. The analysis uses data for 2018–2022 from the income statement. The research hypothesis is that the complexity of Polish energy groups in the Polish energy sector leads to ambiguity in the interpretation of cost data included in stand-alone and consolidated financial statements. Full article

The mathematical theories and methods of fractional calculus are relatively mature, which have been widely used in signal processing, control systems, nonlinear dynamics, financial models, etc. The studies of some basic theories of fractional differential equations can provide more understanding of mechanisms for the applications. In this paper, the expression of the Green function as well as its special properties are acquired and presented through theoretical analyses. Subsequently, on the basis of these properties of the Green function, the existence and uniqueness of positive solutions are achieved for a singular p-Laplacian fractional-order differential equation with nonlocal integral and infinite-point boundary value systems by using the method of a nonlinear alternative of Leray–Schauder-type Guo–Krasnoselskii’s fixed point theorem in cone, and the Banach fixed point theorem, respectively. Some existence results are obtained for the case in which the nonlinearity is allowed to be singular with regard to the time variable. Several examples are correspondingly provided to show the correctness and applicability of the obtained results, where nonlinear terms are controlled by the integrable functions ${\displaystyle \frac{1}{\pi {(lnt)}^{\frac{1}{2}}{(1-lnt)}^{\frac{1}{2}}}}$ and ${\displaystyle \frac{1}{\pi {(lnt)}^{\frac{3}{4}}{(1-lnt)}^{\frac{3}{4}}}}$ in Example 1, and by the integrable functions $\theta ,\overline{\theta }$ and $\phi \left(v\right),\psi \left(u\right)$ in Example 2, respectively. The present work may contribute to the improvement and application of the coupled p-Laplacian Hadamard fractional differential model and further promote the development of fractional differential equations and fractional differential calculus. Full article

Greenflation or inflation for green energy transition in Europe becomes a structural problem of new scarcity and poverty, under Austrian Economics analysis. The current European public agenda on the Green Deal and its fiscal and monetary policies are closer to coercive central planning, against the markets, economic calculus, and Mises’ theorem. In this paper, attention is paid to the green financial bubble and the European greenflation paradox: in order to achieve greater future social welfare, due to a looming climate risk, present wellbeing and wealth is being reduced, causing a real and ongoing risk of social impoverishment (to promote the SGD 13 on climate action, it is violated by SGD 1–3 on poverty and hunger and 7–12 on affordable energy, economic growth, sustainable communities, and production). According to the European Union data, the relations are explained between green transition and public policies (emissions, tax, debt, credit boom, etc.), GDP variations (real–nominal), and the increase of inflation and poverty. As many emissions are reduced, there is a decrease of GDP (once deflated) and GDP per capita, evidencing social deflation, which in turn means more widespread poverty and a reduction of the middle-class. Also, there is a risk of a green-bubble, as in the Great Recession of 2008 (but this time supported by the European Union) and possible stagflation (close to the 1970s). To analyze this problem generated by mainstream economics (econometric and normative interventionism), this research offers theoretical and methodological frameworks of mainline economics (positive explanations based on principles and empirical illustrations for complex social phenomena), especially the Austrian Economics and the New-Institutional Schools (Law and Economics, Public Choice, and Comparative Constitutional Economics). Full article

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. The proposed model builds on existing research by formulating a non-linear coupled fractional boundary value problem with mixed boundary conditions. The primary advantages of our method include its ability to capture the dynamics of complex systems more accurately and its flexibility in handling different types of fractional derivatives. The model’s solution was derived using advanced mathematical techniques, and the results confirmed the existence and uniqueness of the solutions. This approach not only generalizes classical differential equation methods but also offers a robust framework for modeling real-world phenomena governed by fractional dynamics. The study concludes with the validation of the theoretical findings through illustrative examples, highlighting the method’s efficacy and potential for further applications. Full article

The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, however, relies on a set of rigorous assumptions that give rise to several limitations. The non-local property of the fractional derivative (FD) and the identification of fractal characteristics in financial markets have paved the way for the introduction and rapid development of fractional calculus in finance. In comparison to the classical B-S equation, the fractional B-S equations (FBSEs) offer a more flexible representation of market behavior by incorporating long-range dependence, heavy-tailed and leptokurtic distributions, as well as multifractality. This enables better modeling of extreme events and complex market phenomena, The fractional B-S equations can more accurately depict the price fluctuations in actual financial markets, thereby providing a more reliable basis for derivative pricing and risk management. This paper aims to offer a comprehensive review of various FBSEs for pricing European options, including associated solution techniques. It contributes to a deeper understanding of financial model development and its practical implications, thereby assisting researchers in making informed decisions about the most suitable approach for their needs. Full article

Biological and financial models are examples of dynamical systems where both stochastic and historical behavior are important to be considered. The fractional Brownian motion (fBM) is commonly used, sometimes with fractional-order derivatives, to model the combined stochastic and fractional effects. Recently, spectral techniques are used to analyze models with fBM using, e.g., iterated Itô fractional integrals such as the fractional Wiener-Hermite (FWHE). In the current work, FWHE is generalized and adapted to be consistent with the Malliavin calculus approach. The conditions for existence and uniqueness are outlined in addition to the proof of convergence. The solution algorithm is described in detail. Using FWHE, the stochastic fractional model is replaced by a deterministic fractional-order system that can be handled using well-known mathematical tools to evaluate the solution statistics. Analytical solutions can be obtained for many important models such as the fractional stochastic Black–Scholes model. The convergence is studied and compared with the exact solution and high convergence is noticed compared with other techniques. A general numerical algorithm is described to analyze the resultant deterministic system in the case of no feasible analytical solutions. The algorithm is applied to study and simulate the population model with nonlinear losses for different values of the Hurst parameter. The results show the efficiency of FWHE in analyzing practical linear and nonlinear models. Full article

In this work, we propose a fast scheme based on higher order discretizations on graded meshes for resolving the temporal-fractional partial differential equation (PDE), which benefits the memory feature of fractional calculus. To avoid excessively increasing the number of discretization points, such as the standard finite difference or meshfree methods, and, at the same time, to increase the efficiency of the solver, we employ discretizations on spatially non-uniform meshes with an attention on the non-smoothness area of the underlying asset. Therefore, the PDE problem is transformed to a linear system of algebraic equations. We perform numerical simulations to observe and check the behavior of the presented scheme in contrast to the existing methods. Full article

This article is devoted to the applied aspects of using the concept of the time value of money for the purpose of determining the present value of cash flows in conditions of asymmetric distribution of payments and facts of economic life over time. Currently, such situation is standard when doing business and should be thoroughly studied. The purpose of the study is to prove that the method of distribution of payments affects the result of discounting, and that this information is essential when making management decisions and should be disclosed to the user of the information. Based on the basic provisions of the theory of the time value of money and analyzing the specifics of the asymmetric distribution of the described events, the authors come to the conclusion that it is necessary to supplement the cost discounting methodology by including in it a description of the basic approaches to distribution. As such approaches, the use of distribution methods that were called First Payment First Sale (FPFS), First Payment Last Sale (FPLS), and Current Payment Current Sale (CPCS) are proposed. Use of these methods in certain calculations is the main novelty of this article. The difference that arises as a result of the use of different approaches to assessment in the conditions of asymmetric distribution is illustrated with the simulated data. Taking into account a specific approach to the distribution of cash flows leads to a better understanding of the basis for discounting indicators, improves the quality of information and the validity of management decisions based on it, and reduces the risks of choosing the wrong financing strategy. Full article

International oil and gas companies listed in New York must publish the information of oil and gas reserves under the SEC (United States Securities and Exchange Commission) standards every year. For greatly improving the SEC reserve, the SEC reserve value and the SEC reserve substitution rate, in this article not only the SEC reserve equations have been determined but also the SEC reserve value models have been established. The SEC reserve value models have been verified as correct. Based on these models, the multivariate function calculus method, the multivariate function limit method and the function recurrence method have been adopted to research parameter sensitivity differences rules, parameter adjustment directions, parameter adjustment degrees and SEC reserve parameter linkage adjustment rules. The research is significant, because there are great differences between SEC standards and China’s in reserve management mode, reserve estimation method system and financial management system. It is just these differences that cause the frequent adjustment of SEC reserve parameters during the process of SEC reserve submissions each year. As a result, this article reaches some conclusions. Above all, the article has clarified the parameter quantitative conditions that lead to the sensitivity between the SEC reserve and the initial production to begin stronger and weaker than the sensitivity between the SEC reserve and the price in production exponential, hyperbolic and harmonic decline types. Furthermore, the article has clarified the parameter quantitative conditions that lead to the sensitivity between the SEC reserve value and the initial production to begin stronger and weaker than the sensitivity between the SEC reserve value and the price in common production exponential decline types. Moreover, the article has clarified reserve parameter linkage adjustment rules and found the most significant parameter whose least adjustment will cause the largest reserve increase. In addition, the function calculus method adopted to disclose reserve parameter sensitivity rules will expand the parameter sensitivity analysis method that took the previous statistical mapping method as the main analysis method. Full article

Crowdfunding is a successful disruptive innovation of fintech that substitutes financial intermediaries and contributes toward financial inclusion and sustainable development. The present research aimed at exploring the underlying determinant factors that shape the investors’ intentions to fund in a crowdfunding platform, a phenomenon still under-researched in the developing world. To bridge this void in the literature, we investigated how calculus and relational trust mediate the effects of perceived accreditation, blockchain technology, structural assurance, and third-party seal on the investors’ intention using the SEM technique to analyze the data collected from 110 platform investors in Pakistan. Findings suggest that third-party seal and blockchain technology strongly influence the calculus trust. While the investors’ intention to invest is mediated by calculus trust, the relational trust fails to show any mediation effect, suggesting that investors make investment decisions based on what makes sense to them cognitively instead of affectively. The research was concluded with implications for both theory and practice. Full article

Financial derivatives have grown in importance over the last 40 years with futures and options being actively traded on a daily basis throughout the world. The need to accurately price such financial instruments has, thus, also increased, which has given rise to several mathematical models among which is that of Black, Scholes, and Merton whose wide acceptance is partly justified by its ability to price derivatives in mature and well-developed markets. For instruments traded in emerging markets, however, the accurateness of the BSM model is unproven and new proposals need be made to face the pricing challenge. In this paper we develop a model, inspired in conformable calculus, providing greater flexibilities for these markets. After developing the theoretical aspects of the model, we present an empirical application. Full article