Search Results (34)

By leveraging the concept of k-Caputo fractional derivatives for stochastic processes, in this paper, we derive a generalized Hermite–Hadamard inequality tailored to n-times differentiable convex stochastic processes, providing a powerful tool for analyzing systems governed by fractional dynamics in probabilistic settings. Additionally, we establish two new integral identities that serve as the foundation for developing midpoint- and trapezium-type inequalities for $(n+1)$-times differentiable convex stochastic processes. These results not only enrich the theoretical underpinnings of fractional calculus, but also offer practical implications for modeling and understanding complex systems with memory and randomness. The proposed framework opens new avenues for future research in stochastic analysis and fractional calculus, with potential applications in fields such as financial mathematics, engineering, and physics. Full article

Fractional calculus in discrete-time is a recent field that has drawn much interest for dealing with multidisciplinary systems. A result of this tremendous potential, researchers have been using constant and variable-order fractional discrete calculus in the modelling of financial and economic systems. This paper explores the emergence of chaotic and regular patterns of the fractional four-dimensional (4D) discrete economic system with constant and variable orders. The primary aim is to compare and investigate the impact of two types of fractional order through numerical solutions and simulation, demonstrating how modifications to the order affect the behavior of a system. Phase space orbits, the 0-1 test, time series, bifurcation charts, and Lyapunov exponent analysis for different orders all illustrate the constant and variable-order systems’ behavior. Moreover, the spectral entropy (SE) and $C_0$ complexity exhibit fractional-order effects with variations in the degree of complexity. The results provide new insights into the influence of fractional-order types on dynamical systems and highlight their role in promoting chaotic behavior. Additionally, two types of control strategies are devised to guide the states of a 4D fractional discrete economic system with constant and variable orders to the origin within a specified amount of time. MATLAB simulations are presented to demonstrate the efficacy of the findings. Full article

This study introduces Itô-RMSProp, a novel extension of the RMSProp optimizer inspired by Itô stochastic calculus, which integrates adaptive Gaussian noise into the update rule to enhance exploration and mitigate overfitting during training. We embed this optimizer within Gated Recurrent Unit (GRU) networks for stock price forecasting, leveraging the GRU’s strength in modeling long-range temporal dependencies under nonstationary and noisy conditions. Extensive experiments on real-world financial datasets, including a detailed sensitivity analysis over a wide range of noise scaling parameters ($\epsilon$), reveal that Itô-RMSProp-GRU consistently achieves superior convergence stability and predictive accuracy compared to classical RMSProp. Notably, the optimizer demonstrates remarkable robustness across all tested configurations, maintaining stable performance even under volatile market dynamics. These findings suggest that the synergy between stochastic differential equation frameworks and gated architectures provides a powerful paradigm for financial time series modeling. The paper also presents theoretical justifications and implementation details to facilitate reproducibility and future extensions. Full article

In this paper, we investigate the Malliavin differentiability and density smoothness of solutions to stochastic differential equations (SDEs) with non-Lipschitz coefficients. Specifically, we consider equations of the form $d{X}_t= b\left(X_t\right)dt + \sigma \left(X_t\right)d{W}_t, X_0= x_0$ where the drift b(·) and diffusion σ(·) may violate the global Lipschitz condition but satisfy weaker assumptions such as Hölder continuity, linear growth, and non-degeneracy. By employing Malliavin calculus theory, large deviation principles, and Fokker–Planck equations, we establish comprehensive results concerning the existence and uniqueness of solutions, their Malliavin differentiability, and the smoothness properties of density functions. Our main contributions include (1) proving the Malliavin differentiability of solutions under the standard linear growth condition combined with Hölder continuity; (2) establishing the existence and smoothness of density functions using Norris lemma and the Bismut–Elworthy–Li formula; and (3) providing optimal estimates for density functions through large deviation theory. These results have significant applications in financial mathematics (e.g., CIR, CEV, and Heston models), biological system modeling (e.g., stochastic population dynamics and neuronal and epidemiological models), and other scientific domains. Full article

This study presents a self-contained derivation and solution of the Black and Scholes partial differential equation (PDE), replacing the standard Wiener process with a smoothed Wiener process, which is a differentiable stochastic process constructed via normal kernel smoothing. By presenting a self-contained, Itô-free derivation, this study bridges the gap between heuristic financial reasoning and rigorous mathematics, bringing forth fresh insights into one of the most influential models in quantitative finance. The smoothed Wiener process does not merely simplify the technical machinery but further reaffirms the robustness of the Black and Scholes framework under alternative mathematical formulations. This approach is particularly valuable for instructors, apprentices, and practitioners who may seek a deeper understanding of derivative pricing without relying on the full machinery of stochastic calculus. The derivation underscores the universality of the Black and Scholes PDE, irrespective of the specific stochastic process adopted, under the condition that the essential properties of stochasticity, volatility, and of no arbitrage may be preserved. Full article

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