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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E5: Financial Mathematics".

Deadline for manuscript submissions: 5 August 2025 | Viewed by 18690

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Special Issue Editors

Dr. Luca Di Persio

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Department of Computer Science, College of Mathematics, University of Verona, Strada le Grazie 15, 37134 Verona, Italy
Interests: stochastic partial differential equations (SPDEs) in both finite and infinite dimensions; asymptotic expansion of finite/infinite integrals; interacting particle systems; random walk in random media; stochastic mean field games with applications in finance; time series analysis with applications in finance; machine learning and mathematical foundations of neural networks with applications in real markets
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Dr. Matteo Garbelli

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Department of Computer Science, College of Mathematics, University of Verona, Strada le Grazie 15, 37134 Verona, Italy
Interests: adapted optimal transport; probability theory; mean field games; neural networks applications

Special Issue Information

Dear Colleagues,

This Special Issue aims to provide a comprehensive overview of the latest advancements in machine learning methods for finance, emphasizing the critical role of stochastic analysis techniques and the utilization of real financial data. In an era characterized by increasingly complex financial markets and vast amounts of available data, the integration of machine learning methodologies has become paramount for effectively analyzing and navigating the intricacies of financial systems.

This initiative seeks to foster interdisciplinary collaboration between machine learning experts, finance professionals, and data scientists, aiming to explore innovative approaches for addressing contemporary challenges in finance. Topics of interest include, but are not limited to, predictive modelling for market trends, risk assessment and management strategies, algorithmic trading, credit risk modelling, fraud detection, mean field games applications in finance, sentiment analysis, and portfolio optimization.

We particularly welcome submissions that explore innovative approaches to explainable AI (XAI) within the context of finance. Authors are encouraged to address the interpretability, transparency, and trustworthiness of their machine learning models, providing insights into the decision-making process and the factors driving model predictions.

Through this Special Issue, we aim to provide a platform for researchers and practitioners to exchange ideas, share insights, and contribute to the advancement of knowledge at the intersection of machine learning and finance. By showcasing cutting-edge research and innovative methodologies, we strive to facilitate the development of robust and reliable solutions that can enhance decision-making processes and drive positive outcomes in the financial industry.

Authors are invited to submit original research articles, review papers, and case studies that contribute to advancing machine learning methods in finance.

We look forward to hearing from you.

Dr. Luca Di Persio
Dr. Matteo Garbelli
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

machine learning
neural networks
finance
XAI
MFGs
economics
computer science
stochastic analysis
risk
predictive modelling
CR modelling
fraud detection
sentiment analysis

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Published Papers (10 papers)

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Research

27 pages, 1070 KiB

Open AccessArticle

Global Cross-Market Trading Optimization Using Iterative Combined Algorithm: A Multi-Asset Approach with Stocks and Cryptocurrencies

by Kansuda Pankwaen, Sukrit Thongkairat and Worrawat Saijai

Mathematics 2025, 13(8), 1317; https://doi.org/10.3390/math13081317 - 17 Apr 2025

Viewed by 229

Abstract

This study presents an advanced adaptive trading framework that integrates Deep Reinforcement Learning (DRL) with the Iterative Model Combining Algorithm (IMCA) to overcome the critical limitations of static ensemble methods in global portfolio optimization. Using a diverse cross-market dataset of 39 stocks from the US, Australia, Europe, Thailand, and one cryptocurrency (BTC-USD), the research rigorously evaluates models’ adaptability under volatile market conditions. Volatile market conditions—such as COVID-19, SVB crisis, and the 2022 crypto crash—are captured via volatility metrics (e.g., drawdown), with DRL models like PPO/TD3 adapting through dynamic reward signals. This cross-asset integration is particularly critical, as it captures the complex dynamics and correlations between traditional financial markets and emerging digital assets. Although DRL models like PPO and TD3 outperform traditional strategies, they remain vulnerable to market drawdowns and high volatility. IMCA significantly surpasses these models, achieving the highest cumulative return of 29.52% and a superior Sharpe ratio of 0.829 by dynamically recalibrating model weights in response to real-time market dynamics. This study addresses a substantial research gap, highlighting the failure of traditional ensemble models—reliant on static weightings—to adapt to evolving financial conditions, resulting in suboptimal risk-adjusted returns. IMCA offers a dynamic, data-driven approach that continuously optimizes portfolio strategies across fluctuating market regimes, demonstrating its scalability and robustness across diverse asset classes and regional markets, and providing an empirical framework for adaptive portfolio management. Policy recommendations underscore the need for financial institutions to adopt AI-driven adaptive models like IMCA to enhance portfolio resilience, profitability, and responsiveness in uncertain markets. Full article

(This article belongs to the Special Issue Machine Learning and Finance)

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25 pages, 1549 KiB

Open AccessArticle

A Combined Algorithm Approach for Optimizing Portfolio Performance in Automated Trading: A Study of SET50 Stocks

by Sukrit Thongkairat and Woraphon Yamaka

Mathematics 2025, 13(3), 461; https://doi.org/10.3390/math13030461 - 30 Jan 2025

Cited by 1 | Viewed by 831

Abstract

This study investigates portfolio optimization for SET50 stocks using Deep Reinforcement Learning (DRL) algorithms to address market volatility. Five DRL algorithms—Advantage Actor–Critic (A2C), Proximal Policy Optimization (PPO), Deep Deterministic Policy Gradient (DDPG), Soft Actor–Critic (SAC), and Twin Delayed DDPG (TD3)—were evaluated for their effectiveness in managing risk and optimizing returns. We propose an Iterative Model Combining Algorithm (IMCA) that dynamically adjusts model weights based on market conditions to enhance performance. Our results demonstrate that IMCA consistently outperformed traditional strategies, including the Minimum Variance model. IMCA achieved a cumulative return of 14.20% and a Sharpe Ratio of 0.220, compared to the Minimum Variance model’s return of −4.35% and Sharpe Ratio of 0.018. This research highlights the adaptability and robustness of DRL algorithms for portfolio management, particularly in emerging markets like Thailand. It underscores the advantages of dynamic, data-driven strategies over static approaches. Full article

(This article belongs to the Special Issue Machine Learning and Finance)

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35 pages, 1547 KiB

Open AccessArticle

Sustainability, Accuracy, Fairness, and Explainability (SAFE) Machine Learning in Quantitative Trading

by Phan Tien Dung and Paolo Giudici

Mathematics 2025, 13(3), 442; https://doi.org/10.3390/math13030442 - 28 Jan 2025

Cited by 1 | Viewed by 1125

Abstract

The paper investigates the application of advanced machine learning (ML) methodologies, with a particular emphasis on state-of-the-art deep learning models, to predict financial market dynamics and maximize profitability through algorithmic trading strategies. The study compares the predictive capabilities and behavioral characteristics of traditional machine learning approaches, such as logistic regression and support vector machines, with those of highly sophisticated deep learning architectures, including Recurrent Neural Networks (RNNs), Long Short-Term Memory (LSTM) networks, and Gated Recurrent Units (GRUs). The findings underscore the fundamental distinctions between these methodologies, with deeply trained models exhibiting markedly different predictive behaviors and performance, particularly in capturing complex temporal patterns within financial data. A cornerstone of the paper is the introduction and rigorous analysis of a framework to evaluate models, by means of the SAFE framework (Sustainability, Accuracy, Fairness, and Explainability). The framework is designed to address the opacity of black-box ML models by systematically evaluating their behavior across a set of critical dimensions. It also demonstrates how models’ predictive outputs align with the observed data, thereby reinforcing their reliability and robustness. The paper leverages historical stock price data from International Business Machines Corporation (IBM). The dataset is partitioned into a training phase during which the models are calibrated, and a validation phase, used to evaluate the predictive performance of the generated trading signals. The study addresses two primary machine learning tasks: regression and classification. Classical models are utilized for classification tasks, with their outputs directly interpreted as trading signals, while advanced deep learning models are employed for regression, with predictions of future stock prices further processed into actionable trading strategies. To evaluate the effectiveness of each strategy, rigorous backtesting is conducted, incorporating visual representations such as equity curves to assess profitability and key risk metrics like maximum drawdown for risk management. Supplementary performance indicators, including hit rates and the incidence of false positions, are analyzed alongside the equity curves to provide a holistic assessment of each model’s performance. This comprehensive evaluation not only highlights the superiority of cutting-edge deep learning models in predicting financial market trends but also demonstrates the pivotal role of the SAFE framework in ensuring that machine learning models remain trustworthy, interpretable, and aligned with ethical considerations. Full article

(This article belongs to the Special Issue Machine Learning and Finance)

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17 pages, 1745 KiB

Open AccessFeature PaperArticle

Joint Learning of Volume Scheduling and Order Placement Policies for Optimal Order Execution

by Siyuan Li, Hui Niu, Jiani Lu and Peng Liu

Mathematics 2024, 12(21), 3440; https://doi.org/10.3390/math12213440 - 4 Nov 2024

Viewed by 920

Abstract

Order execution is an extremely important problem in the financial domain, and recently, more and more researchers have tried to employ reinforcement learning (RL) techniques to solve this challenging problem. There are a lot of difficulties for conventional RL methods to tackle the order execution problem, such as the large action space including price and quantity, and the long-horizon property. As naturally order execution is composed of a low-frequency volume scheduling stage and a high-frequency order placement stage, most existing RL-based order execution methods treat these stages as two distinct tasks and offer a partial solution by addressing either one individually. However, the current literature fails to model the non-negligible mutual influence between these two tasks, leading to impractical order execution solutions. To address these limitations, we propose a novel automatic order execution approach based on the hierarchical RL framework (OEHRL), which jointly learns the policies for volume scheduling and order placement. OEHRL first extracts the state embeddings at both the macro and micro levels with a sequential variational auto-encoder model. Based on the effective embeddings, OEHRL generates a hindsight expert dataset, which is used to train a hierarchical order execution policy. In the hierarchical structure, the high-level policy is in charge of the target volume and the low-level learns to determine the prices for a series of the allocated sub-orders from the high level. These two levels collaborate seamlessly and contribute to the optimal order execution policy. Extensive experiment results on 200 stocks across the US and China A-share markets validate the effectiveness of the proposed approach. Full article

(This article belongs to the Special Issue Machine Learning and Finance)

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36 pages, 18802 KiB

Open AccessArticle

A Novel Hybrid Model (EMD-TI-LSTM) for Enhanced Financial Forecasting with Machine Learning

by Olcay Ozupek, Reyat Yilmaz, Bita Ghasemkhani, Derya Birant and Recep Alp Kut

Mathematics 2024, 12(17), 2794; https://doi.org/10.3390/math12172794 - 9 Sep 2024

Cited by 3 | Viewed by 4172

Abstract

Financial forecasting involves predicting the future financial states and performance of companies and investors. Recent technological advancements have demonstrated that machine learning-based models can outperform traditional financial forecasting techniques. In particular, hybrid approaches that integrate diverse methods to leverage their strengths have yielded superior results in financial prediction. This study introduces a novel hybrid model, entitled EMD-TI-LSTM, consisting of empirical mode decomposition (EMD), technical indicators (TI), and long short-term memory (LSTM). The proposed model delivered more accurate predictions than those generated by the conventional LSTM approach on the same well-known financial datasets, achieving average enhancements of 39.56%, 36.86%, and 39.90% based on the MAPE, RMSE, and MAE metrics, respectively. Furthermore, the results show that the proposed model has a lower average MAPE rate of 42.91% compared to its state-of-the-art counterparts. These findings highlight the potential of hybrid models and mathematical innovations to advance the field of financial forecasting. Full article

(This article belongs to the Special Issue Machine Learning and Finance)

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27 pages, 666 KiB

Open AccessFeature PaperArticle

Ownership of Cash Value Life Insurance among Rural Households: Utilization of Machine Learning Algorithms to Find Predictors

by Wookjae Heo, Eun Jin Kwak, John Grable and Hye Jun Park

Mathematics 2024, 12(16), 2467; https://doi.org/10.3390/math12162467 - 9 Aug 2024

Viewed by 1064

Abstract

This study examines the determinants of life insurance ownership with a focus on rural areas and farming households in the United States. Utilizing data from online surveys conducted in 2019 and 2021, this paper explores how psychological factors, financial knowledge, and household characteristics influence life insurance ownership. Traditional indicators like wealth, income, and age were evaluated alongside less frequently discussed variables such as farm loans and rural residency. Machine learning techniques, including neural networks, Support Vector Machine modeling, Gradient Boosting, and logistic regression, were employed to identify the most robust predictors of life insurance demand. The findings reveal that farming-associated factors, particularly holding a farm loan and living in a farming household, significantly predict life insurance ownership. The study also highlights the complexity of life insurance demand, showing that financial education and management practices are critical determinants. This research underscores the need for tailored financial risk management strategies for rural and farming households and contributes to a nuanced understanding of life insurance demand in varying contexts. Full article

(This article belongs to the Special Issue Machine Learning and Finance)

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22 pages, 4075 KiB

Open AccessArticle

Methodology for Smooth Transition from Experience-Based to Data-Driven Credit Risk Assessment Modeling under Data Scarcity

by Hengchun Li, Qiujun Lan and Qingyue Xiong

Mathematics 2024, 12(15), 2410; https://doi.org/10.3390/math12152410 - 2 Aug 2024

Viewed by 1047

Abstract

Credit risk refers to the possibility of borrower default, and its assessment is crucial for maintaining financial stability. However, the journey of credit risk data generation is often gradual, and machine learning techniques may not be readily applicable for crafting evaluations at the initial stage of the data accumulation process. This article proposes a credit risk modeling methodology, TED-NN, that first constructs an indicator system based on expert experience, assigns initial weights to the indicator system using the Analytic Hierarchy Process, and then constructs a neural network model based on the indicator system to achieve a smooth transition from an empirical model to a data-driven model. TED-NN can automatically adapt to the gradual accumulation of data, which effectively solves the problem of risk modeling and the smooth transition from no to sufficient data. The effectiveness of this methodology is validated through a specific case of credit risk assessment. Experimental results on a real-world dataset demonstrate that, in the absence of data, the performance of TED-NN is equivalent to the AHP and better than untrained neural networks. As the amount of data increases, TED-NN gradually improves and then surpasses the AHP. When there are sufficient data, its performance approaches that of a fully data-driven neural network model. Full article

(This article belongs to the Special Issue Machine Learning and Finance)

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11 pages, 791 KiB

Open AccessArticle

Novel Machine Learning Based Credit Card Fraud Detection Systems

by Xiaomei Feng and Song-Kyoo Kim

Mathematics 2024, 12(12), 1869; https://doi.org/10.3390/math12121869 - 15 Jun 2024

Cited by 8 | Viewed by 4383

Abstract

This research deals with the critical issue of credit card fraud, a problem that has escalated in the last decade due to the significant increase in credit card usage, largely driven by advances in international trade, e-commerce, and FinTech. With global losses projected to exceed USD 400 billion in the next decade, the urgent need for effective fraud detection systems is apparent. Our study leverages the power of machine learning (ML) and presents a novel approach to credit card fraud detection. We used the European cardholders dataset for model training, addressing the data imbalance issue that often hinders the effectiveness of the learning process. As a key innovative element, we introduce compact data learning (CDL), a powerful tool for reducing the size and complexity of the training dataset without sacrificing the accuracy of the ML system. Comparative experiments have shown that our CDL-adapted feature reduction outperforms various ML algorithms and feature reduction methods. The findings of this research not only contribute to the theoretical foundations of fraud detection but also provide practical implications for the financial sector, which can benefit immensely from the enhanced fraud detection system. Full article

(This article belongs to the Special Issue Machine Learning and Finance)

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16 pages, 2953 KiB

Open AccessArticle

Stochastic Patterns of Bitcoin Volatility: Evidence across Measures

by Georgia Zournatzidou, Dimitrios Farazakis, Ioannis Mallidis and Christos Floros

Mathematics 2024, 12(11), 1719; https://doi.org/10.3390/math12111719 - 31 May 2024

Cited by 1 | Viewed by 2269

Abstract

This research conducted a thorough investigation of Bitcoin volatility patterns using three interrelated methodologies: R/S investigation, simple moving average (SMA), and the relative strength index (RSI). The paper jointly employes the above techniques on volatility range-based estimators to effectively capture the unpredictable volatility patterns of Bitcoin. R/S analysis, SMA, and RSI calculations assess time series data obtained from our volatility estimators. Although Bitcoin is known for its high volatility and price instability, our analysis using R/S analysis and moving averages suggests the existence of underlying patterns. The estimated Hurst exponents for our volatility estimators indicate a level of persistence in these patterns, with some estimators displaying more persistence than others. This persistence underscores the potential of momentum-based trading strategies, reinforcing the expectation of additional price rises after declines and vice versa. However, significant volatility often interrupts this upward movement. The SMA analysis also demonstrates Bitcoin’s susceptibility to external market forces. These observations indicate that traders and investors should modify their risk management approaches in accordance with market circumstances, perhaps integrating a combination of momentum-based and mean-reversion tactics to reduce the risks linked to Bitcoin’s volatility. Furthermore, the existence of robust patterns, as demonstrated by our investigation, presents promising opportunities for investing in Bitcoin. Full article

(This article belongs to the Special Issue Machine Learning and Finance)

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18 pages, 466 KiB

Open AccessArticle

Ensemble Approach Using k-Partitioned Isolation Forests for the Detection of Stock Market Manipulation

by Hugo Núñez Delafuente, César A. Astudillo and David Díaz

Mathematics 2024, 12(9), 1336; https://doi.org/10.3390/math12091336 - 27 Apr 2024

Cited by 2 | Viewed by 1999

Abstract

Stock market manipulation, defined as any attempt to artificially influence stock prices, poses significant challenges by causing financial losses and eroding investor trust. The prevalent reliance on supervised learning models for detecting such manipulations, while showing promise, faces notable hurdles due to the dearth of labeled data and the inability to recognize novel manipulation tactics beyond those explicitly labeled. This study ventures into addressing these gaps by proposing a novel detection framework aimed at identifying suspicious hourly manipulation blocks through an unsupervised learning approach, thereby circumventing the limitations of data labeling and enhancing the adaptability to emerging manipulation strategies. Our methodology involves the innovative creation of features reflecting the behavior of stocks across various time windows followed by the segmentation of the dataset into k subsets. This setup facilitates the identification of potential manipulation instances via a voting ensemble composed of k isolation forest models, which have been chosen for their efficiency in pinpointing anomalies and their linear computational complexity—attributes that are critical for analyzing vast datasets. Evaluated against eight real stocks known to have undergone manipulation, our approach demonstrated a remarkable capability to identify up to 89% of manipulated blocks, thus significantly outperforming previous methods that do not utilize a voting ensemble. This finding not only surpasses the detection rates reported in prior studies but also underscores the enhanced robustness and adaptability of our unsupervised model in uncovering varied manipulation schemes. Through this research, we contribute to the field by offering a scalable and efficient unsupervised learning strategy for stock manipulation detection, thereby marking a substantial advancement over traditional supervised methods and paving the way for more resilient financial markets. Full article

(This article belongs to the Special Issue Machine Learning and Finance)

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