An elementary Recurrent Neural Network that operates on p time lags, called an RNN(p), is the natural generalisation of a linear autoregressive model ARX(p). It is a powerful forecasting tool for variables displaying inherent seasonal patterns across multiple time scales, as is often observed in energy, economic, and financial time series. The architecture of RNN(p) models, characterised by structured feedbacks across time lags, enables the design of efficient training strategies. We conduct a comparative study of learning algorithms for these models, providing a rigorous analysis of their computational complexity and training performance. We present two applications of RNN(p) models in power consumption forecasting, a key domain within the energy sector where accurate forecasts inform both operational and financial decisions. Experimental results show that RNN(p) models achieve excellent forecasting accuracy while maintaining a high degree of interpretability. These features make them well-suited for decision-making in energy markets and other fintech applications where reliable predictions play a significant economic role.
Speculative variance in financial markets is usually traced to disagreement about expected returns. This paper asks whether speculation can arise instead from how traders perceive the covariance between asset returns and the background risks they hedge, a question left open by models built on mean disagreement alone. In a continuous-time economy with rational and non-rational traders, a single bias parameter, motivated by the diagnostic expectations literature, distorts how non-rational traders perceive the covariance between asset returns and the background risks they hedge. When the bias is positive and rational traders stand ready to take the other side of the resulting mispricing, speculative variance remains strictly positive even though all traders agree on expected returns. This variance is a closed-form quadratic in the bias, so a stochastic bias admits an explicit law of motion, and ordinary fluctuations push long-run average risk above the constant-bias benchmark. Financial innovation widens the effect, since each fresh hedging instrument adds a dimension along which covariance can be misjudged, and the speculation it produces looks like hedging demand rather than the volume-heavy trading of disagreement. The same mechanism, sustained by limited arbitrage capital, generates an endogenous price correction and volatility surge once sentiment reverses. Covariance misperception therefore warrants attention as a quiet source of systemic fragility.
In this paper, we study a recently introduced vector-valued version of the Nikaidô–Isoda bifunction that allows Nash games to be reformulated as strong vector equilibrium problems. We show that classical assumptions in Nash equilibrium theory, such as the quasiconvexity of the players’ payoff functions, naturally translate into generalized convexity properties of the vector Nikaidô–Isoda bifunction. This correspondence does not generally hold for the scalar Nikaidô–Isoda bifunction, since it is defined through summation and quasiconvexity is not preserved under sums. Based on this reformulation, we investigate the existence of solutions to strong vector equilibrium problems by means of the finite intersection property and transfer lower continuity, thereby extending and unifying several results in the literature. We further analyze the main properties of the vector Nikaidô–Isoda bifunction and identify classes of games for which this bifunction satisfies generalized quasiconvexity and generalized monotonicity assumptions. Finally, we establish a connection between the classical notion of weak transfer lower continuity in games and the transfer lower continuity of the vector Nikaidô–Isoda bifunction.
This paper develops a comprehensive framework for collective risk measures, tools designed to quantify the aggregate risk stemming from a collective of agents. Crucially, these measures explicitly account for inter-agent cooperation, allowing agents to exchange risk through state-dependent transfers without requiring external capital flows. We review previous works on no-arbitrage in the collective framework, and introduce collective risk measures for both random variables and stochastic processes in discrete time. In the latter case, we allow for time-dependent cooperation and risk sharing, supporting a consistent evaluation of evolving financial positions within a collective framework. Among the several applications of the theory, we study collective super-replication prices and we provide dual characterizations of collective risk measures via families of (martingale) measures.
We consider a competitive exchange economy where the commodity space is a locally convex topological vector space ordered by a closed, generating cone. The model includes finitely many consumers, each with a distinct consumption set, assumed to be a closed subcone of the positive cone. Assuming the positive cone has non-empty semi-interior, we establish the existence of equilibrium. This setting extends earlier frameworks that relied on normed and locally convex spaces and shows how equilibrium theory can be developed in more general topological environments. To this end, we develop a new topology on the commodity space, finer than the original one, under which semi-interior points of the positive cone become interior points. This refinement is a key step that enables the extension of equilibrium existence results to locally convex spaces beyond the normed setting. In addition, the analysis proceeds without assuming any lattice structure on the commodity space or a priori continuity assumptions on preferences, thus broadening the applicability of the equilibrium framework. We conclude with an example that illustrates the validity of our results in an infinite-dimensional locally convex setting where standard assumptions such as lattice structure or a priori continuity of preferences are not imposed.
This study develops a large-scale framework to evaluate whether, and under what conditions, adding cryptocurrencies to equity investment universes improves portfolio performance.We apply four long-only portfolio strategies, Global Minimum Variance, Risk Parity, Most Diversified Portfolio, and Equally Weighted, to 10,000 randomly generated investment universes. These universes consist of baskets containing either only equities or varying combinations of equities and cryptocurrencies. We conduct an out-of-sample analysis on real-world data from 2018 to 2023 to assess the influence of cryptocurrencies on portfolio outcomes. The empirical findings reveal that portfolios constructed from mixed equity and cryptocurrency universes provide a better risk-return profile compared to purely equity-based portfolios, particularly for Risk Parity, Most Diversified, and Equally Weighted.
Continued interest in sustainable investing calls for an axiomatic approach to measures of risk and reward that focus not only on financial returns, but also on measures of environmental and social sustainability, i.e. environmental, social, and governance (ESG) scores. We propose axiomatic definitions for ESG-coherent risk measures and ESG reward–risk ratios based on functions of bivariate random variables that are applied to financial returns and real-time ESG scores, extending the traditional univariate measures to the ESG case. We provide examples, discuss the dual representation, and present an empirical analysis in which the ESG-coherent risk measures and ESG reward–risk ratios are used to rank stocks.
Single-leader multi-follower games model the sequential strategic interactions between one player who chooses first, the leader, and other players, the followers, who respond simultaneously after observing the choice of the leader. In this paper we focus on the concept of subgame perfect equilibrium (SPE for short) in single-leader multi-follower games when the followers are involved in a generalized game and have a non-unique joint best response for each choice of the leader. We first show the connections between the concept of an SPE and those of a pessimistic and optimistic equilibrium, which are usually adopted and extensively studied in this framework. We then prove an existence result for SPEs and a characterization of the set of SPE-outcomes. Specifically, an action profile is shown to be an SPE-outcome if and only if the joint response of the followers to the leader is rational and the leader’s payoff lies between the payoffs achievable in a pessimistic equilibrium and in an optimistic equilibrium. Finally, moving to the broader framework of multi-leader-follower games, we show the existence of SPEs and we identify a subset of SPE-outcomes for a class of multi-leader-follower games where the game played by the leaders involves a generalized weighted potential structure.
In this paper, we adopt a conditional tail risk network methodology for the assessment of transmission channels of risk. In particular, we employ weighted and directed networks to model the mutual influence between banks, with the weights being linked to tail risk measures. More specifically, in the empirical comparison we focus on MES-based pairwise dependence networks and on the parametric Δ CoVaR network estimated on ARMA(1,1)-GARCH(1,1) residuals through SCAD-penalized quantile regression. We use different network indicators to investigate the importance of a bank’s role in both spreading and absorbing risk from other financial institutions. Our analysis focuses on a sample based on banks included in the European Banking Authority 2023 stress test, complemented by two additional systemically relevant European institutions and we consider daily data for the period 2015–2024. The analyses are conducted over four economically meaningful subperiods in order to evaluate the evolution of systemic interconnections across different macro-financial regimes. We find that the choice of tail risk measure leads to substantial differences in network topology, centrality rankings, and community composition. The empirical results demonstrate significant variations in the structural characteristics of the networks through time.
This paper aims to provide a detailed study of an electricity market model initially presented by Aussel et al. (2016). This model can be formulated as a multi-leader-common-follower game, in which N producers act as leaders and an independent system operator acts as follower. The revenue bid function on which each producer operates belongs to a subset of L^2 , and this generates computational difficulties due to the nonsmoothness of the functions. It also means that the solution of the follower is not guaranteed to be unique. For this reason, Aussel et al. (2016) approximate the bid function of each leader with a quadratic function and introduce the concept of a projected solution. This paper proves the existence of a projected solution to the model when N=2 and, under suitable assumptions, for N≥ 3 as well. Furthermore, thanks to the potential structure of the leaders’ game, the projected solution can be determined using KKT conditions. Some numerical tests witness the validity of the approach.
This paper investigates ambiguity aversion in the context of a utility-maximizing investor operating under the LVO-CEV model, which is a novel extension within the Constant Elasticity of Variance (CEV) framework. By embedding a relative entropy penalty, we derive closed-form solutions for optimal portfolio exposure, asset allocation, and consumption under Hyperbolic Absolute Risk Aversion (HARA) utility. These solutions provide clear insights into how ambiguity aversion interacts with key model parameters to shape optimal investment and consumption decisions. We further analyze two representative investor types who adopt commonly observed suboptimal strategies with consumption and quantify their associated utility losses using the Wealth-Equivalent Loss (WEL) metric. Empirically, based on estimates on historical S P 500 index data, we find that in the non-robust case ( ϕ =0 ), the optimal asset allocation peaks at approximately 61 ϕ = 3 ), this value declines to around 30 β . Furthermore, our analysis of wealth-equivalent losses indicates that neglecting ambiguity aversion or wealth-floor constraints can lead to significant utility losses. For instance, disregarding ambiguity aversion alone ( ϕ = 0 ) results in an approximate 8 ϕ = 10 over an eight-year horizon, and this loss rises to approximately 13 ϕ = 0 , F = 0 ). Overall, these findings underscore the importance of robust and integrated portfolio strategies that explicitly account for ambiguity, investor risk preferences, and consumption dynamics in long-term investment planning.
Environmental, Social, and Governance (ESG) factors have become increasingly relevant in financial markets, influencing investment strategies and risk assessments. This article explores the role of raw ESG metrics in predicting the direction of future stock returns, framing return forecasting as a classification problem. We analyse MSCI ACWI index components from 2016 to 2022, focusing on the manufacturing, information, and financial sectors in the USA and Europe. We propose an ESG-oriented data cleaning pipeline and evaluate various machine learning models, finding that XGBoost outperforms other approaches. To assess the predictive power of ESG metrics, we conducted an ablation study, comparing their contribution to benchmark financial variables and past returns. Our results show that ESG and financial variables independently improve classification performance comparably, suggesting a complementary role in return forecasting. Through a SHAP-based feature importance analysis, we examine ESG contributions at the sector-region level, revealing that Environmental and Governance factors are generally the most influential in predictive performance. Our findings suggest that raw ESG metrics contain meaningful predictive value that should not be overlooked.
In this paper, I develop a two-sector growth model with endogenous labour supply in which the non-reproducible factor is employed uniquely in the production of investment goods. Relying on Cobb-Douglas technologies, logarithmic utility of consumption and separate CRRA disutility for labour provision, I show that a centralized solution characterized by a meaningful steady state and the determinacy of equilibrium trajectories requires constant returns to scale only in the production of new equipment. Moreover, I show that a calibrated version of the model tailored on the US economy leads to a marginal rate of transformation between consumption and investment goods lower than one in absolute value, and it is also able to provide a rationale for the procyclical patterns of the relative price of capital goods and the real wage.
This paper proves the existence of the φ -Kantian equilibrium, a general framework for cooperative economic behavior, under weak topological assumptions. Our proof strategy follows a variational approach: we formulate the problem as a Generalized Quasi-Variational Inequality (GQVI) and prove that a solution exists. As a key application, we use this general result to establish the existence of the Walras-Kant equilibrium, a model that integrates competitive market behavior for private goods with Kantian cooperation for public goods.
This study examines the impact of corporate Bitcoin holdings on firm-level risk and return using the Capital Asset Pricing Model (CAPM), Fama–French three-factor (FF3), and Fama–French four-factor (FF4) models. Analyzing Bitcoin exposure through quantity and market value, we find that both measures significantly increase systematic risk, idiosyncratic risk, total volatility, and abnormal returns. Market value exerts a stronger influence on systematic risk, whereas quantity primarily drives idiosyncratic and total risks. The FF3 model yields the largest risk factor coefficients, while the FF4 model indicates that market value reduces firm sensitivity to size and value factors, and quantity enhances exposure to the value factor. Abnormal returns suggest market premiums driven by speculative sentiment. These findings highlight Bitcoin’s dual role as a risk amplifier and return enhancer, with implications for corporate financial strategy, investor decision-making, and digital asset regulation. This study underscores the need to integrate cryptocurrency exposure into asset pricing frameworks.
We consider an insurance market with a finite or infinite number of competitive insurers. Each insurer makes an optimal decision on reinsurance and investment to maximize her expected utility function that depends on her individual wealth as well as the average wealth of her competitors. This decision-making problem incorporating relative performance concerns is modeled as an n-player game and as a mean-field game (MFG) in the case of infinite insurers. We obtain the explicit optimal reinsurance-investment strategy when a Nash equilibrium is reached. The impacts of relative performance concerns on the optimal reinsurance-investment decisions are examined both theoretically and numerically.
While models used to evaluate climate change are inherently simplified, and thus imperfect, representations of reality, they remain indispensable tools for quantifying policy impacts and guiding decision-making. How can we leverage these models while accounting for their potential misspecification -the possibility that none of the existing models is adequate? This paper addresses this question by applying recent advances in decision theory to analyze how concerns about model misspecification influence key climate strategies, including the stringency of emissions targets and the extent of climate overshoot. Using a decision-theoretic framework applied to emulators of IPCC-assessed scenarios, we quantify the trade-offs between mitigation costs and climate-induced economic damages. We show that when decision-makers explicitly account for the possibility of model misspecification, they rationally adopt more ambitious climate goals, characterized by stricter carbon budgets and reduced reliance on emissions overshoot. These results underscore the importance of integrating model misspecification concerns into climate policy design, providing a rigorous foundation for precautionary and robust decision-making under deep uncertainty.
Semi-Markov models provide a more realistic framework for modeling and forecasting real-world processes than conventional Markov models. However, their practical use is severely constrained by the lack of analytical tractability. This paper proposes a viable and tractable alternative based on an expanded-state Markov model with a specific topological structure. We show that the proposed semi-Markov model remains analytically tractable when the process consists of two semi-Markovian states, each represented by at least three Markovian sub-states. Closed-form expressions for the state transition probabilities are derived for both discrete- and continuous-time settings. Empirical analysis using long-term data on U.S. business and stock market cycles demonstrates that the proposed model provides a statistically superior fit compared to conventional Markov models, underscoring its practical relevance and effectiveness.
