Stochastic resetting has attracted significant attention in recent years due to its wide-ranging applications across physics, biology, and search processes. In most existing studies, however, resetting events are governed by an external timer and remain decoupled from the system's intrinsic dynamics. In a recent Letter by Biswas et al, we introduced threshold resetting (TR) as an alternative, event-driven optimization strategy for target search problems. Under TR, the entire process is reset whenever any searcher reaches a prescribed threshold, thereby coupling the resetting mechanism directly to the internal dynamics. In this work, we study TR-enabled search by $N$ non-interacting diffusive searchers in a one-dimensional box $[0,L]$, with the target at the origin and the threshold at $L$. By optimally tuning the scaled threshold distance $u = x_0/L$, the mean first-passage time can be significantly reduced for $N \geq 2$. We identify a critical population size $N_c(u)$ below which TR outperforms reset-free dynamics. Furthermore, for fixed $u$, the mean first-passage time depends non-monotonically on $N$, attaining a minimum at $N_{\mathrm{opt}}(u)$. We also quantify the achievable speed-up and analyze the operational cost of TR, revealing a nontrivial optimization landscape. These findings highlight threshold resetting as an efficient and realistic optimization mechanism for complex stochastic search processes.

Data from national accounts show no effect of change in net saving or consumption, in ratio to market-value capital, on change in growth rate of market-value capital (capital acceleration). Thus it appears that capital growth and acceleration arrive without help from net saving or consumption restraint. We explore ways in which this is possible, and discuss implications for economic teaching and public policy

We study the upper hedging price for contingent claims in market models with strong types of arbitrage: increasing profit, strong arbitrage, and arbitrage of the first kind. The existence of arbitrage may make the price smaller than if it did not exist. For example, when the asset price process has a reflecting boundary, which introduces increasing profit in the market model, the option prices are reduced to those of the corresponding options that knock-out at the boundary. Furthermore, we demonstrate that corporate stock price processes with increasing profit are obtained as a result of corporate stock issuance and repurchase plans.

Static regret to a single expert is often the wrong target for strictly online prediction under non-stationarity, where the best expert may switch repeatedly over time. We study Policy-Controlled Generalized Share (PCGS), a general strictly online framework in which the generalized-share recursion is fixed while the post-loss update controls are allowed to vary adaptively. Its principal instantiation in this paper is PCGS-TF, which uses a causal Transformer as an update controller: after round t finishes and the loss vector is observed, the Transformer outputs the controls that map w_t to w_{t+1} without altering the already committed decision w_t. Under admissible post-loss update controls, we obtain a pathwise weighted regret guarantee for general time-varying learning rates, and a standard dynamic-regret guarantee against any expert path with at most S switches under the constant-learning-rate specialization. Empirically, on a controlled synthetic suite with exact dynamic-programming switching-oracle evaluation, PCGS-TF attains the lowest mean dynamic regret in all seven non-stationary families, with its advantage increasing for larger expert pools. On a reproduced household-electricity benchmark, PCGS-TF also achieves the lowest normalized dynamic regret for S = 5, 10, and 20.

Algorithmic content targeting homogenizes information, with implications for strategic interactions. For example, this increased homogenization was arguably responsible for the run on the Silicon Valley Bank. We argue that existing measures of similarity are inappropriate for studying games -- especially coordination games -- because they do not discipline agents' conditional beliefs. We propose a class of stochastic orders, Concentration Along the Diagonal (CAD), built on agents' conditional beliefs. In canonical binary-action coordination games, greater CAD-similarity is both necessary and sufficient for strategic similarity -- agents adopt the same strategy. We further demonstrate CAD's applicability in congestion games, collective action, and second-price auctions.

Investment in artificial intelligence (AI) has grown rapidly, yet its returns to scientific research remain poorly understood. We study how AI reshapes the production of science using a comprehensive dataset of research proposals submitted to a large international funding agency, including both funded and unfunded projects. Combining keyword extraction with large language model classification, we identify the presence, type, and functional role of AI within each proposal and link these measures to detailed budget allocations, team structure, and subsequent publication outcomes. We find that, in the short run, AI adoption is associated with modest improvements in scientific outcomes concentrated in the upper tail. Instead, its primary effects arise in the organization of research: AI-enabled projects reallocate resources toward human capital, involve larger teams, and undertake a broader set of tasks. These patterns are consistent with a reorganization of the scientific production process rather than immediate efficiency gains, in line with theories of general-purpose technologies. Task-level analyses further show that activities expanded in AI-enabled projects, particularly ideation and experimentation, are increasingly compatible with large language model capabilities, suggesting potential for future productivity gains as these technologies mature.

We investigate stability properties of weak supermartingale optimal transport (WSOT) problems on $\mathbb{R}$. For probability measures $μ,ν\in\mathcal{P}_r$ satisfying $μ\leq_{cd} ν$ (equivalently, $Π_S(μ,ν)\neq\emptyset$), we consider supermartingale couplings $π=μ(d x)π_x(d y)$ and the weak transport functional \[ V_S^C(μ,ν) := \inf_{π\inΠ_S(μ,ν)} \int_\mathbb{R} C(x,π_x)\,μ(d x), \] for some appropriate cost function $C:\mathbb{R}\times\mathcal{P}_r\to\mathbb{R}$. Our first main contribution is an approximation result in adapted Wasserstein distance: under $W_r$-convergence of marginals $(μ^k,ν^k)\to(μ,ν)$ with $μ^k\leq_{cd} ν^k$, any $π\inΠ_S(μ,ν)$ can be approximated by $π^k\inΠ_S(μ^k,ν^k)$ such that $A\mathcal{W}_r(π^k,π)\to0$. As a consequence, we obtain the continuity of the functional $(μ,ν) \mapsto V_S^C(μ,ν)$, and the monotonicity principle for WSOT.

In traditional banking, repeated deposit-and-lend cycles let a single dollar of reserves support multiple dollars of claims. Decentralized finance produces an analogous structure with tokens. Constructing a Token Graph of 10,200 tokens across 200 blockchains, this paper maps the resulting hierarchy and shows that, by late 2025, each dollar of base assets supports $4.7 of total claims. An embedded yield correction disentangles two channels that raw data conflates: a compositional channel, where lending protocols concentrate in deeper tiers and mechanically raise average yields; and a liquidity channel, where each derivation step reduces secondary-market depth and depresses yields in liquidity-sensitive pools. The liquidity channel concentrates in DEX pools and vanishes in lending pools. A yield decomposition shows that the tier gradient operates entirely through fundamental protocol yields, not incentive-token emissions; quantile regressions reveal that the structural associations concentrate in the upper tail of the yield distribution, with near-zero effects at the median. These findings reframe DeFi's "double counting" as a structural risk question and identify liquidity fragmentation as the primary mechanism associated with yield variation across the token hierarchy.

Price controls kill the incentive for arbitrage. We prove a Chaos Theorem: under a binding price ceiling, suppliers are indifferent across destinations, so arbitrarily small cost differences can determine the entire allocation. The economy tips to corner outcomes in which some markets are fully served while others are starved; small parameter changes flip the identity of the corners, generating discontinuous welfare jumps. These corner allocations create a distinct source of cross-market misallocation, separate from the aggregate quantity loss (the Harberger triangle) and from within-market misallocation emphasized in prior work. They also create an identification problem: welfare depends on demand far from the observed equilibrium. We derive sharp bounds on misallocation that require no parametric assumptions. In an efficient allocation, shadow prices are equalized across markets; combined with the adding-up constraint, this collapses the infinite-dimensional welfare problem to a one-dimensional search over a common shadow price, with extremal losses achieved by piecewise-linear demand schedules. Calibrating the bounds to station-level AAA survey data from the 1973--74 U.S. gasoline crisis, misallocation losses range from roughly 1 to 9 times the Harberger triangle.

Systemic financial risk refers to the simultaneous failure or destabilization of multiple financial institutions, often triggered by contagion mechanisms or common exposures to shocks. In this paper, we present a dynamical model of bank leverage (the ratio of asset holdings to equity) a quantity that both reflects and drives risk dynamics. We model how banks, constrained by Value-at-Risk (VaR) regulations, adjust their leverage in response to changes in the price of a single asset, assumed to be held in fixed proportion across banks. This leverage-targeting behavior introduces a procyclical feedback loop between asset prices and leverage. In the dynamics, this can manifest as logistic-like behavior with a rich bifurcation structure across model parameters. By analyzing these coupled dynamics in both isolated and interconnected bank models, we outline a framework for understanding how systemic risk can emerge from seemingly rational micro-level behavior.

Since the formal introduction of its "dual-carbon" strategy in 2020, China has witnessed the concepts of green development and sustainability evolve from policy directives into a broad societal consensus. Within this transformative context, the Environmental, Social, and Governance (ESG) framework has emerged as a critical enabler, mutually reinforcing and synergizing with the national strategic objectives of achieving carbon peak and carbon neutrality. This integration signifies a fundamental shift in corporate philosophy, urging enterprises to transcend a narrow focus on short-term financial metrics. To align with the national vision of ecological civilization and sustainable growth, companies are now expected to proactively fulfill their social responsibilities and pursue long-term, non-financial value creation. This entails a deep integration of ESG principles into the very core of corporate culture and strategy, ensuring their active implementation in daily operations and decision-making processes.

This paper addresses the approximation of the local volatility function in the Cheyette interest rate model. Its main contribution is an explicit analytical formula for approximating local volatility, derived by extending the classical Dupire framework to interest rate models. In particular, an implicit Dupire-like expression for local volatility is first derived for options written on the short rate. This expression is then approximated using a combination of perturbation methods and probabilistic techniques, resulting in a formula expressed in terms of time and strike derivatives of the Bachelier implied variance. The final formula naturally extends to multi-factor Cheyette models and provides a practical tool for model calibration.

How does the endogenous selection of information shape belief formation? In observational settings, individuals only consume information they choose, making it impossible to observe how they would respond to information they actively avoid. We address this identification challenge using a randomized experiment on COVID-19 vaccines in Taiwan. After eliciting subjects' preferences over vaccine-specific reports, we randomly assign them to receive either their chosen or unchosen information, orthogonalizing selection from exposure. We find subjects are more likely to select information about vaccines they already perceive as more effective. Conditional on receiving information, belief updating is substantially larger when the information was self-selected, even after controlling for prior-posterior disagreement. These findings highlight endogenous information demand as a central determinant of persuasion, suggesting that increasing information availability alone may be insufficient when individuals rationally filter out options they perceive as less relevant to their decision-making.

Concentrated Liquidity Market Makers (CLMMs) represent a fundamental innovation in market microstructure, transforming liquidity provision from passive portfolio allocation to active risk management. This evolution creates significant challenges for performance evaluation and strategy optimization, particularly due to the absence of comprehensive historical liquidity data. We address these challenges through a novel methodological framework that reconstructs historical liquidity states from swap transaction data, enabling rigorous backtesting of dynamic liquidity provision strategies. Our parametric reconstruction method achieves high accuracy (approximation errors averaging around 2\%) without relying on historical liquidity snapshots, addressing a critical data gap in decentralized finance research. We apply this framework to evaluate tau-reset strategies--dynamic liquidity reallocation approaches that respond to market movements--across multiple Uniswap v3 pools. Using machine learning to optimize strategy parameters based on market conditions, we identify consistent outperformance (13--23\% higher fees) compared to uniform allocation benchmarks. Our analysis reveals important insights into the risk-return tradeoffs in automated market making, including the critical role of impermanent loss as a dominant risk factor and the effectiveness of asymmetric strategy modifications for capital preservation. These findings contribute to the broader understanding of market microstructure in decentralized exchanges, providing both methodological innovations for performance evaluation and practical insights for liquidity providers navigating this evolving financial landscape.

When AI systems make errors in high-stakes domains like medical diagnosis or autonomous vehicles, a single algorithmic flaw across varying operational contexts can generate highly heterogeneous losses that challenge traditional insurance assumptions. Algorithmic insurance constitutes a novel form of financial coverage for AI-induced damages, representing an emerging market that addresses algorithm-driven liability. However, insurers currently struggle to price these risks, while AI developers lack rigorous frameworks connecting system design with financial liability exposure. We analyze the connection between operational choices of binary classification performance to tail risk exposure. Using conditional value-at-risk (CVaR) to capture extreme losses, we prove that established approaches like maximizing accuracy can significantly increase worst-case losses compared to tail risk optimization, with penalties growing quadratically as thresholds deviate from optimal. We then propose a liability insurance contract structure that mandates risk-aware classification thresholds and characterize the conditions under which it creates value for AI providers. Our analysis extends to degrading model performance and human oversight scenarios. We validate our findings through a mammography case study, demonstrating that CVaR-optimal thresholds reduce tail risk up to 13-fold compared to accuracy maximization. This risk reduction enables insurance contracts to create 14-16% gains for well-calibrated firms, while poorly calibrated firms benefit up to 65% through risk transfer, mandatory recalibration, and regulatory capital relief. Unlike traditional insurance that merely transfers risk, algorithmic insurance can function as both a financial instrument and an operational governance mechanism, simultaneously enabling efficient risk transfer while improving AI safety.

The European Union Emissions Trading System is set to substantially increase the effective carbon price faced by airlines. To quantify the impact of this carbon regulation on the European airline industry, we estimate a two-stage model of airline competition with endogenous route entry, flight frequencies, and pricing using European data on market shares and prices. Counterfactual simulations reveal that the impacts of carbon pricing are highly asymmetric across carrier types and market segments. Consumer surplus declines by up to 25% overall, with medium-haul markets bearing the brunt at up to 90%, while short-haul markets experience positive net welfare gains (including carbon revenue and the social value of avoided emissions) as airlines reallocate capacity toward shorter routes. We find that airline profits decline by 8-45% across scenarios, while carbon tax revenue of $0.9-3.1 billion and a social value of avoided CO2 emissions of $0.5-1.4 billion partially offset the welfare losses. We also show that a hypothetical Wizz Air-Ryanair merger primarily benefits firm profits through network expansion synergies.

This study investigates the relationship between coin demand, payment innovation, COVID-19, and a coin's purchasing power, particularly in emerging countries like Indonesia. The rapid advancement of payment platforms, combined with high adoption during the pandemic, has positioned non-cash payments as a complement or substitute for coin money for transactions. However, there is notably limited coin-money-related research in the economic literature. Employing the autoregressive distributed lag (ARDL) bounds test methodology's cointegration approach using monthly data from 2011 to 2022, our findings reveal a long-term relationship between coin demand and its determinants: payment innovations, the pandemic, coin depreciation, and income. Despite the swift advancement of payment innovations and their usage, coins remain vital to the economy and are unlikely to become obsolete soon. Our study offers essential policy recommendations and enriches the field of knowledge on coin money demand. Policymakers must understand the driving factors of coin demand in both economic and non-economic contexts to improve coin production-related issues and coin circulation policies. Reviewing the Rupiah denomination structure is crucial in addressing the problem of ineffective coin circulation in the economy.

Accurately characterizing the implied volatility curves is a central challenge in option pricing and risk management. The classical SABR model by Hagan et al. has been widely adopted in practice due to its well-defined stochastic volatility structure and its tractable closed-form approximation for Black implied volatility. However, under complex market conditions, its fitting accuracy for implied volatility curves remains limited. To address this issue, this paper proposes an extended model within the SABR framework, referred to as skew-SABR. Specifically, the proposed approach introduces an extension to the stochastic dynamics of the underlying asset price and its variance process, under which a corresponding Black implied volatility expression is derived. By further simplifying and reorganizing the resulting formula, the implied volatility can be expressed in a form that explicitly incorporates a skew parameter, thereby enabling a direct characterization of the asymmetry in the implied volatility curve. The resulting expression preserves the structural simplicity of the Hagan-SABR formula, while significantly enhancing the model's flexibility in capturing complex volatility smile patterns. From a theoretical perspective, the paper provides a systematic analysis of the model specification and the financial interpretation of its parameters. From an empirical perspective, a comprehensive comparison is conducted using data from the Chinese options market over the period 2018--2025. The skew-SABR model is evaluated against the classical Hagan-SABR model, the SVI parameterization, polynomial fitting, and spline-based methods. Numerical results show that, across different market regimes and a wide range of implied volatility curve shapes, the skew-SABR model consistently achieves high and stable fitting accuracy.

This paper revisits and extends the 2013 development by Rockafellar and Uryasev of the Risk Quadrangle (RQ) as a unified scheme for integrating risk management, optimization, and statistical estimation. The RQ features four stochastics-oriented functionals -- risk, deviation, regret, and error, along with an associated statistic, and articulates their revealing and in some ways surprising interrelationships and dualizations. Additions to the RQ framework that have come to light since 2013 are reviewed in a synthesis focused on both theoretical advancements and practical applications. New quadrangles -- superquantile, superquantile norm, expectile, biased mean, quantile symmetric average union, and $φ$-divergence-based quadrangles -- offer novel approaches to risk-sensitive decision-making across various fields such as machine learning, statistics, finance, and PDE-constrained optimization. The theoretical contribution comes in axioms for ``subregularity'' relaxing ``regularity'' of the quadrangle functionals, which is too restrictive for some applications. The main RQ theorems and connections are revisited and rigorously extended to this more ample framework. Examples are provided in portfolio optimization, regression, and classification, demonstrating the advantages and the role played by duality, especially in ties to robust optimization and generalized stochastic divergences.

In the context of containment of default contagion in financial networks, we here study a regulator that allocates pre-shock capital or liquidity buffers across banks connected by interbank liabilities and common external asset exposures. The regulator chooses a nonnegative buffer vector under a linear budget before asset-price shocks realize. Shocks are modeled as belonging to either an $\ell_{\infty}$ or an $\ell_{1}$ uncertainty set, and the design objective is either to enlarge the certified no-default/no-insolvency region or to minimize worst-case clearing losses at a prescribed stress radius. Four exact synthesis results are derived. The buffer that maximizes the default resilience margin is obtained from a linear program and admits a closed-form minimal-budget certificate for any target margin. The buffer that maximizes the insolvency resilience margin is computed by a single linear program. At a fixed radius, minimizing the worst-case systemic loss is again a linear program under $\ell_{\infty}$ uncertainty and a linear program with one scenario block per asset under $\ell_{1}$ uncertainty. Crucially, under $\ell_{1}$ uncertainty, exact robustness adds only one LP block per asset, ensuring that the computational complexity grows linearly with the number of assets. A corollary identifies the exact budget at which the optimized worst-case loss becomes zero. Numerical experiments on the 8-bank benchmark of \cite{Calafiore2025}, on a synthetic core-periphery network, and on a data-backed 107-bank calibration built from the 2025 EBA transparency exercise show large gains over uniform and exposure-proportional allocations. The empirical results also indicate that resilience-maximizing and loss-minimizing interventions nearly coincide under diffuse $\ell_\infty$ shocks, but diverge under concentrated $\ell_1$ shocks.

The institutionalization of stablecoins has led to a paradigm shift in reserve management, accelerated by the 2025 Green Energy and National Infrastructure Underpinning Stablecoins (GENIUS) Act. This study investigates the "Climate-Liquidity Nexus," defined as the structural vulnerability arising from the use of environmentally sustainable but secondary-market-thin assets as collateral for high-velocity digital payment instruments. Utilizing a Vector Error Correction Model (VECM) and GARCH(1,1) volatility frameworks on high-frequency data from 2024 to 2026, we demonstrate that the transition toward green reserves introduces significant "Liquidity Hysteresis." My empirical results indicate that while green bonds fulfill ESG regulatory mandates, they compromise the information-insensitivity of the 1.00 USD peg. Following exogenous climate-finance shocks, the recovery half-life of green-backed stablecoins is found to be 5.4 times longer than that of traditional Treasury-backed counterparts. We find that the "Greenium" paid by issuers acts as a volatility multiplier rather than a safety buffer. These findings suggest that the current regulatory trajectory may inadvertently catalyze systemic fragility during physical risk events, necessitating a redesign of liquidity backstop facilities.

To grow their businesses, entrepreneurs often rely on equity funding. This paper focuses on two elements of entrepreneur-investor equity negotiations: the number of potential investors and the contractual complexity surrounding investor protection. Our approach involves a theoretical model and a series of laboratory experiments that analyze the effects of different bargaining conditions and contractual terms on the equity (ownership) split between entrepreneurs and their investors. We show that the conventional wisdom that entrepreneurs should seek to negotiate with as many investors as possible, while consistent with the theoretical model, is not true in the data. Indeed, negotiating with multiple investors reduces the entrepreneur's profits under most conditions. We also show that investor downside protections may disadvantage early-stage startups, but can be beneficial to later-stage startups. A refinement of belief modeling in multi-party bargaining, as well as a stylized risk allocation framework, reconcile these results with theory predictions. Our findings provide a decision framework for entrepreneurs to optimize their approach to investors and negotiate favorable contractual terms.

Do vulnerabilities in Decentralized Finance (DeFi) destabilize traditional short-term funding markets? While the prevailing ``Contagion Hypothesis'' posits that stablecoin reserve liquidations may transmit distress to traditional markets through fire-sale pressure, we document a short-horizon ``Flight-to-Quality'' pattern in the opposite direction. In the wake of major DeFi exploits, spreads on 3-month AA-rated commercial paper (CP) tend to narrow rather than widen. We interpret this pattern as consistent with a ``liquidity-recycling'' channel: capital leaving DeFi may be re-intermediated into traditional cash-management markets, with regulatory segmentation under SEC Rule 2a-7 making prime-eligible paper a plausible marginal destination. Because we do not directly observe daily fund-level routing into prime money market funds, this mechanism is inferred from pricing patterns and monthly holdings evidence rather than directly identified. The result is specific to exploit-driven operational shocks, this U.S. CP spread, and short event windows.

Probabilistic forecasting is crucial in multivariate financial time-series for constructing efficient portfolios that account for complex cross-sectional dependencies. In this paper, we propose Diffolio, a diffusion model designed for multivariate financial time-series forecasting and portfolio construction. Diffolio employs a denoising network with a hierarchical attention architecture, comprising both asset-level and market-level layers. Furthermore, to better reflect cross-sectional correlations, we introduce a correlation-guided regularizer informed by a stable estimate of the target correlation matrix. This structure effectively extracts salient features not only from historical returns but also from asset-specific and systematic covariates, significantly enhancing the performance of forecasts and portfolios. Experimental results on the daily excess returns of 12 industry portfolios show that Diffolio outperforms various probabilistic forecasting baselines in multivariate forecasting accuracy and portfolio performance. Moreover, in portfolio experiments, portfolios constructed from Diffolio's forecasts show consistently robust performance, thereby outperforming those from benchmarks by achieving higher Sharpe ratios for the mean-variance tangency portfolio and higher certainty equivalents for the growth-optimal portfolio. These results demonstrate the superiority of our proposed Diffolio in terms of not only statistical accuracy but also economic significance.

We develop a statistical framework for risk estimation, inspired by the axiomatic theory of risk measures. Coherent risk estimators -- functionals of P\&L samples inheriting the economic properties of risk measures -- are defined and characterized through robust representations linked to $L$-estimators. The framework provides a canonical methodology for constructing estimators with sound financial and statistical properties, unifying risk measure theory, principles for capital adequacy, and practical statistical challenges in market risk. Numerical illustrations based on simulated and market data demonstrate that coherence of a risk measure does not necessarily carry over to its estimators and show that alternative admissible weight structures within the CRE representation can lead to substantially different capital adequacy outcomes.

The advent of artificial intelligence has significantly enhanced credit scoring technologies. Despite the remarkable efficacy of advanced deep learning models, mainstream adoption continues to favor tree-structured models due to their robust predictive performance on tabular data. Although pretrained models have seen considerable development, their application within the financial realm predominantly revolves around question-answering tasks and the use of such models for tabular-structured credit scoring datasets remains largely unexplored. Tabular-oriented large models, such as TabPFN, has made the application of large models in credit scoring feasible, albeit can only processing with limited sample sizes. This paper provides a novel framework to combine tabular-tailored dataset distillation technique with the pretrained model, empowers the scalability for TabPFN. Furthermore, though class imbalance distribution is the common nature in financial datasets, its influence during dataset distillation has not been explored. We thus integrate the imbalance-aware techniques during dataset distillation, resulting in improved performance in financial datasets (e.g., a 2.5% enhancement in AUC). This study presents a novel framework for scaling up the application of large pretrained models on financial tabular datasets and offers a comparative analysis of the influence of class imbalance on the dataset distillation process. We believe this approach can broaden the applications and downstream tasks of large models in the financial domain.

We study continuous-time mean--variance portfolio selection in markets where stock prices are diffusion processes driven by observable factors that are also diffusion processes, yet the coefficients of these processes are unknown. Based on the recently developed reinforcement learning (RL) theory for diffusion processes, we present a general data-driven RL approach that learns the pre-committed investment strategy directly without attempting to learn or estimate the market coefficients. For multi-stock Black--Scholes markets without factors, we further devise an algorithm and prove its performance guarantee by deriving a sublinear regret bound in terms of the Sharpe ratio. We then carry out an extensive empirical study implementing this algorithm to compare its performance and trading characteristics, evaluated under a host of common metrics, with a large number of widely employed portfolio allocation strategies on S\&P 500 constituents. The results demonstrate that the proposed continuous-time RL strategy is consistently among the best, especially in a volatile bear market, and decisively outperforms the model-based continuous-time counterparts by significant margins.

We study a problem of optimal irreversible investment and emission reduction formulated as a nonzero-sum dynamic game between an investor with environmental preferences and a firm. The game is set in continuous time on an infinite-time horizon. The firm generates profits with a stochastic dynamics and may spend part of its revenues towards emission reduction (e.g., renovating the infrastructure). The firm's objective is to maximize the discounted expectation of a function of its profits. The investor participates in the profits, may decide to invest to support the firm's production capacity and uses a profit function which accounts for both financial and environmental factors. Nash equilibria of the game are obtained via a system of variational inequalities. We formulate a general verification theorem for this system in a diffusive setup and construct an explicit solution in the zero-noise limit. Our explicit results and numerical approximations show that both the investor's and the firm's optimal actions are triggered by moving boundaries that increase with the total amount of emission abatement.

Since the beginning of this century the Colombian monetary authority has conducted monetary policy under a strategy based on setting targets for interest rate and inflation, while allowing the exchange rate of the U.S. dollar in domestic currency to float freely. This paper takes that strategy into account in order to explain inflation. Our econometric results were obtained by applying the Generalized Method of Moments to test the hypotheses derived from the structural form of our model. The main findings indicate: a. the validity of a Phillips curve.That is, a positive relationship between the inflation rate and the output gap, conditional on inflation expectations; b. that the monetary authority has reacted to shocks in inflation and in the output gap by adjusting its policy in the appropriate direction but, up to the end of 2025, without being able to claim that its responses have always been timely and consistently forceful. In other words, it can be said that the monetary authority has not been aggressive in ensuring that observed inflation returns rapidly to levels consistent with the inflation target range.

This paper introduces \emph{biased mean regression}, estimating the \emph{biased mean}, i.e., $\mathbb{E}[Y] + x$, where $x \in \mathbb{R}$. The approach addresses a fundamental statistical problem that covers numerous applications. For instance, it can be used to estimate factors driving portfolio loss exceeding the expected loss by a specified amount (e.g., $ x=\$10 billion$) or to estimate factors impacting a specific excess release of radiation in the environment, where nuclear safety regulations specify different severity levels. The estimation is performed by minimizing the so-called \emph{superexpectation error}. We establish two equivalence results that connect the method to popular paradigms: (i) biased mean regression is equivalent to quantile regression for an appropriate parameterization and is equivalent to ordinary least squares when $x=0$; (ii) in portfolio optimization, minimizing \emph{superexpectation risk}, associated with the superexpectation error, is equivalent to CVaR optimization. The approach is computationally attractive, as minimizing the superexpectation error reduces to linear programming (LP), thereby offering algorithmic and modeling advantages. It is also a good alternative to ordinary least squares (OLS) regression. The approach is based on the \emph{Risk Quadrangle} (RQ) framework, which links four stochastic functionals -- error, regret, risk, and deviation -- through a statistic. For the biased mean quadrangle, the statistic is the biased mean. We study properties of the new quadrangle, such as \emph{subregularity}, and establish its relationship to the quantile quadrangle. Numerical experiments confirm the theoretical statements and illustrate the practical implications.