ABSTRACT We introduce a novel mean‐field game model for multi‐sector economic growth in which a dynamically evolving externality, influenced by the collective action of countries, plays a central role. By integrating environmental considerations in classical growth models, the framework incorporates “common noise” to capture shared uncertainties about the externality variable. We establish the existence and uniqueness of a mean‐field game equilibrium by reformulating the equilibrium conditions as a Forward‐Backward Stochastic Differential Equation via the stochastic maximum principle, first applying a contraction‐mapping argument to guarantee a unique solution, then employing the concept of weak equilibria to prove existence under more general assumptions, and finally invoking a specific monotonicity regime to reaffirm uniqueness. We provide a numerical resolution for a specified model using a fixed‐point approach combined with neural network approximations.
The growing integration of solar power into electricity markets increasingly demands advanced risk management tools to address the inherent variability of solar radiation and its interaction with electricity prices. This paper introduces a novel framework for modeling and pricing new financial instruments designed to link payoffs directly to solar radiation levels, namely SoRad and SoREd. The proposed methodology models solar radiation through a bounded transformation and a Markov-modulated Ornstein-Uhlenbeck diffusion, where the latent regime is a two-state continuous-time Markov chain (CTMC) with month-dependent generator. The model captures persistent switches between high- and low-radiation regimes while preserving tractable conditional moments for pricing. Building on the observed correlation between solar radiation and electricity prices, we derive mean-variance equilibrium prices and hedging rules that combine solar derivatives with electricity futures at the pricing date. The resulting prices are interpreted as inception-date valuations for contracts over a fixed delivery period, not as a full dynamic secondary-market equilibrium. The proposed instruments provide standardized tools for transferring solar radiation risk and reducing residual exposure to weather-driven and electricity-linked revenue fluctuations.
ABSTRACT Assuming a lognormal distribution for the underlying risky asset, we study the limiting distribution of a volatility target index as the rebalancing time step approaches zero. Two limit theorems (a strong law of large numbers and a central limit theorem) are established, and as an application, the exact limiting distribution is derived. We demonstrate that the volatility of the limiting distribution is consistently larger than the target volatility, and study the dependence of the convergence speed on the observation‐window parameter . Besides the exact formula for the drift and the volatility of the limiting distribution, their upper and lower bounds are derived. As a corollary of the exact limiting distribution, we obtain a vega conversion formula which converts the rho sensitivity of a financial derivative on the limiting diffusion to the vega sensitivity of the same financial derivative on the underlying of the volatility target index.
The optimal mean-variance selling problem seeks to determine a dynamically optimal stopping time in the nonlinear problem , where is a geometric Brownian motion with strictly positive drift, the supremum is taken over stopping times of , and is a given and fixed constant. The solution to the problem is known when the horizon is infinite, however, the method of proof developed to solve the problem in that case is not applicable in the case when the horizon is finite. In this paper, we develop a new method of proof, which solves the problem when the horizon is finite. In this way, we find that the dynamically optimal stopping time is given by , where the function can be characterized as a unique solution to a nonlinear Volterra integral equation. We also prove that the dynamically optimal stopping time satisfies the smooth fit principle. To our knowledge, this is the first time that such a nonlinear phenomenon of "dynamic smooth fit" has been derived in the literature. MSC2020 Classification: 60G40, 35R35, 60J65, 90C30, 45G10, 91B06
Environmental, social, and governance (ESG) ratings are pivotal in sustainable investing, informing portfolio construction and regulatory compliance. We develop a game-theoretic principal-agent model involving ESG-conscious investors and competing rating agencies. Investors impose ESG-related portfolio constraints by selecting ratings from different agencies, while agencies strategically issue ESG ratings based on a trade-off between financial incentives related to market share and reputational risks from deviating from ESG fundamentals. The interaction forms an extensive-form game with imperfect information, and we characterize its Stackelberg stable equilibrium. Our analysis shows that (1) agencies may strategically deviate ESG ratings from fundamentals under certain conditions, and such deviations tend to align ratings with asset returns when investors have strong green preferences; (2) asymmetric aversion for deviation generate rating disagreement between multiple agencies even when they have access to the same ESG fundamentals, potentially allowing one agency to monopolize green investors; (3) the magnitude of rating deviation governs a trade-off between investors' pecuniary utility and portfolio's fundamental greenness; and (4) firm-level greenwashing exacerbates rating deviation and further distorts the trade-off between utility and greenness. These results rationalize several empirical findings in the ESG literature and provide insights into designing mechanisms that encourage more informative ESG ratings.
