This paper investigates how additional independent background risks (e.g. fluctuations in financial asset prices, human capital, etc.) affect higher-order stochastic dominance. We show that an appropriately chosen background risk can induce a ranking of prospects under higher-order stochastic dominance according to the magnitude of their higher-order central moments. In addition, we provide an equivalent representation of mean–variance–skewness preferences. This representation can be characterized by the four properties of certainty, downside risk monotonicity, additivity, and skewness-free continuity, and it further strengthens the connection between mean–variance–skewness preferences and third-order stochastic dominance.
This paper investigates a Stackelberg investment and reinsurance game with the ambiguous correlation between the financial and insurance markets. The ambiguous correlation is modeled by constructing a set of non-equivalent prior probability measures. The reinsurer acts as the leader charging reinsurance premiums, while the insurer acts as the follower participating in proportional reinsurance. Both the reinsurer and the insurer invest in the financial market with the aim to maximize the expected utility of their terminal wealth under the worst-case scenario. The Stackelberg equilibrium strategy is derived by solving the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations. Through theoretical analysis and numerical simulation, the influence of ambiguity on the Stackelberg equilibrium strategy is analyzed. We find that both the insurer and reinsurer adopt more conservative strategies in the presence of market correlation ambiguity, with the insurer being more significantly affected by such ambiguity, while the reinsurer is less influenced. When both the insurer and the reinsurer maintain investment in the risky asset, the ambiguity faced by the insurer leads to higher reinsurance premiums, whereas the ambiguity faced by the reinsurer results in lower reinsurance premiums.
This paper develops an infinite-horizon, Lucas-type general equilibrium model of asset markets with a fuzzy decision-making investor. The model yields closed-form intuitive solutions for the equilibrium equity premium and risk-free rate. The solutions clearly separate the equity premium that investors demand for bearing measurable risk from the premium that arises from unmeasurable uncertainty. The magnitude of the premium for risk depends solely on risk aversion, whereas the premium for unmeasurable uncertainty reflects both investor sentiment and the degree of that uncertainty. Our results imply that a pessimistic outlook reduces the risk-free rate and increases the equity premium. Under reasonable parameterizations, the model generates the long-run first moments of the U.S. risk-free rate and equity premium with moderate risk aversion and without inflating the variance of equity returns. This work contributes to the asset-pricing literature by showing that fuzzy sets and membership functions provide a tractable and flexible framework for modeling approximate decision-making in asset markets.
This article proposes the Range Value at Risk under model uncertainty, denoted by RG-VaR, a risk measure that explicitly incorporates both best-case and worst-case scenarios within sublinear expectation framework. We prove that RG-VaR satisfies monotonicity, translation invariance, positive homogeneity, and comonotonic additivity, and we establish its exact relationship to the G-Value at Risk (G-VaR) and G-Expected Shortfall (G-ES). For G-normally distributed risks, we derive closed-form expressions. A counter example demonstrates that G-ES, in either its best-case or worst-case version, can fail to be subadditive. Nevertheless, we show that G-VaR, G-ES, and RG-VaR are subadditive when risks are independent and G-normal. Sensitivity analysis reveals that significance levels are the primary drivers of the risk measures, while volatility uncertainty affects them asymmetrically: upper volatility perturbations dominate worst-case risk, whereas lower volatility changes mainly influence best-case risk. We further propose hypothesis tests for risk measures under model uncertainty and validate G -ES and RG -VaR through numerical simulations and an empirical study on the S P 500 index. The results show that, under the proposed backtesting procedure, worst-case RG -VaR performs best and remains stable across different significance levels. By incorporating model uncertainty, the proposed framework provides both worst-case and best-case range-based risk assessments, offering conservative and optimistic benchmarks for tail risk in complex market environments.
In this work, we extend the Bismut-Elworthy-Li formula to facilitate the sensitivity analysis of market products modeled by semi-linear mean-field stochastic differential equations driven by fractional Brownian motion. We establish an expression representation for the stochastic flow of the solutions in relation to their Malliavin derivatives, after showing the existence, uniqueness, and weak differentiability of these solutions. The framework is applied to analyze the sensitivity of variance swaps to the initial conditions, calculate the vega of the derivatives price with stochastic volatility modeled by mean-field equations, and assess the sensitivity of path-dependent Asian options to the initial states.
We provide a stochastic analysis of an overlapping-generations model under incomplete markets. By casting individual optimization with idiosyncratic income risk into a forward–backward stochastic differential-equation (FBSDE) system, we (i) establish existence and uniqueness of the dynamic general-equilibrium interest rate and (ii) derive analytical and semi-explicit formulas for both the equilibrium interest-rate path and the natural borrowing limit – defined as the discounted expected shortfall of future income. Our FBSDE-based approach yields tractable policy functions and equilibrium mappings without relying on high-dimensional PDE methods, offering clear insights into how income dynamics and demographic structure drive interest-rate fluctuations and credit constraints.
We study the dynamic pricing of discrete goods over a finite selling horizon. One way to capture both the elastic and stochastic reaction of purchases to price is through a model where sellers control the intensity of a counting process, representing the number of sales thus far. The intensity describes the probabilistic likelihood of a sale, and is a decreasing function of the price a seller sets. A classical model for ticket pricing, which assumes a single seller and finite time horizon, is by Gallego and van Ryzin (1994) and it has been widely utilized by airlines, for instance. Extending to more realistic settings where there are multiple sellers, with finite inventories, in competition over a finite time horizon is more complicated both mathematically and computationally. We introduce a dynamic mean field game of this type, and some numerical and existence results. In particular, we analyze the associated coupled system of Hamilton-Jacobi-Bellman and Kolmogorov differential-difference equations, and we prove the existence and uniqueness results under certain conditions. Then, we demonstrate a numerical algorithm to find this solution and provide some insights into the macroeconomic market parameters. Finally, we present a qualitative comparison of our findings with airfare data.
We analyze a continuous-time optimal trade execution problem in multiple assets where the price impact and the resilience can be matrix-valued stochastic processes that incorporate cross-impact effects. In addition, we allow for stochastic terminal and running targets. Initially, we formulate the optimal trade execution task as a stochastic control problem with a finite-variation control process that acts as an integrator both in the state dynamics and in the cost functional. We then extend this problem continuously to a stochastic control problem with progressively measurable controls. By identifying this extended problem as equivalent to a certain linear-quadratic stochastic control problem, we can use established results in linear-quadratic stochastic control to solve the extended problem. This work generalizes [Ackermann, Kruse, Urusov; FinancStoch'24] from the single-asset setting to the multi-asset case. In particular, we reveal cross-hedging effects, showing that it can be optimal to trade in an asset despite having no initial position. Moreover, as a subsetting we discuss a multi-asset variant of the model in [Obizhaeva, Wang; JFinancMark'13].
We consider Merton's problem with proportional transaction costs. It is well known that the optimal investment strategy is characterized by two trading boundaries, the buy boundary and the sell boundary, between which lies the no-trading region. We investigate how these two trading boundaries vary with the transaction cost rates. We show that the cost-adjusted trading boundaries are monotone in the transaction costs. Our result implies the following: (i) the Merton line must lie between the two cost-adjusted trading boundaries; and (ii) when the Merton line is positive, both the buy and sell boundaries are monotone in the transaction cost rates, and consequently the Merton line lies in the no-trading region.
This paper explores the optimal investment problem of a renewal risk model with generalized Erlang distributed interarrival times. The phases of the Erlang interarrival time is assumed to be observable. The price of the risky asset is driven by the constant elasticity of variance model (CEV) and the insurer aims to maximize the exponential utility of the terminal wealth by asset allocation. By solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation, we establish the concavity of the value function and derive an explicit expression for the optimal investment policy when the interest rate is zero. When the interest rate is nonzero, we obtain an explicit form of the optimal investment strategy, along with a semi-explicit expression of the value function, whose concavity is also rigorously proven.
We consider a market of risky financial assets where the participants are an informed trader, a mass of uniformed traders and noisy liquidity providers. We prove the existence of a market-clearing equilibrium when the insider internalizes her power to impact prices. In the price-impact equilibrium the insider strategically reveals a noisier (compared to when the insider takes prices as given) signal, and prices are less reactive to the publicly available information. In contrast to the related literature, we show that in the price-impact equilibrium, the insider's ex-ante welfare monotonically increases in the signal precision. This clarifies when a trader with market power is motivated to both obtain and refine her private information. Furthermore, even though the uniformed traders act as price-takers, the effect of price impact is ex-ante welfare improving for them. By contrast, internalization of price impact may reduce insider ex-ante welfare. This happens provided the insider is sufficiently risk averse and the uninformed traders are sufficiently risk tolerant.
This paper studies continuous-time reinforcement learning in jump-diffusion models by featuring the q-learning (the continuous-time counterpart of Q-learning) under Tsallis entropy regularization. Contrary to the Shannon entropy, the general form of Tsallis entropy renders the optimal policy not necessarily a Gibbs measure. Herein, the Lagrange multiplier and KKT condition are needed to ensure that the learned policy is a probability density function. As a consequence, the characterization of the optimal policy using the q-function also involves a Lagrange multiplier. In response, we establish the martingale characterization of the q-function and devise two q-learning algorithms depending on whether the Lagrange multiplier can be derived explicitly or not. We also study two numerical examples, namely, an optimal liquidation problem in dark pools and a non-LQ control problem. It is interesting to see therein that the optimal policies under the Tsallis entropy regularization can be characterized explicitly, which are distributions concentrated on some compact support. The satisfactory performance of our q-learning algorithms is illustrated in each example.
This paper focuses on a kind of McKean-Vlasov backward stochastic differential equation with Markov regime switching, while the terminal state is constrained in ℝ_+. By virtue of the terminal perturbation method developed by Ji and Peng [7] and Ekeland’s variational principle, we establish a stochastic maximum principle (necessary condition) for the optimal terminal state under Lions derivative. As an application, we explore the backward formulation of continuous-time mean-variance portfolio selection with bankruptcy prohibition under two market regimes (bull market and bear one) and derive the corresponding optimal terminal wealth as well.
An agent solves an exponential utility maximisation problem that is robust to parameter misspecification and where the optimal strategy continuously adapts to new information. The agent invests in a risk-free asset and in risky stocks whose prices follow geometric diffusion processes. The agent does not know the drift parameters of the stock price dynamics, so she considers a set of alternative measures to make the investment problem robust to model misspecification and employs a continuous-time estimator to learn the value of the drift parameters as new information arrives during the investment horizon. For the two risky asset case, the agent’s value function is characterised as the solution to a non-linear PDE. We show that the value function has a stochastic representation and use it to analyse the optimal adaptive-robust strategy and to compare it with various benchmarks.
Climate stress-tests aim at projecting the financial impacts of climate change, covering both transition and physical risks under given macro scenarios. However, in practice, transition risk has been the main focus of supervisory and academic exercises, and existing tools to downscale these macroeconomic projections to the firm level remain limited. We develop a methodology to downscale sector-level trajectories into firm-level projections for credit risk stress-tests. The approach combines probabilistic modeling with stochastic control to capture firm-level uncertainty and optimal decision-making. It can be applied to any transition scenario or sector and highlights how firm-level characteristics such as initial intensity, abatement cost, and exposure to uncertainty shape heterogeneous firm-level responses to the transition. The model explicitly incorporates firm-level business uncertainty through stochastic dynamics on relative emissions and sales, which affect both optimal decisions and resulting financial projections. Firms’ rational behavior is modeled as a stochastic minimization problem, solved numerically through a method we call Backward Sampling. Illustrating our method with the NGFS transition scenarios and three types of companies (Green, Brown and Average), we show that firm-specific intensity reduction strategies yield significantly different financial outcomes compared to assuming uniform sectoral decarbonisation rates. Moreover, investing an amount equivalent to the total carbon tax paid at a given date is limited by its lack of a forward-looking feature, making it insufficient to buffer against future carbon shocks in a disorderly transition. This highlights the importance of firm-level granularity in climate risk assessments. By explicitly modeling firm heterogeneity and optimal decision-making under uncertainty, our methodology complements existing approaches to granular transition risk assessment and contributes to the ongoing development of scenario-based credit risk projections at the firm level.
Financial equilibrium models provide important information on the movement of asset prices in response to subjective beliefs and consumption patterns of economic agents. It is standard to assume that agents are rational and have fixed preferences from the outset—in particular, these do not depend on other agents’ actions in the market. This work deviates from this assumption by allowing the subjective views and consumption clocks of an individual agent to depend on the whole history of the wealth and consumption distribution across agents in the economy. The updating mechanism is generic and may accommodate different behavioural models; for example, it can model herding. In order to analyse existence and uniqueness of equilibrium, we assume that agents have numeraire-invariant preferences, which are rich enough to render any observe agents’ behaviour optimal. The market contains a borrowing and lending account in zero net supply, as well as a stock in positive net supply providing certain dividend stream, exogenously specified. A characterisation of existence and uniqueness of equilibrium in a Brownian setting is provided in terms of stochastic differential equations. The proposed framework naturally allows for equilibria where the risky asset in positive net supply is suboptimal to hold for investment.
This paper develops a representative-agent model where consumption and dividends are cointegrated and examines its asset pricing implications. By specifying the dividend-consumption ratio as a Jacobi process, the model accommodates transitory deviations between dividends and consumption while keeping their ratio stationary. It also yields explicit formulas for equilibrium prices, risk-free rates, and equity premia, revealing countercyclical excess returns, plausible Sharpe ratios, and robust volatility. A calibration to historical U.S. data demonstrates the model’s capacity to match key financial moments, including the equity premium and price-dividend ratios. Overall, dividend-consumption cointegration combines tractability with explanatory power in asset pricing.
This paper investigates the optimal management of a firm’s financial resources by formulating a joint capital injection and dividend optimization problem in which the surplus process evolves as a general integrable Lévy process accommodating both positive and negative jumps. I characterize the value function of the control problem as the unique viscosity solution to a Hamilton-Jacobi-Bellman variational inequality, under semi-state constraints. A key result reveals a dichotomy in optimal capital injection strategies: either it is optimal to inject capital only when the surplus becomes negative up to a specific threshold, or it is never optimal to inject at all. The nature of the negative jumps – finite versus infinite variation – affects the boundary conditions and the structure of the optimal strategy. In the second part, assuming finite activity of negative jumps, I provide a complete characterization of the optimal dividend strategy, identifying a threshold above which immediate dividend payments are optimal.
