This paper investigates a stochastic differential investment and reinsurance game with delay between two competitive constant absolute risk aversion (CARA) insurers in a defaultable market. Each insurer purchases a proportional reinsurance contract to transfer the claim risk and invests its wealth in three assets: a risk-free asset, a risky asset whose price process is described by the Heston local-stochastic volatility model, and a defaultable bond. Introducing the delay feature, we obtain the wealth process modeled by the stochastic differential delay equation. The competitive relationship between the two insurers is characterized by the non-zero-sum stochastic differential game in which two insurers consider the relative performance measured by the difference in their terminal wealth. The objective of each insurer is to maximize the expected CARA utility of the combination of its terminal wealth and the relative performance with delay. Applying the dynamic programming approach, we obtain the Hamilton-Jacobi-Bellman equation. Then, we employ the perturbation method to solve complex nonlinear partial differential equations and obtain the asymptotic solutions for the Nash equilibrium strategies as well as the corresponding value functions in the post-default case and pre-default case. Furthermore, sensitivity analysis is conducted to explain the effects of model parameters on the equilibrium strategy.

Generalized Model of a Stochastic Common Property Fishery Differential Game: A Numerical Study

SDG-CDA: Stackelberg Differential Games and Combinatorial Double Auctions-Based Pricing Mechanism in Cloud–Edge Environment

ABSTRACT In this study, we propose a dynamic event‐triggered (DET) optimal control method for nonlinear continuous‐time multiplayer nonzero‐sum (NZS) games with time delays using adaptive dynamic programming (ADP). To mitigate the adverse effects of time delays on the system state, a Lyapunov–Krasovskii function that incorporates historical state information is integrated into the cost function. To reduce computational burden and enhance control efficiency, a DET mechanism is introduced to formulate coupled Hamilton–Jacobi (HJ) equations. The ADP technique is utilized to derive event‐triggering optimal strategies by employing a single‐critic neural network architecture. The weights of the neural network are adjusted according to the rules obtained from a gradient descent procedure, and these adjustments are enhanced by employing experience replay (ER) methods to mitigate the stringent conditions of the persistent excitation. The Lyapunov method is applied to ensure the stability of the system state and the convergence of the neural network weights. Finally, the effectiveness of the proposed control method is confirmed via a simulation involving a three‐player differential game.

With the rapid advancement of space technology, satellites are playing an increasingly vital role in fields such as Earth observation, communication and navigation, space exploration, and military applications. Efficiently deploying satellite missions under multi-objective, multi-constraint, and dynamic environments has become a critical challenge in the current aerospace domain. This paper integrates the concepts of game theory and proposes a distributed collaborative task model suitable for on-orbit satellite mission planning. A two-player impulsive maneuver game model is constructed using differential game theory. Based on the ideas of Nash equilibrium and distributed collaboration, multi-agent technology is applied to the distributed collaborative task planning, achieving collaborative allocation and countermeasure strategies for multi-objective and multi-satellite scenarios. Experimental results demonstrate that the method proposed in this paper exhibits good adaptability and robustness in multiple impulse scheduling, maneuver strategy iteration, and heterogeneous resource utilization, providing a feasible technical approach for mission planning and game confrontation in satellite clusters.

We consider a differential game of many pursuers and one evader. The motion of the players is described by a linear differential equations. Control functions of the players are subject to generalized integral constraints. The position of the pursuit and evader at some time t is described by x(t) and y(t) respectively, z(t) represents the state of the game express as z(t) = y(t) - x(t). Game is said to be completed if z(t) = 0, that is if the position of pursuer and evader is the same. Pursuer tries to complete the game and evader pursues the opposite goal. We construct a formula for guaranteed pursuit time and prove that it an optimal pursuit time. To this end, we construct the optimal strategies of the players. Lastly, we demonstrate our results with a numerical example.

Pursuit Differential Game Problem with Component-wise Constraints in $$l_\infty $$

Abstract In this paper, we model a differential game played à la Stackelberg between a regulator and a polluting firm in a stock pollution context. The regulator can be a single body deciding on the emission standard and the probability of inspection overtime as functions of the pollution stock. Alternatively, the regulator can delegate the inspection activities to a local agency that maximizes revenues coming from fines net of inspection costs. Although the objective of the agency departs from social welfare, decentralization can be welfare improving, crucially depending on the type of strategic interaction between the local agency and the polluting firm, as well as on the firm anticipating the effects of current pollution decisions on future regulatory policy. Up to our kowledge, this is the first paper dealing with hierarchical regulation in a stock pollution context.

Deriving control policies for a make-to-order manufacturing system is often predicated on a well-specified probabilistic model that governs demand realization. In practice, however, such a model may be a simplification of the actual scenario because of tractability considerations. Consequently, policies obtained under such simplifications may perform poorly if the assumed model does not accurately capture reality. In this paper, we propose a modeling paradigm that can generate control policies based on a simplified model, accounting for possible model errors that may result. The make-to-order system offers multiple products and has an outsourcing mechanism. Our focus is on addressing deliberate model simplification for the demand realization process. We formulate a robust control problem that takes the form of a two-player, zero-sum game. Because the original formulation is not tractable enough, we further develop an approximating problem under the heavy-traffic assumption that effectively results in a stochastic differential game. The solution to this game then translates into an implementable control policy for the original make-to-order system. We supplement the proposed modeling paradigm with a simulation-based method for selecting an appropriate uncertainty set. Numerical experiments expose, among other things, the value of building robustness into decision making. This paper was accepted by Barış Ata, stochastic models and simulation. Funding: X. Zhu’s work is supported by the National Natural Science Foundation of China (NSFC) [Grants 72401124 and 72394363/72394360]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.01820 .

This paper investigates a signal online channel green hotel supply chain comprising a green small and medium-sized hospitality enterprise (GSMHE) responsible for offline electronic Word-of-Mouth (e-WOM) promotion and green service efforts, and an online travel agency (OTA) engaged in online e-WOM promotion activities. Both offline and online e-WOM efforts significantly contribute to the overall e-WOM level of the green hotel supply chain. Due to the inherent characteristic of reviews posting, the impacts of these efforts on e-WOM exist time lags. Therefore, this study aims to explore the influence of time lags and examine the feasibility of a cooperative program in this context. This paper develops a differential game model that incorporates time lags. By analyzing and comparing the equilibrium strategies and profits under three scenarios, we demonstrate that a cost-sharing model can be successfully implemented when specific relationships regarding the time lags of the GSMHE are satisfied.

In this work, we investigate a simple motion differential game involving one pursuer and one evader, where the dynamics of the pursuer and the evader are governed by first-order and second-order differential equations, respectively. The control functions for both players are assumed to satisfy generalized integral constraints, which limit their maneuverability over time. Our main focus is on the analysis of pursuit under the <i>l</i>-capture condition. Specifically, we demonstrate that pursuit is completed when the <i>l</i>-distance between the positions of the pursuer <i>x(t)</i> and the evader <i>y(t)</i> becomes less than or equal to a given threshold <i>l</i>, that is, <img width="125" height="16" src="http://article.sciencepublishinggroup.com/journal/347/2261158/image001.png" />, at some finite time <i>t</i>. We derive an explicit formula for the guaranteed time of pursuit, which holds for all admissible strategies of the evader. Additionally, we show that an optimal escape strategy for the evader, known as <i>l</i>-escape, can also be realized at this same optimal time. This simultaneous realization of pursuit and escape conditions indicates a balance in the strategies of both players, thus leading to an optimal pursuit time. Our results contribute to the broader understanding of differential games and optimal control theory.

Implementing hybrid carbon policies is crucial for supply chains’ low-carbon transition. However, the downstream retailer is often passive in low-carbon strategies, leading to fair issues that may influence the decision-making of channel members. Therefore, this study integrates green technology, remanufacturing, retailer’s fairness concerns, low-carbon preference, and hybrid carbon policies into a manufacturer-led supply chain through differential game theory. Then, the equilibrium solutions for each member are analyzed under the centralized case and decentralized case involving a cost-sharing contract for low-carbon promotion. Our results show that centralized decision-making can optimize both the economic and environmental performances of channel members; retailer’s fairness concerns can enhance low-carbon promotional efforts and the cost-sharing ratio for such initiatives, but do not impact low-carbon production efforts. Additionally, a threshold exists on the relationship between retailer’s fairness concerns and the cost-sharing ratio; increased low-carbon preference motivates more efforts in low-carbon production and promotion. Moreover, stricter carbon policies motivate the manufacturer to increase low-carbon efforts, but the retailer tailors its low-carbon promotional strategy according to the unit carbon emissions of products to maintain an adequate level of low-carbon goodwill.

This paper investigates the robust tracking control of unmanned aerial vehicles (UAVs) against external time-varying disturbances. First, by introducing a virtual position controller, we innovatively decouple the UAV dynamics into independent position and attitude error subsystems, transforming the robust tracking problem into two zero-sum differential games. This approach contrasts with conventional methods by treating disturbances as strategic “players”, enabling a systematic framework to address both external disturbances and model uncertainties. Second, we develop an integral reinforcement learning (IRL) framework that approximates the optimal solution to the Hamilton–Jacobi–Isaacs (HJI) equations without relying on precise system models. This model-free strategy overcomes the limitation of traditional robust control methods that require known disturbance bounds or accurate dynamics, offering superior adaptability to complex environments. Third, the proposed recursive Ridge regression with a forgetting factor (R3F2 ) algorithm updates actor-critic-disturbance neural network (NN) weights in real time, ensuring both computational efficiency and convergence stability. Theoretical analyses rigorously prove the closed-loop system stability and algorithm convergence, which fills a gap in existing data-driven control studies lacking rigorous stability guarantees. Finally, numerical results validate that the method outperforms state-of-the-art model-based and model-free approaches in tracking accuracy and disturbance rejection, demonstrating its practical utility for engineering applications.

ABSTRACT With the diffusion and integration of digital technologies (DTs) in the innovation ecosystem (IE), an increasing number of enterprises have embarked on the ‘fast track’ of digitalisation. When integrating some advanced DTs, they often find a gap between practical effects and expectations. It is imperative to address this challenge to enhance integration efficiency and promote the digitisation of IEs. This paper explores the DTs integration strategies between the focal and nodal firms in the innovation ecosystem by constructing a differential game model, including the Nash non-cooperative game, the Stackelberg game, and the cooperative game, to clarify their strategic choices and influencing factors. Furthermore, it elucidates the mechanisms of decision-making and influence through numerical simulations. Our findings indicate that the cooperation strategy for DTs integration is the most efficient for enterprises compared to independent innovation and cost-sharing strategies in IEs. Furthermore, DTs integration, social benefits and incentive mechanisms of revenue distribution and cost-sharing are positively associated with the social digitisation level. However, the cost of effort for DTs integration and the discount rate have a negative impact on it. These results convey profound insights for practitioners and researchers, indicating that cooperation strategies, incentive methods and the balance of revenue and cost play a pivotal role in satisfying personalised demands and improving DTs’ integration and diffusion efficiency.

The longitudinal decision-making control of connected vehicle platoons is critical to ensuring safety and efficiency. This study proposes a differential game-based control strategy for vehicle platoons, incorporating control variables such as vehicle state, accel-eration, and engine delay. A differential game framework is formulated, establishing the objective function and state transition equations based on platooning goals and spacing policies. The longitudinal control problem is then transformed into an optimal control problem, and the Nash equilibrium strategy is derived by solving the Hamilton-Jacobi-Bellman equation. A double-predecessor-following topology is further introduced to refine the objective function. Simulation results in a longitudinal platoon environment demonstrate that, under the TPF topology, the control model reduces spacing error by 48.59%, offering improved stability, accuracy, and safety. This differential game-based strategy effectively meets platooning control objectives and enhances the stability of connected vehicle longitudinal control.

To play or not to play: A characterization of the marginal contribution of the opponent in a class of LQ differential games

A Large-Population Stochastic Differential Game With Terminal State Constraint and Common Noise

Asset pricing models have played an important role in the development of modern financial theory. From the Capital Asset Pricing Model (CAPM) to the Arbitrage Pricing Theory (APT), these classical models provide the basic framework for understanding and predicting asset prices. However, most traditional asset pricing models are based on the efficient market hypothesis, which assumes that all investors are rational, have homogeneous expectations, and can process all available information quickly and accurately. Despite their success in some respects, these models have proved inadequate in explaining and predicting market anomalies. For example, market bubbles, excessive volatility, and frequent deviations of asset prices from their fundamental values cannot be adequately explained by traditional rational expectations models. In order to more accurately capture the complex market dynamics, especially given the heterogeneous beliefs and their interactions between investors, this study uses of Partial Differential Equations (PDEs), deeply analyses the mechanism of heterogeneous beliefs influence. This study not only improves our understanding of how financial markets operate, but also provides a solid foundation for the development of effective risk management strategies and policy design. This study contributes to building a more robust and inclusive financial environment to better cope with future challenges and uncertainties.

In this paper, we first establish an Itô formula for a finite quadratic variation process [Formula: see text] expanding [Formula: see text] when [Formula: see text] is of class [Formula: see text] in space and is absolutely continuous in time. Second, via a Fukushima–Dirichlet decomposition we obtain an explicit chain rule for [Formula: see text], when [Formula: see text] is a continuous semimartingale and [Formula: see text] is a “quasi-strong solution” (in the sense of approximation of classical solutions) of a parabolic PDE. Those results are applied in a companion paper to establish a verification theorem in stochastic differential games.

This paper examines a non-zero-sum stochastic differential reinsurance-investment game between two competitive insurers under the -maximin mean-variance criterion. Both insurers can purchase proportional reinsurance and invest in a financial market consisting of one risk-free asset and one risky asset, and each insurer is concerned with its terminal surplus and relative performance compared to its competitor. The insurers aim to maximize the -maximin mean-variance utility, which allows them to exhibit different attitudes towards model ambiguity. By solving the extended Hamilton-Jacobi-Bellman (HJB) equations for both insurers, we derive the -robust equilibrium reinsurance and investment strategies. Finally, several numerical examples are provided to illustrate the impact of some model parameters on the equilibrium strategies.