Correction to: The Impacts of Reward Frequency and Reward Conditionality on Sustainable Use of Common Resources: a Stochastic Common Property Differential Game Model

This paper presents a novel approach to address the problem of intercepting non-cooperative targets with multiple satellites in Earth orbit. The multi-satellite interception problem is formulated as a multi-player pursuit–evasion game that explicitly accounts for stochastic disturbances and control constraints. By combining differential game theory with stochastic optimization techniques, the paper derives optimal interception trajectories that ensure safety and performance under modeling uncertainties. A linear exponential quadratic cost functional is established, and corresponding Nash equilibrium strategies are obtained to determine the optimal control laws. Numerical simulations validate the effectiveness and robustness of the proposed approach in achieving reliable interception performance.

This paper investigates the non-zero-sum stochastic differential game problem of optimal asset-liability management (ALM) between insurers and reinsurers. Combining the investment- reinsurance problem with the ALM, this paper innovatively studies the investment-reinsurance ALM non-zero-sum game within the framework of the constant elasticity of variance (CEV) model under common shock dependence. We allow the insurance companies to purchase proportional reinsurance from reinsurance companies, and both companies can invest in a financial market composed of one risk-free asset and one risk asset whose price process follows the CEV model. By innovatively introducing a class of stochastic liability processes associated with the volatility of risky assets, we obtain the dynamic evolution of net wealth. Using stochastic control theory, we obtain an explicit expression of Nash equilibrium strategy for the problem and a closed-form expression for the corresponding equilibrium value function by maximizing the expected utility from the terminal net wealth. Finally, we illustrate the effect of relevant parameters on the optimal investment-reinsurance strategy by means of numerical examples.

A Stochastic Differential Oligopoly Game for Housing Characteristics in Investment Decisions

Non-zero-sum linear quadratic stochastic differential games with jump diffusion and input delay: an asymmetric information framework

Parameter estimation for stochastic dynamic systems is crucial yet challenging due to inherent randomness and noise. Traditional deterministic methods often fail to cope with such uncertainties. Inspired by the deterministic tracking approach, we propose a new parameter estimation method—stochastic dynamic problem based parameter estimation method (SDPE), which firstly transforms the original stochastic optimal control problem into a stochastic differential game problem by introducing a perturbation term, and then realizes the simultaneous estimation of the system parameters and the optimal control function. We prove the consistency of the SDPE estimator. Numerical simulations demonstrate that this method exhibits superior performance in terms of accuracy and robustness compared to maximum likelihood estimation (MLE), likelihood-based methods (EM) and Bayesian inference methods (MCMC), particularly under varying noise levels and sample sizes. Furthermore, we apply SDPE to a stochastic optimal control SIRD model to analyze COVID-19 data from Florida, USA. Results demonstrate that SDPE effectively captures time-varying transmission, recovery, and mortality rates, providing a powerful tool for modeling and understanding complex epidemic dynamics.

This paper is concerned with a three-level multi-leader-follower incentive Stackelberg game with $H_\infty$ constraint. Based on $H_2/H_\infty$ control theory, we firstly obtain the worst-case disturbance and the team-optimal strategy by dealing with a nonzero-sum stochastic differential game. The main objective is to establish an incentive Stackelberg strategy set of the three-level hierarchy in which the whole system achieves the top leader's team-optimal solution and attenuates the external disturbance under $H_\infty$ constraint. On the other hand, followers on the bottom two levels in turn attain their state feedback Nash equilibrium, ensuring incentive Stackelberg strategies while considering the worst-case disturbance. By convex analysis theory, maximum principle and decoupling technique, the three-level incentive Stackelberg strategy set is obtained. Finally, a numerical example is given to illustrate the existence of the proposed strategy set.

Finite-Agent Stochastic Differential Games on Large Graphs: I. The Linear-Quadratic Case

Open-loop and closed-loop saddle points of infinite dimensional linear–quadratic stochastic differential games with Poisson jumps

A jump–diffusion Stackelberg stochastic differential game in optimal carbon abatement strategies with green subsidy

A stochastic differential game based pollution management study of regional alliance

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

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 .

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

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.

Mean-Field Linear-Quadratic Nonzero Sum Stochastic Differential Games with Overlapping Information

To facilitate power system decarbonization, optimizing clean energy integration has emerged as a critical pathway for establishing sustainable power infrastructure. This study addresses the multi-timescale operational challenges inherent in power networks with high renewable penetration, proposing a novel stochastic dynamic programming framework that synergizes intraday microgrid dispatch with a multi-phase carbon cost calculation mechanism. A probabilistic carbon flux quantification model is developed, incorporating source–load carbon flow tracing and nonconvex carbon pricing dynamics to enhance environmental–economic co-optimization constraints. The spatiotemporally coupled multi-microgrid (MMG) coordination paradigm is reformulated as a continuous state-action Markov game process governed by stochastic differential Stackelberg game principles. A communication mechanism-enabled multi-agent twin-delayed deep deterministic policy gradient (CMMA-TD3) algorithm is implemented to achieve Pareto-optimal solutions through cyber–physical collaboration. Results of the measurements in the MMG containing three microgrids show that the proposed approach reduces operation costs by 61.59% and carbon emissions by 27.95% compared to the least effective benchmark solution.

There is a very large literature on applications of stochastic control of jump diffusions and a smaller literature on such games. In many applications it is natural to assume that the arrival intensity is controlled, but except for two long-forgotten papers the literature instead assumes that it is the jump sizes that are controlled. The more natural assumption is typically avoided because a failed Lipschitz condition means that the classical existence and uniqueness proofs cannot be used. We here derive an asymptotic Markov equilibrium of the game with controlled jump intensities and show that it, at least in an example, is very similar to the Markov equilibrium of an analog game with controlled jump sizes. The paper thus makes two contributions: It supplies a way to solve some optimization problems and games with controlled jump intensities and it shows that the commonly used formulation with controlled jump sizes is quite defensible for at least some classes of games.