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We consider a (non) zero-sum differential game between two insurers and determine a robust reinsurance contract from joint interests of the insurers and the reinsurer under the framework of Stackelberg differential game, that is, a hybrid stochastic differential game framework is introduced in this article. We assume that each party can invest in a financial market consisting of a risk-free asset and a risky asset whose price evolution follows a stochastic volatility model, and the reinsurer charges the premium according to both the expected value premium principle and the variance premium principle. Since the performance-related capital inflow or outflow feature is introduced, the insurers and reinsurer all aim to maximize their own time-inconsistent mean-variance criterion with delay. Using the dynamic programming approach, we derive the equilibrium investment strategy, the robust equilibrium reinsurance contract, and the corresponding equilibrium value functions under two premium principles with certain conditions, respectively. Some interesting properties of the equilibrium investment strategy are proved explicitly, and several numerical examples and sensitivity analysis are presented to demonstrate the effects of model parameters on the equilibrium reinsurance contract. [ABSTRACT FROM AUTHOR]

This study aims to explore allocation strategies for idle emergency supplies in a "demander–platform–supplier" supply chain system along with government regulation during the post-disaster recovery period. Allocation of emergency supplies is a complex task that encompasses resource allocation before and after disasters. It is essential to reduce losses in disaster-stricken areas and support development during post-disaster recovery. However, there is often an excessive supply of emergency materials and a mismatch between supply and demand sides in downstream supply chains, which may lead to severe waste and difficulties in recovering surplus materials. This paper takes idle emergency resource sharing level and corporate social responsibility goodwill as endogenous variables. The allocation approaches are dynamically evaluated by incorporating random elements that influence the endogenous variables. Three stochastic differential games are introduced to examine the interactions between the players. The centralized decision-making satisfies the consistency of overall and individual rationalities at any time in the emergency material allocation process, promoting the optimal sharing levels of emergency materials and overall profits. The decentralized decision-making with cost-sharing contracts achieves local optima and increases the dual marginal effect of the emergency industry chain. This paper incorporates the sharing economy into emergency management, showing how technology-driven sharing platforms can optimize resource utilization. The results suggest introducing cost-sharing contracts between demanders and suppliers can enhance collaboration and effort, leading to better resource allocation and increased efficiency. It contributes to sustainability by promoting efficient resource utilization through idle emergency resource sharing. By optimizing allocation strategies and enhancing corporate social responsibility, the study fosters the long-term viability and resilience of the supply chain system in post-disaster management. [ABSTRACT FROM AUTHOR]

In this paper, we study a zero-sum stochastic differential game with the following salient features: (i) the system state is dictated by a hybrid diffusion, (ii) both players use impulse controls, and (iii) the game takes place on an infinite time horizon. First, the dynamic programming principle for the problem is proven. Then, the lower and upper value functions of the game are characterized as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, which turns out to be a coupled system of variational inequalities with bilateral obstacles. Moreover, a verification theorem as a sufficient condition to identify a Nash equilibrium is established. The Nash equilibrium strategies for the two players, indicating when and how it is optimal to intervene, are given in terms of the obstacle part of the HJBI equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with a linear quadratic (LQ) nonzero-sum stochastic differential game for regime switching diffusions with mean-field interactions. The salient features of this paper include that the concept of strategies is first adopted in the LQ nonzero-sum game and conditional mean-field terms appear in the state equation and cost functionals. First, a candidate optimal feedback control-strategy pair for the two players is formally constructed based on solutions of four coupled Riccati equations. Then, we verify that the formal optimal pair is indeed a Nash equilibrium for the game by a delicate multi-step completion of squares. The four Riccati equations introduced in this paper are new in the literature. Uniqueness of solutions to the Riccati equations for the general case and existence of solutions for a special case are obtained. Finally, a numerical example is reported to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper explores the optimal management of defined contribution pension plans under multiple risks and strategic interactions using a stochastic differential game between two managers, accounting for model misspecification. Managers who are both competitive and averse to ambiguity can invest in a financial market that contains a risk-free asset, an inflation-linked bond, a market index, and a stock. These managers may have varying levels of aversion to ambiguous risks. The inflation index is computed using the Fisher equation. The market index and stock prices are described by a class of non-Markovian multivariate stochastic covariance models, with the market risk prices being reliant on a multivariate affine-diffusion factor process. Each manager aims to maximize the expected utility of inflation-adjusted terminal wealth relative to their competitors under the worst-case scenario of different measures, ensuring that the investment strategy is robust to model uncertainty. The robust Nash equilibrium investment strategies' explicit expressions, density generator processes under the well-defined worst-case scenarios, and corresponding value functions are derived using a backward stochastic differential equation approach to address this robust non-Markovian game. The admissibility of the robust equilibrium policies is confirmed under certain technical conditions. Finally, we provide numerical examples to demonstrate the effects of model parameters on robust investment policies and clarify the economic significance of our theoretical findings. [ABSTRACT FROM AUTHOR]

This paper takes corporate social responsibility goodwill and consumers' reference low-carbon level as endogenous variables of joint carbon emission reduction in the "supplier–manufacturer–retailer–consumer" supply chain system. The joint carbon emission reduction strategies of this four-tier system are analyzed from a dynamic perspective by considering random factors that affect the endogenous variables. Three stochastic differential games are proposed to examine the mechanism between each player, namely the cooperative model, Nash non-cooperative model, and Stackelberg master–slave model. Compared to the Nash non-cooperative game, the manufacturer/supplier-led Stackelberg master–slave game leads to Pareto improvement in the profits of the entire supply chain system and each player. The cooperative game demonstrates the highest expected emission reduction and corporate social responsibility goodwill, but also the highest variance. More importantly, the reference low-carbon level embraces consumers' subjective initiative in the dynamic of carbon emission reduction. This level is an internal benchmark used to compare against the observed low-carbon level. This paper provides a theoretical foundation for strategic decision-making in emission reduction, contributing to sustainable development. By addressing environmental, economic, and social sustainability, it promotes climate action through carbon reduction strategies and offers policy recommendations aligned with the Sustainable Development Goals. [ABSTRACT FROM AUTHOR]

In recent years, problems of nonzero-sum investment and reinsurance games have received a lot of attention from scholars. However, for practical consideration, there is also consumption in the operation process of insurers, which is rarely taken into account by scholars. This paper integrated the consumption problem into nonzero-sum investment-reinsurance problems, and studied the nonzero-sum investment, consumption, and reinsurance game between two competitive insurers. Each insurer was allowed to purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset and a risky asset, and the price process of the risky asset was described by the constant elasticity of variance (CEV) model. Moreover, the consumption behavior of each insurer was also considered. The main objective of each insurer was to maximize the utility of his terminal surplus and accumulated consumption relative to that of his competitor. Based on the stochastic differential game theory, we obtained the Hamilton-Jacobi-Bellman (HJB) equations for both insurers and derived the equilibrium strategies and the equilibrium value function. Numerical examples were given in the end to illustrate the influence of model parameters on the equilibrium strategies. [ABSTRACT FROM AUTHOR]

We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have q players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form Ito-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents { γ 1 , γ 2 , ... } under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel approach to study the production of consumption goods within a price floor carbon emission trading scheme. A stochastic Stackelberg game is set up between the environment authority (Player 1), and the manufacturer/producer (Player 2). In such a hybrid emission trading scheme, the authority sets an emissions target and producers will choose to participate in carbon trading on this basis, subject to a stochastic demand with information delayed. When the stochastic demand is determined by the market sale price and the scale of carbon emission reduction, we apply the maximum principle to obtain both the optimal production and market price. We show when the demand is determined by both the market sales price and carbon emission reduction, manufacturers are persuaded to set the market sales price first (ex ante incentive) and then proceed to a carbon trading scheme, thus providing a backward-looking advantage to manufacturers. Consequently, the manufacturer and the environmental authority can find a subgame-perfect Nash equilibrium via backward induction. [ABSTRACT FROM AUTHOR]

Information drives trading and hence affects price dynamics. We study how an informed trader optimally trades, how multiple traders compete with each other, and how their competition affects price dynamics as well as their inventories in the linear price impact model framework. We formulate the informed trading problem as a stochastic optimal control problem and a stochastic differential game in different cases. By virtue of different methods, problems are solved explicitly in all cases. In the case of a single trader, the risk-averse informed trader trades less in the beginning and then speeds up to reduce exposure to the volatility of dynamic information. In addition, the risk-averse trader reveals information more slowly and thus contributes less to price efficiency. When multiple risk-neutral traders compete with information, regardless of the competition structure, they tend to trade more rapidly in the early stage to fill orders at a good price. Competition and the presence of a leader help promote price efficiency, and competition also leads to the empirically observed mean-reverting behavior of price dynamics and order flows. Furthermore, among risk-averse informed traders, inventory management triggers predatory trading and active information leakage during informed trading, and the positions of traders with the same level of risk aversion tend to be identical and mean-reverting to zero under multiple rounds of competition. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic differential investment consumption and proportional reinsurance game problem for two competitive insurers with constant absolute risk aversion (CARA) utilities. The utility of an insurer depends not only on his absolute wealth and consumption but also his relative wealth and consumption when compared to the other insurer. The surplus processes of both insurers are governed by the diffusion approximated process of the classical Cramér-Lundberg model. Each insurer is allowed to purchase a proportional reinsurance treaty and invest his surplus into a financial market consisting of one risk-free asset and one risky asset to manage his insurance risk. Moreover, the consumption behavior of each insurer is also considered. The main objective of each insurer is to maximize the utility of his terminal surplus and accumulated consumption relative to that of his competitor. Applying the techniques of stochastic dynamic programming, we obtain the Hamilton-Jacobi-Bellman (HJB) equations for both insurers. We establish the equilibrium reinsurance-investment-consumption strategies and the corresponding equilibrium value functions of both insurers by solving the HJB equations. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium reinsurance-investment-consumption strategies. Numerical simulation results indicate that the insurer's investment, consumption and reinsurance strategies increase with the increasing of the sensitivity parameter to competition; and decrease with the increasing of the constant absolute risk aversion coefficient. [ABSTRACT FROM AUTHOR]

This paper studies two-player stochastic differential games with diffusion control on finite time horizons. The players' state processes are described in terms of controlled martingale problems, and we allow for correlation between the players' states, described by some correlation coefficient. We consider zero-sum ranking games in the sense that the criterion to optimize only depends on the difference of the two players' state processes at the finite time horizon. We explicitly compute saddle points of the game depending on the correlation coefficient. In particular, for correlation coefficients smaller than some explicit threshold, the game has a value and a saddle point (in strict controls). For correlation coefficients exceeding the threshold, we introduce the larger class of relaxed controls that can be interpreted as mixed strategies. We show that the game has a value and a saddle point in relaxed controls for all correlation coefficients. [ABSTRACT FROM AUTHOR]

This work is devoted to Markovian-switching systems. In particular, backward stochastic differential equations (BSDEs), forward-backward stochastic differential equations (FBSDEs), such equations with mean-field interactions, and related nonzero-sum stochastic mean-field games. First, BSDEs with Markovian switching, FBSDEs with Markovian-switching, and FBSDEs with both mean-field interactions and regime-switching are examined. Unique solvability of the underlying equations is obtained under monotonicity conditions without assuming non-degeneracy condition for the forward equation. Then the existence of open-loop Nash equilibrium points for nonzero-sum linear-quadratic stochastic differential games with random coefficients is investigated. [ABSTRACT FROM AUTHOR]

This paper focuses on the stochastic disturbance on quality control strategies with a two-echelon supply chain. A state equation is described to the change of product quality, and the stochastic differential game models with quality disturbance are designed according to different cooperation modes between the manufacturer and the retailer. The paper aims to investigate the relationship between the quality disturbance with quality control strategies of the supply chain members. After the construction and analysis of the evolutionary process of the product quality to stochastic differential model, the optimal equilibrium solutions and subsidy ratio is solved, by using the dynamic programming method. The result shows: The product quality is inversely proportional to the quality investment cost and the depreciation rate, and is proportional to the marginal revenue of the quality; The cooperative game model has the better performance in improving the input of the supply chain members to product quality. [ABSTRACT FROM AUTHOR]

We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have q players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form Ito-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents {γ1,γ2,…} under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI.

Due to the conjunction of huge expenditure and latent inefficiencies, competitive advertising has always been occupying a front‐and‐center place in marketing, among which a bilinear system is one of the most important. Because much real business is two giants playing among other emerging challengers and the players can only inspect the market through a related noisy process by the random sampling survey, the sales‐advertising dynamics is a partially observed nonzero‐sum stochastic differential game. However, by the intrinsic difficulties of mathematics, there is no extant research to augment the analysis on the infinite‐time horizon, which is of particular significance to implementing the online sales campaign and evaluating the consequence of endless competition. Motivated by this, controllability and stability analysis are explored first for the iterative approximation to bilinearity. Then, the simulation provides a rapid prototyping and automatic approach for the most efficient strategies. Moreover, four implications have been discovered by comparison with eight examples as the application of four encounters on finite‐ and infinite‐time horizons. Several new insights have been revealed for those struggling to develop successful advertising strategies. [ABSTRACT FROM AUTHOR]

From the previous work, when solving the LQ optimal control problem with random coefficients (SLQ, for short), it is remarkably shown that the solution of the backward stochastic Riccati equations is not regular enough to guarantee the robustness of the feedback control. As a generalization of SLQ, interesting questions are, "how about the situation in the differential game?", "will the same phenomenon appear in SLQ?". This paper will provide the answers. In this paper, we consider a closed-loop two-person zero-sum LQ stochastic differential game with random coefficients (SDG, for short) and generalize the results of Lü–Wang–Zhang into the stochastic differential game case. Under some regularity assumptions, we establish the equivalence between the existence of the robust optimal feedback control strategy operators and the solvability of the corresponding backward stochastic Riccati equations, which leads to the existence of the closed-loop saddle points. On the other hand, the problem is not closed-loop solvable if the solution of the corresponding backward stochastic Riccati equations does not have the needed regularity. [ABSTRACT FROM AUTHOR]

In the field of advertising, there are always situations in which individuals or companies advertise their products in order to find marketing opportunities and attract customers in a competitive environment. Consider a duopoly market in a mature product category where total sales are distributed between the two firms, denoted as firm 1 and firm 2, which compete for the market share through advertising spending. There are two firms in a market and the profit of firm 1 and that of 2 are respectively: max u1 {E0[∫T0 (q1x(t)- (t) - c1/2 u12 (t))}e-pt dt+q1Te-pT x (T)]}, (1) maxu1 {E0[∫T0(q2(1-x(t))-c2/2u²2(t))e-pt dt+q2Te-pT (1-x(T))]}. (2) Market shares of firms 1 and 2 at time tas x(t) and 1-x (t), respectively. The dynamics of firm 1's market share is governed by dx (t) = (u1(t)√1-x(t) -u2(t)√x(t))dt + σx (t)dB (t), x(t0) = x0. (3) Thus, each firm seeks to maximize its expected, discounted profit stream subjec to the market share dynamics. Material and Methods A feedback solution which allows the firm to choose their advertising rates contingent upon the state of the game is a realistic approach to the problem. A feedback Nash equilibrium solution to the game (1)-(3) has to satisfy the following conditions: {∂/∂tV1 (t,x)+1/2x² ∂²/∂x²V1 (t,x) + maxu1 {∂/∂xV1(t,x)(u1(t)√1-x(t) - u2(t)√x(t)) +(q1x(t)-c1/2u1²(t))e-pt} = 0, ∂/∂tV2 (t,x) + 1/2x² ∂²/∂x²V2(t,x) + maxu2 {∂/∂xV2(t,x) (u1(t)√1-x(t)-u2(t)√x(t)) +(q2(1-x(t))-c2/2u²2(t))e-pt} = 0, V1(T,x)=q1Te-pTx,V2,(T,x)=q2Te-pT (1-x). Determining the equilibrium strategy for this problem requires solving a set of nonlinear partial differential equations called the Hamilton-Jacobi-Bellman equation. Another purpose of this paper is to propose an efficient and suitable computational method for solving the Hamilton-Jacobi-Bellman equation. The proposed method for solving the problem is a combination of the collocation method based on the Chelyshkov polynomials differential operator matrix and the policy iteration method. The collocation method has the disadvantage that it requires solving a system of nonlinear algebraic equations resulting from the implementation of the method. The advantage of using the policy iteration algorithm is that instead of finding the solution to a nonlinear partial differential equation, it is enough to solve a sequence of linear partial differential equations. Results and discussion Since for firms having different parameter values, the solutions are more complicated. First we will consider the case of two symmetric firms. The case of asymmetric firms will be dealt after that. We will summarize the analytical results of comparative statistics with symmetric and asymmetric firms in the Tables 1 and 2. When there is a marginal increase in the value of advertising (r increases) or a reduction in its cost (c decreases), then, the amount of advertising increases. However, contrary to what one would expect to see in a monopoly model of advertising, the value function decreases. Advertising does not increase the size of the marketing pie but only affects its allocation. Thus, the increase in advertising causes a decrease in the value function. Conclusion In this study, we have applied a combined Chelyshkov collocation method with policy iteration to solve HJB PDE arising in finite-horizon stochastic competitive advertising differential games numerically. The method is based on the exponential Chelyshkov functions and policy iteration method. The method was utilized to convert the nonlinear HJB PDE into a sequence of linear algebraic equations systems which can be solved easily. This is one of the strengths of the proposed method compared to the collocation method. It should be noted that this method can be expanded to solve HJB PDEs in the other applications of stochastic differential games with feedback information. [ABSTRACT FROM AUTHOR]

Based on the perspective of government and enterprises, we explore the cooperative strategy and cost-sharing problem of cooperative open sharing of data between government and enterprises. In order to accurately analyze the data-opening strategies of government and enterprises, stochastic differential game theory is applied to construct the Nash non-cooperative game, Stackelberg master-slave game and cooperative game models with government and enterprises as game subjects to obtain the optimal open data effort, the optimal trajectory of social data open sharing level and the optimal benefit function of government and enterprises in three scenarios. Combined with numerical simulations to analyze the sensitivity of the relevant parameters affecting the level of social data openness, the results of the study revealed the following: ① When the government's income distribution ratio is greater than 1/3, the benefits of the government and the enterprises under the Stackelberg master-slave game and the effort to open and share data are greater than in the Nash non-cooperative situation; in the case of a cooperative game, the degree of effort and total revenue of both parties reach the Pareto optimal state. ② When the government's income distribution ratio is greater than 1/3, the expectation and variance of the open data and shared stock under the cost-sharing situation and the corresponding limit value are all greater than the value in the Nash non-cooperative situation, and in the cooperative game, the expectation and variance of open data and shared stock and its corresponding limit value are the greatest. ③ The government and enterprises coexist with profit and risk under the influence of random interference factors, and high profit means high risk. This research provides a theoretical basis and practical guidance for promoting the open sharing of government and enterprise data.

Forest carbon sinks integrated into the carbon trading is an important way to solve the funding shortage of carbon emission reduction and is a major measure to achieve carbon sequestration and thereby carbon neutrality. We propose a carbon credit trading mechanism based on forest carbon sinks and build an optimal carbon credit trading model on forest companies and manufacturers. The scale of carbon sink forest and trading price is determined by stochastic differential game method. Reasonable emission reduction responsibilities are set to achieve coordination between forest carbon sequestration and economic development. The results show that farsighted forest company is more likely to participate in carbon credit trade when the total amount of forest carbon sequestration is small, while shortsighted forest company is more likely to participate in carbon credit trade when the total amount of forest carbon sequestration is large. Low emission reduction responsibility in carbon credit trade is conductive to increasing forest carbon sequestration, and improving the enthusiasm of shortsighted forest company to participate in the trade, which reflects the superiority of the proposed carbon credit trading mechanism. This implies that the forest company should undertake low emission reduction responsibilities to achieve a win–win situation for economic and environmental benefits. [ABSTRACT FROM AUTHOR]

In this paper, supposing that the price process of the risky asset is described by a CEV stochastic volatility model, we investigate a stochastic differential investment and proportional reinsurance game problem with delay between two competing insurers. Each insurer’s risk process is described by the diffusion approximated process of the classical Cramér-Lundberg model. Each insurer can purchase the proportional reinsurance to mitigate their claim risks; and can invest in one risk-free asset and one risky asset whose price dynamics follows the CEV model. The main objective of each insurer is to maximize the utility of his terminal surplus relative to that of his competitor. For the representative cases of the mean-variance utility and exponential utility, we derive the explicit equilibrium reinsurance and investment strategies by applying the techniques of differential game theory and stochastic control theory. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium reinsurance and investment strategies. Numerical simulation results indicate that: whether delay information and the elasticity parameter is considered or not will greatly affect the final equilibrium reinsurance strategy and optimal investment strategy. The more value of wealth at an earlier time is considered, the insurer will be more cautious and rational in their investment; however, the investment strategy of the insurer with the relative performance concern is riskier than that without the concern; meanwhile, the elasticity parameter will significantly affect the investment strategy of the insurer, and its influence trend varies with the price of risky assets. [ABSTRACT FROM AUTHOR]

Technological developments play a crucial role in allowing governments and industries to meet carbon emission targets, whilst maintaining cost effectiveness. Mathematical modeling related to climate change has often included technology (including technology transfer between nations) as an effective policy instrument. However, such models often incorporate technology as an exogenous variable, highlighting the need to further interrogate the role of technology, its dynamics and limitations on reducing international pollution levels to improve sustainability, energy reliability and subsequent policy initiatives. Hence, in this study, we consider technology as an endogenous variable within a broader trans-boundary industrial pollution problem with random interference factors to obtain a closed-loop (Markov perfect) Nash equilibrium. We then articulate the Nash non-cooperative and cooperative equilibria via a stochastic linear quadratic differential game paradigm and prove the stability of a cooperative game by using Pareto optimal solution. We show that under such strategies to control carbon pollution a cooperative game is more efficient than a non-cooperative game, emphasizing the importance of technology transfer and collaboration between nations, subsequently serving as a mutual benefit for multi-lateral efforts to reduce global carbon emissions. In doing so, our study highlights the role of government subsidy incentives when collaborating with industry to encourage the integration of carbon-reducing technologies, whilst simultaneously increasing each country's net revenue. Hence, our study provides a novel insight and framework for policymakers when encouraging industry to use carbon capturing and storage technologies. We also emphasize that efforts to coordinate emissions control should be pursued jointly to ensure mutual benefit for government and industry alike. [ABSTRACT FROM AUTHOR]

This paper presents and examines a problem in which two insurance companies apply non-proportional reinsurance to control risk. Additionally, each firm pays out dividends. The situation is modelled as a zero-sum stochastic differential game between the two companies. The goal of one company is to maintain business competitive advantage over the other by sustaining or increasing the difference between the respective liquid reserves of the two companies while the second company aims to minimise that difference. A verification theorem is formulated, proved and subsequently employed to derive the saddle point components. For the case of the payoff with a non-zero running cost function, we are able to solve explicitly the differential game. Numerical simulations are presented to illustrate the results as well as the economic interpretation. [ABSTRACT FROM AUTHOR]

In this paper, we regard policyholders, insurance companies, and government departments to be an anti-fraud supervision system, and we explore the supervision of motor vehicle insurance fraud from the perspective of a tripartite game. Taking into consideration the bad reputation records of policyholders as a state variable, through continuous accumulation in effective time, it creates a continuous growth-type warning effect on policyholders, and thus, effectively curbs policyholder fraud and false supervision by insurance companies. At the same time, by considering the influence of random factors on the anti-fraud game of motor vehicle insurance, in this paper, we establish a stochastic differential game model to explore the optimal strategy, the optimal income level, and the expectation and variance of the insured's bad reputation record stock under the conditions of with and without government supervision. Finally, through a simulation analysis, it is found that the game with government supervision is more conducive to reduce the insured's fraud intensity, and the simulation proves the impact of different parameters on system stability. [ABSTRACT FROM AUTHOR]

The ecological compensation mechanism emerges as a solution to reduce pollution for aquatic product supply chain. To comprehend the dynamic pollution management strategies and coordination under the ecological compensation mechanism, this study develops a stochastic differential game between farmers and firms from the perspective of extreme weather crisis. The subjects' optimal decisions are explored in three cases including cooperative case, noncooperative case, and land transfer-cost sharing contract case. Then we simulate theoretical results and illustrate a case study to demonstrate the applicability of ecological compensation collaboration. The results indicate that the extreme weather crisis changes the evolutionary trend and steady state of pollution accumulation. Moreover, the land transfer-cost sharing contract could realize the Pareto improvement in high-risk environment especially after the crisis. But its coordination relies on the ecological compensation coefficient, which gets larger with the interval of [ 4.9 , 6.3 ] after the crisis, compared to the range of [ 4 , 5.4 ] before the crisis. Notably, the contract and ecological compensation mechanism exhibit a complementary relationship in response to the crisis. The findings could provide new inspirations for aquatic product supply chain to balance production and environmental protection. • Extreme weather crises have the potential to alter the trends and steady states of pollution accumulation. • The land transfer-cost sharing contract is more effective in high-risk environments. • A larger ecological compensation coefficient enhances the coordination of the contract. • A stochastic stopping model is employed to characterize extreme weather crises. [ABSTRACT FROM AUTHOR]

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We study the competition of two strategic agents for liquidity in the benchmark portfolio tracking setup of Bank et al. (Math Financial Economics 11(2):215–239 2017). Specifically, both agents track their own stochastic running trading targets while interacting through common aggregated temporary and permanent price impact à la Almgren and Chriss (J Risk 3:5–39 2001). The resulting stochastic linear quadratic differential game with terminal state constraints allows for a unique and explicitly available open-loop Nash equilibrium. Our results reveal how the equilibrium strategies of the two players take into account the other agent's trading targets: either in an exploitative intent or by providing liquidity to the competitor, depending on the relation between temporary and permanent price impact. As a consequence, different behavioral patterns can emerge as optimal in equilibrium. These insights complement and extend existing studies in the literature on predatory trading models examined in the context of optimal portfolio liquidation games. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of market competition among manufacturers pursuing two types of green technology innovation: product innovation and process innovation, which provides a novel perspective on how green technological innovations contribute to enhancing product greenness. Using a two-tier supply chain framework, the study examines the differences in cost-sharing across various types of innovation and the impact of game players on product greenness. A stochastic differential game model involving a shared supplier and two competing manufacturers is constructed. Main findings are as follows: (1) In green product innovation, when market preference for greenness is relatively low, the formation of cost-sharing contracts is less influenced by the intensity of greenness-based market competition. In such scenarios, government subsidies emerge as a crucial tool to effectively guiding the market mechanism towards achieving its intended objectives. (2) As the market competition intensifies, manufacturers' incentives towards suppliers will shift from reducing costs to increasing demand. (3) When the intensity of green competitive is low, and supplier process innovation efficiency is high, manufacturers should bear a greater share of costs; Conversely, suppliers should bear more costs. When supplier process innovation efficiency is higher and market competition is stronger, suppliers should bear more costs if consumer green preferences are lower. (4) Over time, considering consumer preferences, a marginal increase in price exceeds the marginal increment in greenness with respect to contributing to stabilise price fluctuations. Conversely, it tends to raise the mean value of the price. (5) The increase in market competition intensity accelerates suppliers' and manufactures' strategy adjustments for green product innovation, but decelerates the strategy adjustment for suppliers' green process innovation., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2024 Elsevier Ltd. All rights reserved.)

The implementation of ecological compensation for cross-regional domestic waste treatment is beneficial for balancing the interests of waste generation districts and waste treatment districts. This paper introduces dynamic differential games to capture the temporal evolution and adjustment of strategies, examining dynamic game strategies under different compensation scenarios (non-horizontal ecological compensation, horizontal ecological compensation and fully shared ecological compensation). A new interest distribution model is proposed, with empirical analysis conducted using Shanghai as a case study to validate the reliability and feasibility of the model. The findings indicate: (1) Government intervention combined with horizontal ecological compensation significantly improves waste management efficiency and urban welfare. The compensation mechanism fosters regional cooperation, optimizes resource allocation, reduces decision-making conflicts and enhances overall processing effectiveness. (2) Without a compensation mechanism, high costs in export regions and low benefits in import regions result in low cooperation willingness. The ecological compensation mechanism enhances willingness to cooperate by fairly distributing costs and benefits, optimizing long-term cooperation and overall gains. (3) Complete shared ecological compensation is optimal, though partial shared mechanisms are also effective in practice. Proper interest and ecological compensation ratios can significantly improve waste management efficiency and urban environment, strengthening long-term cooperation. The study theoretically expands the dynamic optimization and cooperation mechanism analysis in waste management, provides solutions for environmental policy formulation in cross-regional waste handling and offers new perspectives and tools for addressing complex issues in cross-regional environmental governance, including targeted policy recommendations with practical significance for enhancing cross-regional waste management., Competing Interests: Declaration of conflicting interestsThe authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

From the previous work, when solving the LQ optimal control problem with random coefficients (SLQ, for short), it is remarkably shown that the solution of the backward stochastic Riccati equations is not regular enough to guarantee the robustness of the feedback control. As a generalization of SLQ, interesting questions are, “how about the situation in the differential game?”, “will the same phenomenon appear in SLQ?”. This paper will provide the answers. In this paper, we consider a closed-loop two-person zero-sum LQ stochastic differential game with random coefficients (SDG, for short) and generalize the results of Lü–Wang–Zhang into the stochastic differential game case. Under some regularity assumptions, we establish the equivalence between the existence of the robust optimal feedback control strategy operators and the solvability of the corresponding backward stochastic Riccati equations, which leads to the existence of the closed-loop saddle points. On the other hand, the problem is not closed-loop solvable if the solution of the corresponding backward stochastic Riccati equations does not have the needed regularity.

In the Internet era, consumers prefer products with the attributes of social responsibility. Supply chain enterprises strengthen corporate social responsibility (CSR) management for their own development. To improve CSR throughout the supply chain, it requires coordination and cooperation among the members of the supply chain. In this paper, we consider a three-tier supply chain system consisting of a supplier, a manufacturer, and a retailer and use stochastic differential game to study the CSR coordination of the supply chain. The following indicators are investigated under four decision situations, such as the optimal level of CSR effort for the supply chain members, the optimal value of profit for the supply chain members and the supply chain system, and the expectation and variance of CSR goodwill. Some important results are obtained. (i) Compared with decentralized decision-making, the optimal level of CSR effort increases for the supplier and the manufacturer under local alliance decision-making without cost sharing, whereas the optimal level of CSR effort remains unchanged for the retailer. (ii) Compared with local alliance decision-making without cost sharing, the optimal level of CSR effort remains unchanged for the supplier and the manufacturer under local alliance decision-making with cost sharing. When the sum of the marginal profit for the supplier and the manufacturer is greater than half of the marginal profit for the retailer, the optimal level of CSR effort increases for the retailer. (iii) Compared with local alliance decision-making with cost sharing, the optimal level of CSR effort increases for the supply chain members under overall alliance decision-making. (iv) From decentralized decision-making to local alliance decision-making without cost sharing, to local alliance decision-making with cost sharing, and then to overall alliance decision-making, the optimal value of profit increases for the supply chain members and the supply chain system. Also, the expectation and variance of CSR goodwill increase. [ABSTRACT FROM AUTHOR]

We investigate a class of zero-sum linear-quadratic stochastic differential games on a finite time horizon governed by slow and fast state equations. The slow-fast nature of the problem can be leveraged to reformulate the associated generalized Riccati equation as a deterministic singular perturbation problem. In doing so, we show that as the evolutionary speed of the slow and fast processes diverge, the existence of a solution to the associated generalized Riccati equation is guaranteed by the existence of a solution to a decoupled pair of differential and algebraic Riccati equations with a reduced order of dimensionality. Furthermore, we are able to formulate a pair of asymptotic estimates to the value function of the game problem by constructing an approximate feedback strategy and observing the limiting value function. [ABSTRACT FROM AUTHOR]

Climate change and greenhouse gas emission reduction have become pressing concerns in recent years. Carbon trading systems and emission-reduction cost-sharing contracts are important emission-reduction measures, under the two mechanisms, this paper considers a dynamic emission-reduction technology investment decision-making problem in a dyadic supply chain consisting of a manufacturer and a retailer. In considering the influence of consumers’ low-carbon preferences on market demand as well as the impact of uncertainty on carbon emission-reduction behaviour, this paper (1) constructs the investment game model under cost-sharing coordination between manufacturers and retailers; (2) adopts differential game and dynamic optimisation methods to obtained investment strategies for manufacturers and retailers under cost-sharing contracts.; and (3) uses a numerical simulation method to simulate the path evolution process of each state variable and, by analysing the sensitivity of various parameters, to determine the influence of various parameters on the decision making of emission reduction among stakeholders. The study finds that under the carbon trading system, cost-sharing contracts have a regulatory effect on enterprise emission-reduction investment and enterprise profits, and that the impact of regulatory effects increases over time. Likewise, the evolution path of the parameters used for various indicators presents a strengthened trend over time. The results show that it is necessary to enhance the cooperative development and exchange of carbon emission-reduction technology among enterprises.

In the Internet era, consumers prefer products with the attributes of social responsibility. Supply chain enterprises strengthen corporate social responsibility (CSR) management for their own development. To improve CSR throughout the supply chain, it requires coordination and cooperation among the members of the supply chain. In this paper, we consider a three-tier supply chain system consisting of a supplier, a manufacturer, and a retailer and use stochastic differential game to study the CSR coordination of the supply chain. The following indicators are investigated under four decision situations, such as the optimal level of CSR effort for the supply chain members, the optimal value of profit for the supply chain members and the supply chain system, and the expectation and variance of CSR goodwill. Some important results are obtained. (i) Compared with decentralized decision-making, the optimal level of CSR effort increases for the supplier and the manufacturer under local alliance decision-making without cost sharing, whereas the optimal level of CSR effort remains unchanged for the retailer. (ii) Compared with local alliance decision-making without cost sharing, the optimal level of CSR effort remains unchanged for the supplier and the manufacturer under local alliance decision-making with cost sharing. When the sum of the marginal profit for the supplier and the manufacturer is greater than half of the marginal profit for the retailer, the optimal level of CSR effort increases for the retailer. (iii) Compared with local alliance decision-making with cost sharing, the optimal level of CSR effort increases for the supply chain members under overall alliance decision-making. (iv) From decentralized decision-making to local alliance decision-making without cost sharing, to local alliance decision-making with cost sharing, and then to overall alliance decision-making, the optimal value of profit increases for the supply chain members and the supply chain system. Also, the expectation and variance of CSR goodwill increase.

In this paper, we regard policyholders, insurance companies, and government departments to be an anti-fraud supervision system, and we explore the supervision of motor vehicle insurance fraud from the perspective of a tripartite game. Taking into consideration the bad reputation records of policyholders as a state variable, through continuous accumulation in effective time, it creates a continuous growth-type warning effect on policyholders, and thus, effectively curbs policyholder fraud and false supervision by insurance companies. At the same time, by considering the influence of random factors on the anti-fraud game of motor vehicle insurance, in this paper, we establish a stochastic differential game model to explore the optimal strategy, the optimal income level, and the expectation and variance of the insured’s bad reputation record stock under the conditions of with and without government supervision. Finally, through a simulation analysis, it is found that the game with government supervision is more conducive to reduce the insured’s fraud intensity, and the simulation proves the impact of different parameters on system stability.

This paper investigates a mixed leader-follower differential games problem, where the model involves two players with the same hierarchy in decision making and each player has two controls which act as a leader and a follower, respectively. Specifically, we solve a follower problem with unconstrained controls and obtain the corresponding Nash equilibrium. Then a leader problem with constrained controls is tackled and a pair of optimal constrained controls are presented by a projection mapping. Furthermore, the control weights are allowed to be singular. In this case, we first investigate the uniform convexity of the cost functional whose corresponding states are fully-coupled forward-backward stochastic differential equation. After that, the minimizing sequence of solutions with non-degenerate control weights are constructed to study the weak convergence of the corresponding cost functionals. Finally, two examples are addressed for non-singular and singular cases, respectively. [ABSTRACT FROM AUTHOR]

We analyze a class of stochastic differential games of singular control, motivated by the study of a dynamic model of interbank lending with benchmark rates. We describe Pareto optima for this game and show how they may be achieved through the intervention of a regulator, whose policy is a solution to a singular stochastic control problem. Pareto optima are characterized in terms of the solutions to a new class of Skorokhod problems with piecewise‐continuous free boundary. Pareto optimal policies are shown to correspond to the enforcement of endogenous bounds on interbank lending rates. Analytical comparison between Pareto optima and Nash equilibria provides insight into the impact of regulatory intervention on the stability of interbank rates. [ABSTRACT FROM AUTHOR]

We consider the interior Hölder regularity of spatial gradient of viscosity solution to the parabolic normalized p (x , t) -Laplace equation u t = (δ i j + (p (x , t) − 2) u i u j | D u | 2 ) u i j with some suitable assumptions on p (x , t) , which arises naturally from a two-player zero-sum stochastic differential game with probabilities depending on space and time. [ABSTRACT FROM AUTHOR]

A stochastic differential game theoretic model has been proposed to determine optimal behavior of a fish while migrating against water currents both in rivers and oceans. Then, a dynamic objective function is maximized subject to two stochastic dynamics, one represents its location and another its relative velocity against water currents. In relative velocity stochastic dynamics, a Cucker–Smale type stochastic differential equation is introduced under white noise. As the information regarding hydrodynamic environment is incomplete and imperfect, a Feynman type path integral under 8 / 3 Liouville-like quantum gravity surface has been introduced to obtain a Wick-rotated Schrödinger type equation to determine an optimal strategy of a fish during its migration. The advantage of having Feynman type path integral is that, it can be used in more generalized nonlinear stochastic differential equations where constructing a Hamiltonian–Jacobi–Bellman (HJB) equation is impossible. The mathematical analytic results show exact expression of an optimal strategy of a fish under imperfect information and uncertainty. [ABSTRACT FROM AUTHOR]

In this study we provide a more robust transboundary industrial pollution reduction strategy for global emission collaborations. We consider the dynamics of each country's quantity of pollution as a Brownian motion with Jumps to capture the systematic jumps caused by surprise effects arising from policy uncertainties within the economy. When the output of each country's domestic consumption good production is proportional to the level of pollution emissions, we apply optimal control theory to find the Nash noncooperative, cooperative and Stackelberg optimal emission paths. To formulate this problem we allow each country's discounted stream of net revenues to be maximized via a Stochastic Differential Game (SDG). We then articulate the Nash noncooperative equilibria, cooperative equilibria and Stackelberg equilibria via a feedback control strategy. We show that the outcome of the game depends on the parameters of the game and the type of equilibrium one considers. Furthermore, in this continuous-time differential game paradigm model we show that the feedback Stackelberg equilibrium will not coincide with the feedback Nash noncooperative equilibrium. In this setting, if the first mover advantage of the leader (Player I) disappears then both equilibria coincide. [ABSTRACT FROM AUTHOR]

Considering the fact that the development of low carbon economy calls for the low carbon technology sharing between interested enterprises, this paper study a stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises. In the paper, we consider the random interference factors that include the uncertain external environment and the internal understanding limitations of decision maker. In the model, superior enterprises and inferior enterprises are separated entities, and they play Stacklberg master-slave game, Nash non-cooperative game, and cooperative game, respectively. We discuss the feedback equilibrium strategies of superior enterprises and inferior enterprises, and it is found that some random interference factors in sharing system can make the variance of improvement degree of low carbon technology level in the cooperation game higher than the variance in the Stackelberg game, and the result of Stackelberg game is similar to the result of Nash game. Additionally, a government subsidy incentive and a special subsidy that inferior enterprises give to superior enterprises are proposed.

In this paper, a nonzero-sum stochastic differential reinsurance game is studied. A model including controls for the market share (advertising), investment, and reinsurance policies is considered. A jump–diffusion process is used to represent insurance claims. Necessary conditions that would lead to the Nash equilibrium can be found in the duopoly game we consider. Cases with and without controls on the reinsurance are discussed separately. Closed-form solutions for optimal strategies are derived by applying the Hamilton–Jacobi–Bellman equation. Numerical examples are given to validate the correctness of our results. [ABSTRACT FROM AUTHOR]

This paper considers the problem of two-player zero-sum stochastic differential game with both players adopting impulse controls in finite horizon under rather weak assumptions on the cost functions (c and χ not decreasing in time). We use the dynamic programming principle and viscosity solutions approach to show existence and uniqueness of a solution for the Hamilton–Jacobi–Bellman–Isaacs (HJBI) partial differential equation (PDE) of the game. We prove that the upper and lower value functions coincide. [ABSTRACT FROM AUTHOR]

Considering the multiple effects of product greenness and carbon emission trading, this article constructs a differential game model to study the collaborative innovation between enterprises in a green supply chain composed of one manufacturer and one supplier. This article examines the optimal green technology innovation strategies, optimal revenues of manufacturers and suppliers as well as the overall green supply chain revenues in the Nash non-cooperative game model, the Stackelberg master–slave game model, and cooperative game model, respectively. Through the comparative analysis of equilibrium results, it is found that the optimal strategies of suppliers and manufacturers as well as the total revenue of the green supply chain in the cooperative games are better than those in the non-cooperative game scenario. Meanwhile, undertaking part of suppliers' green innovation cost by manufacturers is the Pareto optimization mechanism that increases the total revenue of the green supply chain and their respective revenues. The research results of this article provide theoretical support for green supply chain enterprises to formulate incentive mechanisms for scientific strategy-making and reasonable revenue distribution. [ABSTRACT FROM AUTHOR]