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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]

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]

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]

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.

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.

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]

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]

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.)

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]

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.

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]

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.

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]

We consider a general nonzero-sum impulse game with two players. The main mathematical contribution of this paper is a verification theorem that provides, under some regularity conditions, a suitable system of quasi-variational inequalities for the payoffs and the strategies of the two players at some Nash equilibrium. As an application, we study an impulse game with a one-dimensional state variable, following a real-valued scaled Brownian motion, and two players with linear and symmetric running payoffs. We fully characterize a family of Nash equilibria and provide explicit expressions for the corresponding equilibrium strategies and payoffs. We also prove some asymptotic results with respect to the intervention costs. Finally, we consider two further nonsymmetric examples where a Nash equilibrium is found numerically. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with a non-zero sum differential game problem of an anticipated forward-backward stochastic differential delayed equation under partial information. We establish a maximum principle and a verification theorem for the Nash equilibrium point by virtue of the duality and convex variation approach. We study a linear-quadratic system under partial information and present an explicit form of the Nash equilibrium point. We derive the filtering equations and prove the existence and uniqueness for the Nash equilibrium point. As an application, we solve a time-delayed pension fund management problem with nonlinear expectation to measure the risk and obtain the explicit solution.

Abstract We study a zero-sum stochastic differential switching game in infinite horizon. We prove the existence of the value of the game and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities with bilateral obstacles. We also obtain a verification theorem which provides an optimal strategy of the game. Finally, some numerical examples with two regimes are given. [ABSTRACT FROM AUTHOR]

This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

In this paper, we discuss a cooperative stochastic differential game for the transboundary industrial pollution problems of Three Gorges Reservoir Area. Base on the stochastic optimal control theory, we derive the Hamilton-Jacobi-Bellman equations for the cooperative games. Furthermore we solve the Hamilton-Jacobi-Bellman equations by using a fitted finite volume method. Finally, an empirical study base on the datum of Three Gorges Reservoir Area is given to demonstrate the efficiency and usefulness of the numerical method. [ABSTRACT FROM AUTHOR]

We consider optimal investment and dividend problem of an insurer, where the insurer decides dividend payment policy and invests his surplus into the financial market to manage his risk exposure. The insurer’s control problem, with the presence of model uncertainty, is formulated as zero-sum, forward–backward games between insurer and market. In the framework of game theory, we develop the games between insurer and market to the more general forward–backward stochastic differential games, where the system is governed by forward–backward stochastic differential equations; the control processes are regular-singular controls; and the informations available to the two players are asymmetric partial informations. Then the maximum principles are established to give sufficient and necessary optimality conditions for the saddle points of the general forward–backward games. Finally, we apply the maximum principles to solve the optimal investment and dividend problem of an insurer under model uncertainty. [ABSTRACT FROM AUTHOR]

Two insurance companies I 1 , I 2 with reserves R 1 ( t ) , R 2 ( t ) compete for customers, such that in a suitable differential game the smaller company I 2 with R 2 ( 0 ) < R 1 ( 0 ) aims at minimizing R 1 ( t ) − R 2 ( t ) by using the premium p 2 as control and the larger I 1 at maximizing by using p 1 . Deductibles K 1 , K 2 are fixed but may be different. If K 1 > K 2 and I 2 is the leader choosing its premium first, conditions for Stackelberg equilibrium are established. For gamma-distributed rates of claim arrivals, explicit equilibrium premiums are obtained, and shown to depend on the running reserve difference. The analysis is based on the diffusion approximation to a standard Cramér-Lundberg risk process extended to allow investment in a risk-free asset.

This paper is concerned with a non-zero sum differential game problem of an anticipated forward-backward stochastic differential delayed equation under partial information. We establish a maximum principle and a verification theorem for the Nash equilibrium point by virtue of the duality and convex variation approach. We study a linear-quadratic system under partial information and present an explicit form of the Nash equilibrium point. We derive the filtering equations and prove the existence and uniqueness for the Nash equilibrium point. As an application, we solve a time-delayed pension fund management problem with nonlinear expectation to measure the risk and obtain the explicit solution. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the uncertainty factors that impact the collection strategy of a remanufacturing closed-loop supply chain. The basis of this type of supply chain is the collection of used products, without which the remanufacturing of new products would be out of the question. In this paper, we considered the effect of stochastic disturbance on the collection process in a closed-loop supply chain consisting of a manufacturer and a retailer, in which the manufacturer is responsible for collecting the used products. Considering stochastic disturbance, in this research, we proposed a stochastic collection model, and derived the feedback control strategies for both the manufacturer and retailer. Next, we investigated the evolutionary path and probability distribution of the stochastic return rate. Finally, we proposed a contract to coordinate the decentralized closed-loop supply chain. The results showed that the manufacturer would improve the level of return effort as the intensity of stochastic disturbance increases. However, the disturbance would not affect the pricing strategies of the supply chain members. As a result of the stochastic disturbance, the return rate always hovered around the expected return rate, while the expectation and variance of the return rate remained stable from a long-term perspective. The expected value and variance of the return rate could decrease or increase with time, depending on the value of the initial return rate of the system. [ABSTRACT FROM AUTHOR]

This work considers cooperative advertising in a manufacturer-retailer supply chain. While the manufacturer is the Stackelberg leader, the retailer is the follower. Using Sethi model it models the dynamic effect of the manufacturer and retailer's advertising efforts on sale. It uses optimal control technique and stochastic differential game theory to obtain the players' advertising strategies and the long-run value of the awareness share. Further, it models the relationship between the payoffs of both players and the awareness share. The work shows that with the provision of subsidy the retail advertising effort increases while the manufacturer's advertising effort reduces. It further shows that the total channel payoff is higher for subsidised retail advertising. However, the subsidy can only be possible if the rate of growth of the manufacturer's payoff is twice higher than that of the retailer. [ABSTRACT FROM AUTHOR]

We study a class of stochastic target games where one player tries to find a strategy such that the state process almost surely reaches a given target, no matter which action is chosen by the opponent. Our main result is a geometric dynamic programming principle, which allows us to characterize the value function as the viscosity solution of a nonlinear partial differential equation. Because abstract measurable selection arguments cannot be used in this context, the main obstacle is the construction of measurable almost optimal strategies. We propose a novel approach where smooth supersolutions are used to define almost-optimal strategies of Markovian type, similarly as in verification arguments for classical solutions of Hamilton-Jacobi-Bellman equations. The smooth supersolutions are constructed by an extension of Krylov's method of shaken coefficients. We apply our results to a problem of option pricing under model uncertainty with different interest rates for borrowing and lending. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic asset allocation and reinsurance game between two insurance companies with contagious claims, where the insurance claim of one insurer can simultaneously affect the claim intensities of itself and its competitor. This clustering feature of claims is modelled by a mutual-excitation Hawkes process with exponential decays. Furthermore, we assume that the management of the insurance company wants to maximise the expected utility of the relative difference between its terminal surplus and that of its competitor at a fixed time point. The Nash equilibrium strategies have been constructed by solving the Hamilton–Jacobi–Bellman equations, where the explicit formulas of the optimal allocation policies have been derived to be independent of the claim intensities. We also introduce an iterative scheme based on the Feynman–Kac formula to compute the optimal proportional reinsurance policies numerically, where the existence and uniqueness of the solution to the fixed point equation and the convergence of the iterative numerical algorithm are proved rigorously. Finally, numerical examples are presented to show the effect of claim intensities on the optimal controls. • Jump diffusion process handles optimal stochastic differential game problem between two insurers. • Contagious claims exist through a mutual-exciting Hawkes process. • Nash equilibrium strategies can be obtained based on the dynamic programming principle. • The optimal allocation policies do not depend on the claim intensities. • The optimal proportional reinsurance policies highly depend on the claim intensities. [ABSTRACT FROM AUTHOR]

Increasing traffic and bandwidth requirements bring challenges to the next generation wireless networks (5G). As one of the main technology in 5G networks, Ultra-Dense Network (UDN) can be used to improve network coverage. In this paper, a radio over fiber based model is proposed to solve the load balancing problem in ultra-dense network. Stochastic differential game is introduced for the load balancing algorithm, and optimal load allocated to each access point (RAP) are formulated as Nash Equilibrium. It is proved that the optimal load can be achieved and the stochastic differential game based scheme is applicable and acceptable. Numerical results are given to prove the effectiveness of the optimal algorithm. [ABSTRACT FROM AUTHOR]

This paper provides a survey on several numerical approximation schemes for stochastic control problems that arise from actuarial science and finance. The problems to be considered include dividend optimization, reinsurance game, and quantile hedging for guaranteed minimum death benefits. To better describe the complicated financial markets and their inherent uncertainty and randomness, the so-called regime-switching models are adopted. Such models are more realistic and versatile, however, far more complicated to handle. Due to the complexity of the construction, the regime-switching diffusion systems can only be solved in very special cases. In general, it is virtually impossible to obtain closed-form solutions. We use Markov chain approximation techniques to construct discrete-time controlled Markov chains to approximate the value function and optimal controls. Examples are presented to illustrate the applicability of the numerical methods. [ABSTRACT FROM AUTHOR]

This paper addresses a new differential game problem with forward-backward doubly stochastic differential equations. There are two distinguishing features. One is that our game systems are initial coupled, rather than terminal coupled. The other is that the admissible control is required to be adapted to a subset of the information generated by the underlying Brownian motions. We establish a necessary condition and a sufficient condition for an equilibrium point of nonzero-sum games and a saddle point of zero-sum games. To illustrate some possible applications, an example of linear-quadratic nonzero-sum differential games is worked out. Applying stochastic filtering techniques, we obtain an explicit expression of the equilibrium point. [ABSTRACT FROM PUBLISHER]

An asset allocation problem of a member of a defined contribution (DC) pension fund is discussed in a hidden, Markov regime-switching, economy using backward stochastic differential equations, (BSDEs). A risk-based approach is considered, where the member selects an optimal asset mix with a view to minimizing the risk described by a convex risk measure of his/her terminal wealth. Firstly, filtering theory is adopted to transform the hidden, Markov regime-switching, economy into one with complete observations and to develop, (robust), filters for the hidden Markov chain. Then the optimal asset allocation problem of the member is formulated as a two-person, zero-sum stochastic differential game between the member and the market in the economy with complete observations. The BSDE approach is then used to solve the game problem and to characterize the saddle point of the game problem. An explicit expression for the optimal asset mix is obtained in the case of a convex risk measure with quadratic penalty and it can be considered a generalized version of the Merton ratio. An explicit expression for the optimal strategy of the market is also obtained, which leads to a risk-neutral wealth dynamic and may provide some insights into asset pricing in the economy with inflation risk and regime-switching risk. Numerical examples are provided to illustrate financial implications of the BSDE solution. [ABSTRACT FROM AUTHOR]

Abstract: A stochastic minimax optimal control strategy for partially observable uncertain quasi-Hamiltonian systems is proposed. First, the stochastic optimal control problem of a partially observable nonlinear uncertain quasi-Hamiltonian system is converted into that of a completely observable linear uncertain system based on a theorem due to Charalambous and Elliot. Then, the converted stochastic optimal control problem is solved by a minimax optimal control strategy based on stochastic averaging method and stochastic differential game. The worst-case disturbances and the optimal controls are obtained by solving a Hamilton-Jacobi-Isaacs (HJI) equation. As an example, the stochastic minimax optimal control of a partially observable Duffing–van der Pol oscillator with uncertain disturbances is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy. [Copyright &y& Elsevier]

Abstract: Separating the management system into several Local-Worlds, the multi-players cooperation game and non-cooperation game in a graph are constructed to describe the interaction between agents in a certain Local-World and between different Local-Worlds respectively. Furthermore, the corresponding analytic processes that consist of two different game theory models are specified, then the profits, the optimal decision of agent, in this system with cooperative stochastic differential game model between Agents and non-cooperative stochastic differential game model between Super-Agents considered simultaneously, are given. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a maximum principle for both zero-sum and nonzero-sum games. The most distinguishing feature, compared with the existing literature, is that the game systems are described by forward–backward stochastic differential equations. This kind of games is motivated by linear-quadratic differential game problems with generalized expectation. We give a necessary condition and a sufficient condition in the form of maximum principle for the foregoing games. Finally, an example of a nonzero-sum game is worked out to illustrate that the theories may find interesting applications in practice. In terms of the maximum principle, the explicit form of an equilibrium point is obtained. [Copyright &y& Elsevier]

We study a stochastic differential game between two insurance companies who employ reinsurance to reduce the risk of exposure. Under the assumption that the companies have large insurance portfolios compared to any individual claim size, their surplus processes can be approximated by stochastic differential equations. We formulate competition between the two companies as a game with a single payoff function which depends on the surplus processes. One company chooses a dynamic reinsurance strategy in order to maximize this expected payoff, while the other company simultaneously chooses a dynamic reinsurance strategy so as to minimize the same quantity. We describe the Nash equilibrium of this stochastic differential game and solve it explicitly for the case of maximizing/minimizing the exit probability. [ABSTRACT FROM AUTHOR]

We consider a risk minimization problem in a continuous-time Markovian regime-switching financial model modulated by a continuous-time, observable and finite-state Markov chain whose states represent different market regimes. We adopt a particular form of convex risk measure, which includes the entropic risk measure as a particular case, as a measure of risk. The risk-minimization problem is formulated as a Markovian regime-switching version of a two-player, zero-sum stochastic differential game. One important feature of our model is to allow the flexibility of controlling both the diffusion process representing the financial risk and the Markov chain representing macro-economic risk. This is novel and interesting from both the perspectives of stochastic differential game and stochastic control. A verification theorem for the Hamilton-Jacobi-Bellman (HJB) solution of the game is provided and some particular cases are discussed. [ABSTRACT FROM AUTHOR]

A minimax optimal control strategy for quasi-Hamiltonian systems with bounded parametric and/or external disturbances is proposed based on the stochastic averaging method and stochastic differential game. To conduct the system energy control, the partially averaged Itô stochastic differential equations for the energy processes are first derived by using the stochastic averaging method for quasi-Hamiltonian systems. Combining the above equations with an appropriate performance index, the proposed strategy is searching for an optimal worst-case controller by solving a stochastic differential game problem. The worst-case disturbances and the optimal controls are obtained by solving a Hamilton-Jacobi-Isaacs (HJI) equation. Numerical results for a controlled and stochastically excited Duffing oscillator with uncertain disturbances exhibit the efficacy of the proposed control strategy. [ABSTRACT FROM AUTHOR]