Stochastic Differential Games Research Articles

Ergodic risk-sensitive stochastic differential games with reflecting diffusions in a bounded domain

In this article, we study risk-sensitive stochastic differential games for controlled reflecting diffusion processes in a smooth bounded domain. We analyze the ergodic cost evaluation criterion for both nonzero-sum games and zero-sum games. Using principal eigenvalue approach, we establish the existence of Nash/saddle-point equilibria for relevant cases.

Conversion of Dual-Use Technology: A Differential Game Analysis under the Civil-Military Integration

Civil-military integration enables the symmetry development of dual-use technologies in the defense and civilian areas. In this paper, we used stochastic differential game theory to build theoretical models of the dual-use technology conversion between a military firm and a civilian firm. The models include three directions of dual-use technology conversion, namely, spin-off, spin-in, and mix. In particular, we incorporate the dynamic development of dual-use technology into the research framework. Our results show that the order of the Pareto optimality results is mix, spin-in, spin-out, including the best effort, the highest revenue, and the most technology value-added. The results also indicate that: (1) Some random interference factors can affect the transfer efforts, such as the coefficient of technological innovation capability and cost coefficients. (2) The military subsidies factor is an inventive mechanism that can promote dual-use transfer from the civilian firm to the military. (3) The military enterprise only earns more than the civilian firm in spin-in. (4) The growth of models in the dual-use technology conversion process faces increased technology uncertainty.

Stochastic differential reinsurance games in diffusion approximation models

In this paper, we study some noncooperative and cooperative stochastic differential games between two insurers with reinsurance controls. The surplus processes are modeled by diffusion approximation processes and the insurers can purchase reinsurance dynamically (i.e. control the drift and diffusion terms continuously over time). We consider two types of reinsurance: quota-share (QS) reinsurance and excess-of-loss (XL) reinsurance. In the noncooperative game, one insurer tries to minimize the probability that the surplus difference of the two insurers reaches a low target before it hits a high target, while the other aims to maximize the probability. We consider two cases of the game: in the first case, both insurers purchase XL reinsurance; and in the second case, one insurer purchases QS reinsurance while the other purchases XL reinsurance. In some parameter cases, we solve the game by finding the value function and Nash equilibrium strategy explicitly. In some parameter cases, the value function and Nash equilibrium strategy do not exist and we find the sup-value and sub-value functions of the game. We also establish and solve a cooperative game in which both insurers make joint efforts to minimize the probability that the sum of surplus processes reaches a low target before it hits a high target. Numerical examples and economic implications are given to illustrate the results.

Robust optimal investment and reinsurance for an insurer with inside information

This paper studies a robust optimal investment–reinsurance problem for an insurer who possesses inside information on the financial market and the insurance business under model uncertainty. The insurer’s surplus process and the risky asset process in the financial market are assumed to be correlated jump diffusion processes with random coefficients. The inside information is modelled by a general random variable related to the future realizations of the surplus process and the risky asset process. Under the criterion of expected utility maximization, the optimal investment–reinsurance problem with inside information is transformed into an adapted stochastic control problem. By incorporating Knightian uncertainty into the model, we establish an adapted stochastic differential game problem with a nonstandard performance functional, which aims to select the robust optimal investment–reinsurance strategy for the insurer with inside information. The stochastic maximum principles are adopted to derive a characterization of the optimal strategy. Some interesting special cases are discussed and the corresponding optimal strategies are derived.

A Mean-Field Linear-Quadratic Stochastic Stackelberg Differential Game with one Leader and Two Followers

This paper is concerned with a linear-quadratic (LQ) stochastic Stackelberg differential game with one leader and two followers, where the game system is governed by a mean-field stochastic differential equation (MF-SDE). By maximum principle and verification theorem, the open-loop Stackelberg solution is expressed as a feedback form of the state and its mean with the help of three systems of Riccati equations.

Nonzero-Sum Stochastic Differential Reinsurance Games with Jump–Diffusion Processes

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.

Explicit Characterization of Feedback Nash Equilibria for Indefinite, Linear-Quadratic, Mean-Field-Type Stochastic Zero-Sum Differential Games with Jump-Diffusion Models

We consider the indefinite, linear-quadratic, mean-field-type stochastic zero-sum differential game for jump-diffusion models (I-LQ-MF-SZSDG-JD). Specifically, there are two players in the I-LQ-MF-SZSDG-JD, where Player 1 minimizes the objective functional, while Player 2 maximizes the same objective functional. In the I-LQ-MF-SZSDG-JD, the jump-diffusion-type state dynamics controlled by the two players and the objective functional include the mean-field variables, i.e., the expected values of state and control variables, and the parameters of the objective functional do not need to be (positive) definite matrices. These general settings of the I-LQ-MF-SZSDG-JD make the problem challenging, compared with the existing literature. By considering the interaction between two players and using the completion of the squares approach, we obtain the explicit feedback Nash equilibrium, which is linear in state and its expected value, and expressed as the coupled integro-Riccati differential equations (CIRDEs). Note that the interaction between the players is analyzed via a class of nonanticipative strategies and the “ordered interchangeability” property of multiple Nash equilibria in zero-sum games. We obtain explicit conditions to obtain the Nash equilibrium in terms of the CIRDEs. We also discuss the different solvability conditions of the CIRDEs, which lead to characterization of the Nash equilibrium for the I-LQ-MF-SZSDG-JD. Finally, our results are applied to the mean-field-type stochastic mean-variance differential game, for which the explicit Nash equilibrium is obtained and the simulation results are provided.

Partial Information Stochastic Differential Games for Backward Stochastic Systems Driven by Lévy Processes

In this paper, we consider a partial information two-person zero-sum stochastic differential game problem, where the system is governed by a backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. A sufficient condition and a necessary one for the existence of the saddle point for the game are proved. As an application, a linear quadratic stochastic differential game problem is discussed.

Robust optimal R&D investment under technical uncertainty in a regime-switching environment

ABSTRACT In this paper, we extend the classic optimal research and development (R&D) investment model into the regime-switching environment. We formulate a robust model to obtain the maximal net value function of the R&D project under a family of real-world measures, which can also be regarded as a stochastic differential game. Then, the verification argument is provided. Though we do not find the closed-form solution, we give a numerical simulation to further study the qualities of the solution to our model.

Maximum Principle for Non-Zero Sum Stochastic Differential Game with Discrete and Distributed Delays

This technical note is concerned with the maximum principle for a non-zero sum stochastic differential game with discrete and distributed delays. Not only the state variable, but also control variables of players involve discrete and distributed delays. By virtue of the duality method and the generalized anticipated backward stochastic differential equations, the author establishes a necessary maximum principle and a sufficient verification theorem. To explain theoretical results, the author applies them to a dynamic advertising game problem.

Non-Zero Sum Stochastic Differential Investment And Financial Market Game Under Model Uncertainty

Non-Zero Sum Stochastic Differential Investment And Financial Market Game Under Model Uncertainty

Linear-Quadratic Mixed Stackelberg–Nash Stochastic Differential Game with Major–Minor Agents

In this paper, we study a controlled linear-quadratic-Gaussian large population system combining three types of interactive agents mixed, which are respectively, major leader, minor leaders, and minor followers. In reality, they may represent three typical types of participants involved in market price formation: major supplier, minor suppliers and minor producers. The Stackelberg–Nash–Cournot (SNC) approximate equilibrium is derived from the combination of a major–minor mean-field game (MFG) and a leader–follower Stackelberg game. Although all agents are of forward states in that only initial conditions are specified in their dynamics, our SNC analysis provides an MFG framework that is naturally in a forward–backward state in that both initial and terminal conditions are specified. This result differs from those reported in the literature on standard MFG frameworks, mainly as a result of the adoption of a Stackelberg structure. Through variational analysis, the consistency condition system can be represented by some fully-coupled forward–backward-stochastic-differential-equations with a high-dimensional block structure in an open-loop case. To sufficiently address the related solvability, we also derive the feedback form of the SNC approximate. equilibrium strategy via some coupled Riccati equations. Our study includes various mean-field game models as its special cases.

A Reinsurance and Investment Game between Two Insurers under the CEV Model

In this paper, the problem of nonzero-sum stochastic differential game between two competing insurance companies is considered, i.e., the relative performance concerns. A certain proportion of reinsurance can be taken out by each insurer to control his own risk. Moreover, each insurer can invest in a risk-free asset and risk asset with the price dramatically following the constant elasticity of variance (CEV) model. Based on the principle of dynamic programming, a general framework regarding Nash equilibrium for nonzero-sum games is established. For the typical case of exponential utilization, we, respectively, give the explicit solutions of the equilibrium strategy as well as the equilibrium function. Some numerical studies are provided at last which assist in obtaining some economic explanations.

Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market

We apply dynamic programming principle to discuss two optimal investment problems by using zero-sum and nonzero-sum stochastic game approaches in a continuous-time Markov regime-switching environment within the frame work of behavioral finance. We represent different states of an economy and, consequently, investors’ floating levels of psychological reactions by a D-state Markov chain. The first application is a zero-sum game between an investor and the market, and the second one formulates a nonzero-sum stochastic differential portfolio game as the sensitivity of two investors’ terminal gains. We derive regime-switching Hamilton–Jacobi–Bellman–Isaacs equations and obtain explicit optimal portfolio strategies with Feynman–Kac representations of value functions. We illustrate our results in a two-state special case and observe the impact of regime switches by comparative results.

Transboundary Emission Under Stochastic Differential Game

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.

On the convergence of closed-loop Nash equilibria to the mean field game limit

This paper continues the study of the mean field game (MFG) convergence problem: In what sense do the Nash equilibria of $n$-player stochastic differential games converge to the mean field game as $n\rightarrow\infty $? Previous work on this problem took two forms. First, when the $n$-player equilibria are open-loop, compactness arguments permit a characterization of all limit points of $n$-player equilibria as weak MFG equilibria, which contain additional randomness compared to the standard (strong) equilibrium concept. On the other hand, when the $n$-player equilibria are closed-loop, the convergence to the MFG equilibrium is known only when the MFG equilibrium is unique and the associated “master equation” is solvable and sufficiently smooth. This paper adapts the compactness arguments to the closed-loop case, proving a convergence theorem that holds even when the MFG equilibrium is nonunique. Every limit point of $n$-player equilibria is shown to be the same kind of weak MFG equilibrium as in the open-loop case. Some partial results and examples are discussed for the converse question, regarding which of the weak MFG equilibria can arise as the limit of $n$-player (approximate) equilibria.

A Differential Game of Ecological Compensation Criterion for Transboundary Pollution Abatement under Learning by Doing

This paper studies a stochastic differential game of transboundary pollution abatement between two kinds of ecological compensation and the abatement policy, in which the learning by doing is taken into account. Emission and pollution abatement between upstream and downstream region in the same basin is a Stackelberg game, and the downstream regions provide economic compensation for pollution abatement in the upstream region. We discuss the feedback Nash equilibrium strategies of proportional compensation and investment compensation, and it is found that an appropriate ecological compensation ratio can improve the investment level of pollution abatement in the two regions by accumulating experience in the process of learning by doing. In the long term, the investment compensation mechanism is an effective transboundary pollution abatement measure that can continuously reduce the water pollution stock in the upstream and downstream.

Non-zero-sum reinsurance and investment game between two mean-variance insurers under the CEV model

This paper considers a non-zero-sum stochastic differential game between two competitive mean-variance insurers, who aim to seek the time-consistent reinsurance and investment strategies. These two insurers are allowed to purchase proportional reinsurance to mitigate individual claim risks; and can invest in one risk-free asset and one risky asset whose price dynamics follows the constant elasticity of variance model. The main objective of each insurer is to maximizing the mean-variance utility of his relative terminal wealth with respect to that of his competitor. Applying the techniques of stochastic control theory, we derive the Nash equilibrium reinsurance and investment strategies explicitly and present the corresponding verification theorem. Furthermore, Nash equilibrium strategies and value functions are also provided under the diffusion approximation model. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium reinsurance and investment strategies and draw some economic interpretations from these results.

Nonzero-Sum Stochastic Differential Games Between an Impulse Controller and a Stopper

We study a two-player nonzero-sum stochastic differential game, where one player controls the state variable via additive impulses, while the other player can stop the game at any time. The main goal of this work is to characterize Nash equilibria through a verification theorem, which identifies a new system of quasivariational inequalities, whose solution gives equilibrium payoffs with the correspondent strategies. Moreover, we apply the verification theorem to a game with a one-dimensional state variable, evolving as a scaled Brownian motion, and with linear payoff and costs for both players. Two types of Nash equilibrium are fully characterized, i.e. semi-explicit expressions for the equilibrium strategies and associated payoffs are provided. Both equilibria are of threshold type: in one equilibrium players’ intervention are not simultaneous, while in the other one the first player induces her competitor to stop the game. Finally, we provide some numerical results describing the qualitative properties of both types of equilibrium.

Nonzero-sum risk-sensitive stochastic differential games with discounted costs

We study nonzero-sum stochastic differential games with risk-sensitive discounted cost criteria. Under fairly general conditions on drift term and diffusion coefficients, we establish a Nash equilibrium in Markov strategies for the discounted cost criterion. We achieve our results by studying relevant systems of coupled HJB equations.