Nonzero-sum Stochastic Differential Games Research Articles

Nonzero-Sum Stochastic Impulse Games with an Application in Competitive Retail Energy Markets

We study a nonzero-sum stochastic differential game with both players adopting impulse controls, on a finite time horizon. The objective of each player is to maximize her total expected discounted profits. The resolution methodology relies on the connection between Nash equilibrium and the corresponding system of quasi-variational inequalities (QVIs in short). We prove, by means of the weak dynamic programming principle for the stochastic differential game, that the equilibrium expected payoff of each player is a constrained viscosity solution to the associated QVIs system in the class of linear growth functions. We also introduce a family of equilibrium expected payoffs converging to our equilibrium expected payoff of each player, and which is characterized as the unique constrained viscosity solutions of an approximation of our QVIs system. This convergence result is useful for numerical purpose. We apply a probabilistic numerical scheme which approximates the solution of the QVIs system to the case of the competition between two electricity retailers. We show how our model reproduces the qualitative behavior of electricity retail competition.

Partially-Observed Bilinear Nonzero-Sum Stochastic Differential Game with Affine-Quadratic Discounted Payoff and Application to Competitive Advertising

Partially-Observed Bilinear Nonzero-Sum Stochastic Differential Game with Affine-Quadratic Discounted Payoff and Application to Competitive Advertising

A Stochastic Control Approach for Constrained Stochastic Differential Games with Jumps and Regimes

We develop an approach for two-player constraint zero-sum and nonzero-sum stochastic differential games, which are modeled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems for two different types of constraints, which lead us to find real-valued and stochastic Lagrange multipliers, respectively. Then, we illustrate our results for a nonzero-sum game problem with the stochastic maximum principle technique. Our application is an example of cooperation between a bank and an insurance company, which is a popular, well-known business agreement type called Bancassurance.

Linear-Quadratic Non-Zero Sum Backward Stochastic Differential Game With Overlapping Information

This article studies a linear quadratic non-zero sum stochastic differential game with overlapping information, where the state dynamics are described by a backward stochastic differential equation and the information obtained by two players has a common part but no inclusion relation. The open-loop Nash equilibrium strategy is given by some conditional mean-field stochastic differential equations. In addition, coupled Riccati equations are introduced to express the state feedback form of the Nash equilibrium strategy.

Nonzero-sum impulse games with regime switching

This paper is concerned with a two-player nonzero-sum stochastic differential game, where both players use impulse controls and the state process evolves as a regime switching diffusion. Based on a system of variational inequalities, a verification theorem as a sufficient criterion for optimality is established. Nash equilibrium strategies for the two players, indicating when and how it is optimal to intervene, are given in terms of the obstacle parts of the variational inequalities.

Approximating solution of stochastic differential games for distributed control of a water network

In this paper, our objective is to design a distributed optimal control for pumping stations operating in a Water Distribution Network (WDN), where we would like to satisfy consumer demands with minimum energy consumption. The WDN has been modeled using graph theory and stochastic differential equations. This leads to a non-zero sum stochastic differential game. We have approximated the solution of the aforementioned game using Markov chain approximation and combined it with Shapley's algorithm so as to obtain Minimax mixed strategies. Minimax solution can be obtained as a distributed computation at the pumping stations without any knowledge of the costs incurred by the other pumping stations. Simulation results on the water network show convergence to an approximate Minimax solution.

Maximum Principle for General Partial Information Nonzero Sum Stochastic Differential Games and Applications

Maximum Principle for General Partial Information Nonzero Sum Stochastic Differential Games and Applications

Linear-quadratic non-zero sum differential game for mean-field stochastic systems with asymmetric information

In this study, we consider a class of asymmetric information linear-quadratic non-zero sum stochastic differential game problems. The system dynamics are governed by a forward linear mean-field stochastic differential equation and the cost functional is quadratic. In order to facilitate some practical applications, the mean-field terms for the state process and control process are considered in both the system dynamics and cost functional. Using the classical calculus of variation and dual methods, the open-loop Nash equilibrium point can be expressed by introducing an auxiliary mean-field forward backward stochastic differential equation, which comprises one forward and two backward components. Due to some Riccati equations and ordinary differential equations that possess unique solutions, the Nash equilibrium point can also be represented in feedback form for several special cases under asymmetric information. The corresponding filtering equations are derived, and the existence and uniqueness of the solutions are proved. An investment problem in finance is discussed to demonstrate the good performance of the theoretical results.

An Impulse-Regime Switching Game Model of Vertical Competition

We study a new kind of nonzero-sum stochastic differential game with mixed impulse/switching controls, motivated by strategic competition in commodity markets. A representative upstream firm produces a commodity that is used by a representative downstream firm to produce a final consumption good. Both firms can influence the price of the commodity. By shutting down or increasing generation capacities, the upstream firm influences the price with impulses. By switching (or not) to a substitute, the downstream firm influences the drift of the commodity price process. We study the resulting impulse-regime switching game between the two firms, focusing on explicit threshold-type equilibria. Remarkably, this class of games naturally gives rise to multiple potential Nash equilibria, which we obtain thanks to a verification-based approach. We exhibit three candidate types of equilibria depending on the ultimate number of switches by the downstream firm (zero, one or an infinite number of switches). We illustrate the diversification effect provided by vertical integration in the specific case of the crude oil market. Our analysis shows that the diversification gains strongly depend on the pass-through from the crude price to the gasoline price.

Time-inconsistent linear-quadratic non-zero sum stochastic differential games with random jumps

ABSTRACT We study a kind of time-inconsistent linear-quadratic non-zero sum stochastic differential game problems with random jumps. The time-inconsistency arises from the presence of a quadratic term of the expected state and a state-dependent term, as well as the time-dependent weight of each term in the cost functionals. We define the time-consistent Nash equilibrium point for this kind of problems and establish a general sufficient condition for it through a flow of forward-backward stochastic differential equations with random jumps. In the situation of one-dimensional state and deterministic coefficients, a Nash equilibrium point is given explicitly by some flows of Riccati-like and linear ordinary differential equations. We apply the truncation method to obtain the existence and uniqueness of solutions to them for a specific case. Moreover, an investment and consumption problem is solved and some numerical examples are further provided to illustrate the application of our theoretic results.

Mean-field linear-quadratic stochastic differential games in an infinite horizon

This paper is concerned with two-person mean-field linear-quadratic non-zero sum stochastic differential games in an infinite horizon. Both open-loop and closed-loop Nash equilibria are introduced. The existence of an open-loop Nash equilibrium is characterized by the solvability of a system of mean-field forward-backward stochastic differential equations in an infinite horizon and the convexity of the cost functionals, and the closed-loop representation of an open-loop Nash equilibrium is given through the solution to a system of two coupled non-symmetric algebraic Riccati equations. The existence of a closed-loop Nash equilibrium is characterized by the solvability of a system of two coupled symmetric algebraic Riccati equations. Two-person mean-field linear-quadratic zero-sum stochastic differential games in an infinite horizon are also considered. Both the existence of open-loop and closed-loop saddle points are characterized by the solvability of a system of two coupled generalized algebraic Riccati equations with static stabilizing solutions. Mean-field linear-quadratic stochastic optimal control problems in an infinite horizon are discussed as well, for which it is proved that the open-loop solvability and closed-loop solvability are equivalent.

Dynamic Equilibrium of Market Making with Price Competition

In this paper, we discuss the dynamic equilibrium of market making with price competition and incomplete information. The arrival of market sell/buy orders follows a pure jump process with intensity depending on bid/ask spreads among market makers and having a looping countermonotonic structure. We solve the problem with the nonzero-sum stochastic differential game approach and characterize the equilibrium value function with a coupled system of Hamilton–Jacobi nonlinear ordinary differential equations. We prove, do not assume a priori, that the generalized Issac’s condition is satisfied, which ensures the existence and uniqueness of Nash equilibrium. We also perform some numerical tests that show our model produces tighter bid/ask spreads than those derived using a benchmark model without price competition, which indicates the market liquidity would be enhanced in the presence of price competition of market makers.

Mean-field backward–forward stochastic differential equations and nonzero sum stochastic differential games

We study a general class of fully coupled backward–forward stochastic differential equations of mean-field type (MF-BFSDE). We derive existence and uniqueness results for such a system under weak monotonicity assumptions and without the non-degeneracy condition on the forward equation. This is achieved by suggesting an implicit approximation scheme that is shown to converge to the solution of the system of MF-BFSDE. We apply these results to derive an explicit form of open-loop Nash equilibrium strategies for nonzero sum mean-field linear-quadratic stochastic differential games with random coefficients. These strategies are valid for any time horizon of the game.

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.

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

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.

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.