Stochastic Differential Games Research Articles

Adaptive Stabilization of Noncooperative Stochastic Differential Games

Adaptive Stabilization of Noncooperative Stochastic Differential Games

Pareto efficiency of infinite-horizon cooperative stochastic differential games with Markov jumps and Poisson jumps

Pareto efficiency of infinite-horizon cooperative stochastic differential games with Markov jumps and Poisson jumps

Robust asset-liability management games for n players under multivariate stochastic covariance models

This paper investigates a non-zero-sum stochastic differential game among n competitive CARA asset-liability managers, who are concerned about the potential model ambiguity and aim to seek the robust investment strategies. The ambiguity-averse managers are subject to uncontrollable and idiosyncratic random liabilities driven by generalized drifted Brownian motions and have access to an incomplete financial market consisting of a risk-free asset, a market index and a stock under a multivariate stochastic covariance model. The market dynamics permit not only stochastic correlation between the risky assets but also path-dependent and time-varying risk premium and volatility, depending on two affine-diffusion factor processes. The objective of each manager is to maximize the expected exponential utility of his terminal surplus relative to the average among his competitors under the worst-case scenario of the alternative measures. We manage to solve this robust non-Markovian stochastic differential game by using a backward stochastic differential equation approach. Explicit expressions for the robust Nash equilibrium investment policies, the density generator processes under the well-defined worst-case probability measures and the corresponding value functions are derived. Conditions for the admissibility of the robust equilibrium strategies are provided. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium investment strategies and draw some economic interpretations from these results.

Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

This paper describes a kind of linear quadratic uncertain stochastic hybrid differential game system grounded in the framework of subadditive measures, in which the system dynamics are described by a hybrid differential equation with Wiener–Liu noise and the performance index function is quadratic. Firstly, we introduce the concept of hybrid differential games and establish the Max–Min Lemma for the two-player zero-sum game scenario. Next, we discuss the analysis of saddle-point equilibrium strategies for linear quadratic hybrid differential games, addressing both finite and infinite time horizons. Through the incorporation of a generalized Riccati differential equation (GRDE) and guided by the principles of the Itô–Liu formula, we prove that that solving the GRDE is crucial and serves as both a sufficient and necessary precondition for identifying equilibrium strategies within a finite horizon. In addition, we also acquire the explicit formulations of equilibrium strategies in closed forms, alongside determining the optimal values of the cost function. Through the adoption of a generalized Riccati equation (GRE) and applying a similar approach to that used for the finite horizon case, we establish that the ability to solve the GRE constitutes a sufficient criterion for the emergence of equilibrium strategies in scenarios extending over an infinite horizon. Moreover, we explore the dynamics of a resource extraction problem within a finite horizon and separately delve into an H∞ control problem applicable to an infinite horizon. Finally, we present the conclusions.

On the Population Size in Stochastic Differential Games

On the Population Size in Stochastic Differential Games

Optimal stopping of BSDEs with constrained jumps and related zero-sum games

In this paper, we introduce a non-linear Snell envelope which at each time represents the maximal value that can be achieved by stopping a BSDE with constrained jumps. We establish the existence of the Snell envelope by employing a penalization technique and the primary challenge we encounter is demonstrating the regularity of the limit for the scheme. Additionally, we relate the Snell envelope to a finite horizon, zero-sum stochastic differential game, where one player controls a path-dependent stochastic system by invoking impulses, while the opponent is given the opportunity to stop the game prematurely. Importantly, by developing new techniques within the realm of control randomization, we demonstrate that the value of the game exists and is precisely characterized by our non-linear Snell envelope.

Mean-field ranking games with diffusion control

AbstractWe consider a stochastic differential game, where each player continuously controls the diffusion intensity of her own state process. The players must all choose from the same diffusion rate interval $$[\sigma _1, \sigma _2]$$ [ σ 1 , σ 2 ] , and have individual random time horizons that are independently drawn from the same distribution. The players whose states at their respective time horizons are among the best $$p \in (0,1)$$ p ∈ ( 0 , 1 ) of all terminal states receive a fixed prize. We show that in the mean field version of the game there exists an equilibrium, where the representative player chooses the maximal diffusion rate when the state is below a given threshold, and the minimal rate else. The symmetric n-fold tuple of this threshold strategy is an approximate Nash equilibrium of the n-player game. Finally, we show that the more time a player has at her disposal, the higher her chances of winning.

Robust non-zero-sum stochastic differential game of two insurers with common shock and CDS transaction

Robust non-zero-sum stochastic differential game of two insurers with common shock and CDS transaction

Pareto optimality of stochastic cooperative differential game with general delays in finite horizon

The Pareto optimality of stochastic cooperative differential game with a discrete delay, a moving-average delay and a noisy memory process is studied. We establish two sets of equivalent necessary and sufficient conditions for Pareto efficient strategies. The first set comes from reduction to a discrete delayed Pareto optimality, while the second set given by Malliavin derivative is derived by the decomposition of the adjoint equation and the relationship between two Hamiltonian functions. As applications, we use the theoretical results to an indefinite linear quadratic (LQ) Pareto game with general delays.

Stochastic differential game of joint emission reduction in the supply chain based on CSR and carbon cap-and-trade mechanism

This paper considers both carbon emission reduction and corporate social responsibility (CSR) goodwill as endogenous variables in the joint emission reduction process of the supply chain system. The joint emission reduction strategy of the three-tier supply chain system is examined from a long-term perspective utilizing stochastic differential game models. Three models, namely the cooperative game, the Nash non-cooperative game, and the Stackelberg master–slave game, are developed to analyse and compare the equilibrium strategies under each model. The results indicate that the manufacturer-led Stackelberg master–slave game can achieve the Pareto improvement in the profits of the supplier, manufacturer, retailer, and the entire supply chain system.

A Decision Support System Based on Stochastic Differential Game Model in Pollution Control Chain

A Decision Support System Based on Stochastic Differential Game Model in Pollution Control Chain

Stochastic differential reinsurance game for two competitive insurers with ambiguity-aversion under mean-variance premium principle

Stochastic differential reinsurance game for two competitive insurers with ambiguity-aversion under mean-variance premium principle

Stochastic Adaptive Linear Quadratic Differential Games

Game theory is playing more and more important roles in understanding complex systems and in investigating intelligent machines with various uncertainties. As a starting point, we consider the classical two-player zero-sum linear-quadratic stochastic differential games, but in contrast to most of the existing studies, the coefficient matrices of the systems are assumed to be unknown to both players, and consequently it is necessary to study adaptive strategies of the players, which may be termed as adaptive games and which has rarely been explored in the literature. In this paper, by introducing a suitable information structure for adaptive games, we will show that a theory can be established on adaptive strategies that are designed based on both the certainty equivalence principle and the diminishing excitation technique. Under almost the same physical structure conditions as those in the traditional known parameters case, it is shown that the closed-loop adaptive game systems will be globally stable and asymptotically reach the Nash equilibrium.

A Stackelberg–Nash equilibrium with investment and reinsurance in mixed leadership game

In this paper, we investigate the optimal reinsurance and investment problem from joint interests of the insurer and the reinsurer in the framework of the mixed leadership game, where both of them can invest their wealth in the financial market, and the liabilities are correlated with the risky asset. More specifically, we assume that the insurer is the leader to determine the amount he/she invests into the risky asset but acts as the follower to decide the optimal reinsurance retention level, while the reinsurer is the leader to set the reinsurance premium price and acts as a follower to determine his/her investment strategy. Both the insurer and the reinsurer aim to maximize the expected utility of their terminal wealth, and the explicit Stackelberg-Nash equilibrium is derived by solving the corresponding Hamilton-Jacobi-Bellman (HJB) equations. In order to better understand the effect of the mixed leadership game on the optimal reinsurance and investment problem, we also consider the optimization problem within the framework of the stochastic Stackelberg differential game shown in [Bai, Y., Zhou, Z., Xiao, H., Gao, R. & Zhong, F. (2021). A Stackelberg reinsurance-investment game with asymmetric information and delay. Optimization 70(10), 2131–2168.], where the reinsurer is the leader while the insurer acts as the follower for both the reinsurance and investment strategies. Some theoretical analyses and numerical examples are provided to show the economic intuition and insights of the results.

Zero-Sum Stochastic Linear-Quadratic Stackelberg Differential Games with Jumps

Zero-Sum Stochastic Linear-Quadratic Stackelberg Differential Games with Jumps

Stackelberg Stochastic Differential Games in Feedback Information Pattern with Applications

Stackelberg Stochastic Differential Games in Feedback Information Pattern with Applications

Research on military-civilian collaborative innovation of science and technology based on a stochastic differential game model.

The construction of an integrated national strategic system and capability is an essential goal of implementing the strategy of military-civilian integration in the contemporary era. And the collaborative innovation of military-civilian S&T is an inevitable choice to achieve this goal. Due to the dynamic, complex, and stochastic characteristics of military-civilian S&T collaborative innovation, the level of S&T innovation is highly volatile. This paper takes the internal and external stochastic disturbance factors of military-civilian S&T collaborative innovation as the perspective, studies the strategy selection problem of military-civilian S&T collaborative innovation under military domination, constructs a differential game model to explore the innovation strategies under the non-cooperative model without military subsidies, the non-cooperative model with military subsidies, and the collaborative model. Finally, we use numerical experiments to verify the validity of the conclusions. The study shows that: (1) Within a reasonable range of values of the benefit distribution coefficient, the system can achieve the Pareto optimum, and the collaborative model is conducive to improving the S&T innovation level and the optimum benefit level of the system. (2) Military subsidies can increase the benefits of the system and the parties involved to achieve Pareto improvement. (3) The level of S&T innovation under the collaborative model has dynamic evolutionary characteristics of maximum expectation and variance. As the intensity of disturbance increases, the stability of the system may be destroyed. Risk-averse civil enterprises prefer the cooperative mode, whereas risk-averse civil enterprises prefer the non-cooperative model.

Non-zero-sum stochastic differential games on investment, consumption and proportional reinsurance

Non-zero-sum stochastic differential games on investment, consumption and proportional reinsurance

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.

Inventory Management Problems in Stochastic Differential Game Models: Markov Chain Approximation and Optimal Policies

Inventory Management Problems in Stochastic Differential Game Models: Markov Chain Approximation and Optimal Policies