Stochastic Differential Games Research Articles

Dynamic asset allocation with relative wealth concerns in incomplete markets

In dynamic portfolio choice problems, stochastic state variables such as stochastic volatility lead to adjustments of the optimal stock demand referred to as hedge terms or Merton-Breeden terms. By deriving an explicit solution in a two-agent framework with a stochastic opportunity set, we show that relative wealth concerns give rise to new hedge terms beyond the ordinary ones. This is because the agents hedge against both exogenous changes in the state variable and endogenous decisions of the other agent. Depending on the parametrization of the model, these new terms can significantly change the investors’ hedging demands. We also show that both heterogeneity in risk aversion or relative wealth concerns can have similar effects on the heterogeneity in portfolio decisions. Formally, we study a non-cooperative, non-zero sum stochastic differential game for which we prove a verification theorem in a setting with an unspanned state variable.

Time-consistent non-zero-sum stochastic differential reinsurance and investment game under default and volatility risks

This paper investigates a non-zero-sum stochastic differential game between two mean–variance insurers. These two insurers are concerned about their terminal wealth and the relative performance compared with each other. We assume that they can buy proportional reinsurance and invest in a financial market consisting of a risk-free asset, a stock and a defaultable bond. The price process of stock is driven by the constant elasticity of variance (CEV) model and the defaultable bond recovers a proportion of value at default. So, these two insurers are faced with insurance risk, volatility risk and default risk. The non-zero-sum goal of these insurers is to maximize the mean–variance utility of a weighted value of their terminal and relative wealths. We solve the mean–variance problem in the time-consistent case and establish the extended Hamilton−Jacobi−Bellman systems for the post-default case and the pre-default case, respectively. Furthermore, we derive the closed form solutions of the Nash equilibrium reinsurance and investment strategies for these two insurers. In the end of this paper, we calibrate the parameters based on real data and several numerical examples are provided to illustrate the effects of economic parameters on the equilibrium strategies.

A closed-loop saddle point for zero-sum linear-quadratic stochastic differential games with mean-field type

This paper is concerned with a linear-quadratic zero-sum differential game with mean-field type. The notions of explicit and implicit strategy laws are proposed. Based on them, a closed-loop formulation for saddle points in the mixed-strategy-law form is established. In order to construct the saddle point, the classical entire completion-of-square argument is developed to a four-step-completion-of-square argument, and a pair of optimality conditions is proposed.

A kind of non-zero sum mixed differential game of backward stochastic differential equation

This paper is concerned with a non-zero sum mixed differential game problem described by a backward stochastic differential equation. Here the term “mixed” means that this game problem contains a deterministic control v_{1} of Player 1 and a random control process v_{2} of Player 2. By virtue of the classical variational method, a necessary condition and an Arrow’s sufficient condition for the mixed stochastic differential game problem are presented. A linear–quadratic mixed differential game problem is discussed, and the corresponding Nash equilibrium point is explicitly expressed by the solution of mean-field forward–backward stochastic differential equation. The most distinguishing feature, compared with the existing literature, is that the optimal state process of the linear–quadratic game satisfies a linear mean-field backward stochastic differential equation. Finally, a home mortgage and wealth management problem is given to illustrate our theoretical results.

Two-player zero-sum stochastic differential games with regime switching

This paper is concerned with the two-player zero-sum stochastic differential game in a regime switching model with an infinite horizon. The state of the system is characterized by a number of diffusions coupled by a continuous-time finite-state Markov chain. Based on the dynamic programming principle (DPP), the lower and upper value functions are shown to be the unique viscosity solutions of the associated lower and upper Hamilton–Jacobi–Bellman–Isaacs (HJBI) equations, respectively. Moreover, the lower and upper value functions coincide under the Isaacs’ condition, which implies that the game admits a value. All the proofs in this paper are markedly different from those for the case when there is no regime switching.

A Mean Field Game-Based Distributed Edge Caching in Fog Radio Access Networks

In this paper, the edge caching optimization problem in fog radio access networks (F-RANs) is investigated. Taking into account time-variant user requests and ultra-dense deployment of fog access points (F-APs), we propose a distributed edge caching scheme to jointly minimize the request service delay and fronthaul traffic load. Considering the interactive relationship among F-APs, we model the optimization problem as a stochastic differential game (SDG) which captures the dynamics of F-AP states. To address both the intractability problem of the SDG and the caching capacity constraint, we propose to solve the optimization problem in a distributive manner. Firstly, a mean field game (MFG) is converted from the original SDG by exploiting the ultra-dense property of F-RANs, and the states of all F-APs are characterized by a mean field distribution. Then, an iterative algorithm is developed that enables each F-AP to obtain the mean field equilibrium and caching control without extra information exchange with other F-APs. Secondly, a fractional knapsack problem is formulated based on the mean field equilibrium, and a greedy algorithm is developed that enables each F-AP to obtain the final caching policy subject to the caching capacity constraint. Simulation results show that the proposed scheme outperforms the baselines.

A non-zero-sum reinsurance-investment game with delay and asymmetric information

<p style='text-indent:20px;'>In this paper, we investigate a non-zero-sum stochastic differential reinsurance-investment game problem between two insurers. Both insurers can purchase proportional reinsurance and invest in a financial market that contains a risk-free asset and a risky asset. We consider the insurers' wealth processes with delay to characterize the bounded memory feature. For considering the effect of asymmetric information, we assume the insurers have access to different levels of information in the financial market. Each insurer's objective is to maximize the expected utility of its performance relative to its competitor. We derive the Hamilton-Jacobi-Bellman (HJB) equations and the general Nash equilibrium strategies associated with the control problem by applying the dynamic programming principle. For constant absolute risk aversion (CARA) insurers, the explicit Nash equilibrium strategies and the value functions are obtained. Finally, we present some numerical studies to draw economic interpretations and find the following interesting results: (1) the insurer with less information completely ignores its own risk aversion factor, but imitates the investment strategy of its competitor who has more information on the financial market, which is a manifestation of the herd effect in economics; (2) the difference between the effects of different delay weights on the strategies is related to the length of the delay time in the framework of the non-zero-sum stochastic differential game, which illustrates that insurers should rationally estimate the correlation between historical performance and future performance based on their own risk tolerance, especially when decision makers consider historical performance over a long period of time.

Stochastic Linear Quadratic Stackelberg Differential Game with Overlapping Information

This paper is concerned with the stochastic linear quadratic Stackelberg differential game with overlapping information, where the diffusion terms contain the control and state variables. Here the term “overlapping” means that there are common part between the follower’s and the leader’s information, while they have no inclusion relation. Optimal controls of the follower and the leader are obtained by the stochastic maximum principle, the direct calculation of the derivative of the cost functional and stochastic filtering. A new system of Riccati equations is introduced to give the state estimate feedback representation of the Stackelberg equilibrium strategy, while its solvability is a rather difficult open problem. A special case is then studied and is applied to the continuous-time principal-agent problem.

An empirical study of transboundary air pollution of the Beijing-Tianjin region

In this paper, we present a stochastic differential game to model the transboundary air pollution problems of the Beijing-Tianjin region with emission permits trading. By using stochastic optimal control theory, we obtain the Hamilton-Jacobi-Bellman equation satisfied by the value function for the cooperative games. Next, we solve the Hamilton-Jacobi-Bellman equation by using a fitted finite volume method. Finally, the efficiency and the usefulness of the fitted finite volume method are illustrated by an empirical study.

Stochastic Differential Game Model Analysis of Emission-Reduction Technology Under Cost-Sharing Contracts in the Carbon Trading Market

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.

Pareto Optima for a Class of Singular Control Games

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

From the master equation to mean field game limit theory: Large deviations and concentration of measure

We study a sequence of symmetric $n$-player stochastic differential games driven by both idiosyncratic and common sources of noise, in which players interact with each other through their empirical distribution. The unique Nash equilibrium empirical measure of the $n$-player game is known to converge, as $n$ goes to infinity, to the unique equilibrium of an associated mean field game. Under suitable regularity conditions, in the absence of common noise, we complement this law of large numbers result with nonasymptotic concentration bounds for the Wasserstein distance between the $n$-player Nash equilibrium empirical measure and the mean field equilibrium. We also show that the sequence of Nash equilibrium empirical measures satisfies a weak large deviation principle, which can be strengthened to a full large deviation principle only in the absence of common noise. For both sets of results, we first use the master equation, an infinite-dimensional partial differential equation that characterizes the value function of the mean field game, to construct an associated McKean–Vlasov interacting $n$-particle system that is exponentially close to the Nash equilibrium dynamics of the $n$-player game for large $n$, by refining estimates obtained in our companion paper. Then we establish a weak large deviation principle for McKean–Vlasov systems in the presence of common noise. In the absence of common noise, we upgrade this to a full large deviation principle and obtain new concentration estimates for McKean–Vlasov systems. Finally, in two specific examples that do not satisfy the assumptions of our main theorems, we show how to adapt our methodology to establish large deviations and concentration results.

Application of optimal control to the dynamic advertising decisions for supply chain with multiple delays

This paper is concerned with application problem of optimal control to a class of dynamic advertising models with multiple delays. Here, a dynamic model with state and control delays is introduced to describe the impacts of advertising delayed and memory effects on the evolution of brand goodwill. In the decentralized and centralized systems, the optimal advertising strategies of supply chain members are presented by utilizing the nonzero-sum differential game governed by differential equation with multiple delays in state and control variables. Special effort is made to analyse the geometrical shapes of advertising strategy and provide the optimal decision structure of supply chain system. Furthermore, nonzero-sum stochastic differential game is applied to explore the dynamic advertising decision problems of supply chain system with the multiple delays and stochastic cases. Finally, numerical examples are exploited to illustrate the effectiveness of the proposed results.

Robust Utility Maximization Under Model Uncertainty via a Penalization Approach

This paper addresses the problem of utility maximization under uncertain parameters. In contrast with the classical approach, where the parameters of the model evolve freely within a given range, we constrain them via a penalty function. We show that this robust optimization process can be interpreted as a two-player zero-sum stochastic differential game. We prove that the value function satisfies the Dynamic Programming Principle and that it is the unique viscosity solution of an associated Hamilton–Jacobi–Bellman–Isaacs equation. We test this robust algorithm on real market data. The results show that robust portfolios generally have higher expected utilities and are more stable under strong market downturns. To solve for the value function, we derive an analytical solution in the logarithmic utility case and obtain accurate numerical approximations in the general case by three methods: finite difference method, Monte Carlo simulation, and Generative Adversarial Networks.

A Game Theoretical Approach to Homothetic Robust Forward Investment Performance Processes in Stochastic Factor Models

This paper studies an optimal forward investment problem in an incomplete market with model uncertainty, in which the dynamics of the underlying stocks depends on the correlated stochastic factors. The uncertainty stems from the probability measure chosen by an investor to evaluate the performance. We obtain directly the representation of the power robust forward performance process in factor-form by combining the zero-sum stochastic differential game and ergodic BSDE approach. We also establish the connections with the risk-sensitive zero-sum stochastic differential games over an infinite horizon with ergodic payoff criteria, as well as with the classical power robust expected utility for long time horizons.

Zero-sum risk-sensitive stochastic differential games with reflecting diffusions in the orthant

We study zero-sum games with risk-sensitive cost criterion on the infinite horizon where the state is a controlled reflecting diffusion in the nonnegative orthant. We consider two cost evaluation criteria: discounted cost and ergodic cost. Under certain assumptions, we establish the existence of saddle point equilibria. We obtain our results by studying the corresponding Hamilton–Jacobi–Isaacs equations. For the ergodic cost criterion, exploiting the stochastic representation of the principal eigenfunction, we have completely characterized saddle point equilibrium in the space of stationary Markov strategies.

Stochastic hybrid differential games and match race problems

We discuss the general framework of a stochastic two-player, hybrid differential game, and we apply it to the modelling of a “match race” between two sailing boats, namely a competition in which the goal of both players is to proceed in the windward direction, while trying to slow down the other player. We provide a convergent approximation scheme for the computation of the value function of the game, and we validate the approach on some typical racing scenarios.

On the bounded and stabilizing solution of a generalized Riccati differential equation arising in connection with a zero‐sum linear quadratic stochastic differential game

SummaryWe study a class of coupled nonlinear matrix differential equations arising in connection with the solution of a zero‐sum two‐player linear quadratic (LQ) differential game for a dynamical system modeled by an Itô differential equation subject to random switching of its coefficients. The system of differential equations under consideration contains as special cases the game‐theoretic Riccati differential equations arising in the solution of the H∞ control problem from the deterministic and stochastic cases. A set of sufficient conditions that guarantee the existence of the bounded and stabilizing solution of this kind of Riccati differential equations is provided. We show how such stabilizing solution is involved in the construction of the equilibrium strategy of a zero‐sum LQ stochastic differential game on an infinite‐time horizon and give as a byproduct the solution of such a control problem.

Analysis of Reverse Channels in a Stochastic Closed-loop Supply Chain

We study a closed-loop supply chain in which used products can be collected by a manufacturer, a retailer, or a third-party for the purpose of recycling. Since the collection process can be noisy, the system dynamic is modeled by a stochastic differential equation. We formulate stochastic differential game models for each of the three reverse channels, derive their feedback control strategies, and compare the performance of the three reverse channels. We find that the manufacturer collection is the most efficient reverse channel. This is different from the known result in a static setting that has the retailer collection as the best channel. Furthermore, we extend our model to consider the impact of the reverse logistics cost and the collection cost coefficient. The retailer and the third-party collection could also be the best reverse channel when the retailer has the advantage of a low reverse logistics cost, or the third-party has the advantage of a high collection capacity, respectively.

Anticipated backward stochastic differential equations and their applications to zero-sum stochastic differential games

In this article, we are concerned with the zero-sum stochastic differential game problems with the cost functional depending not only on but also on The main approach is the anticipated backward stochastic differential equations under weaker conditions. For these equations, we prove the existence and uniqueness theorem and the comparison theorems. Then, by using these techniques, we get the saddle-point strategy for the zero-sum stochastic differential games when the Isaacs’ condition holds.