Class Of Stochastic Differential Games Research Articles

Global existence for quadratic FBSDE systems and application to stochastic differential games

In this note, we extend some recent results on systems of backward stochastic differential equations (BSDEs) with quadratic growth to the case of coupled forward-backward stochastic differential equations (FBSDEs). We work in a Markovian setting, and use results from the quadratic BSDE literature together with PDE techniques to obtain a-priori estimates which lead to an existence result. We also identify a general class of stochastic differential games whose corresponding FBSDE systems are covered by our main existence result. This leads to the existence of Markovian Nash equilibria for such games.

Interbank lending with benchmark rates: Pareto optima for a class of singular control games

AbstractWe 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.

Dynamic Contract Design for Systemic Cyber Risk Management of Interdependent Enterprise Networks

The interconnectivity of cyber and physical systems and Internet of things has created ubiquitous concerns of cyber threats for enterprise system managers. It is common that the asset owners and enterprise network operators need to work with cybersecurity professionals to manage the risk by remunerating them for their efforts that are not directly observable. In this paper, we use a principal-agent framework to capture the service relationships between the two parties, i.e., the asset owner (principal) and the cyber risk manager (agent). Specifically, we consider a dynamic systemic risk management problem with asymmetric information where the principal can only observe cyber risk outcomes of the enterprise network rather than directly the efforts that the manager expends on protecting the resources. Under this information pattern, the principal aims to minimize the systemic cyber risks by designing a dynamic contract that specifies the compensation flows and the anticipated efforts of the manager by taking into account his incentives and rational behaviors. We formulate a bi-level mechanism design problem for dynamic contract design within the framework of a class of stochastic differential games. We show that the principal has rational controllability of the systemic risk by designing an incentive compatible estimator of the agent's hidden efforts. We characterize the optimal solution by reformulating the problem as a stochastic optimal control program which can be solved using dynamic programming. We further investigate a benchmark scenario with complete information and identify conditions that yield zero information rent and lead to a new certainty equivalence principle for principal-agent problems. Finally, case studies over networked systems are carried out to illustrate the theoretical results obtained.

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.

Nash equilibrium approximation of some class of stochastic differential games: A combined Chebyshev spectral collocation method with policy iteration

This study relates to nonzero-sum stochastic games with perfect information. It proposes an efficient combined Chebyshev spectral collocation method (CSCM) with the policy iteration (PI) algorithm for solving nonlinear coupled Hamilton–Jacobi (HJ) equations. The proposed approach is comprised of two steps. First, the PI algorithm is used to reduce the nonlinear coupled HJ equations to a sequence of linear uncoupled PDEs. Then, these equations are approximated by the CSCM. The CSCM＋PI is especially useful when the CSCM fails due to the increasing number of collocation points for solving the associated system of nonlinear algebraic equations. The main advantage of the resulting method is that it converts nonlinear coupled HJ equations to the systems of linear algebraic equations, which can be solved readily. Convergence analysis of this method is also provided in detail. To confirm the accuracy and validity of the proposed computational algorithm, several illustrative examples are presented.

Certainty equivalence principle in stochastic differential games: An inverse problem approach

SummaryThis paper aims to characterize a class of stochastic differential games, which satisfy the certainty equivalence principle beyond the cases with quadratic, linear, or logarithmic value functions. We focus on scalar games with linear dynamics in the players' strategies and with separable payoff functionals. Our results are based on the resolution of an inverse problem that determines strictly concave utility functions of the players so that the game satisfies the certainty equivalence principle. Besides establishing necessary and sufficient conditions, the results obtained in this paper are also a tool for discovering new closed‐form solutions, as we show in two specific applications: in a generalization of a dynamic advertising model and in a game of noncooperative exploitation of a productive asset.

Stochastic Differential Games: A Sampling Approach via FBSDEs

The aim of this work is to present a sampling-based algorithm designed to solve various classes of stochastic differential games. The foundation of the proposed approach lies in the formulation of the game solution in terms of a decoupled pair of forward and backward stochastic differential equations (FBSDEs). In light of the nonlinear version of the Feynman–Kac lemma, probabilistic representations of solutions to the nonlinear Hamilton–Jacobi–Isaacs equations that arise for each class are obtained. These representations are in form of decoupled systems of FBSDEs, which may be solved numerically.

Stochastic Differential Games for Which the Open-Loop Equilibrium is Subgame Perfect

It is generally admitted that a correct forecasting of uncertain variables needs Markov decision rules. In a dynamic game environment, this belief is reinforced if one focuses on credible actions of the players. Usually, subgame perfectness requires equilibrium strategies to be constructed on Markov rules. It comes as a surprise that there are interesting classes of stochastic differential games where the equilibrium based on open-loop strategies is subgame perfect. This fact is well known for deterministic games. We explore here the stochastic case, not dealt with up to now, identifying different game structures leading to the subgame perfectness of the open-loop equilibrium.

Structures of Stochastic Differential Games for Which the Open-Loop Equilibrium Is Subgame Perfect

It is generally admitted that a correct forecasting of uncertain variables needs Markov decision rules. In a dynamic game environment, this belief is reinforced if one focus on credible actions of the players. Usually, subgame perfectness requires equilibrium strategies being constructed on Markov rules. It comes as a surprise that there are interesting classes of stochastic differential games where the equilibrium based on open loop strategies is subgame perfect. This fact is well known for deterministic games. We explore here the stochastic case, not dealt with up to now, identifying different game structures leading to the subgame perfectness of the open-loop equilibrium.

Certainty Equivalence Principle in Stochastic Differential Games: An Inverse Problem Approach

This paper aims to characterize a class of stochastic differential games which satisfy the certainty equivalence principle. This means that the Markov Perfect Nash Equilibrium is also an equilibrium of the associated deterministic game. By focusing on a model scalar game with linear dynamics in the players’ strategies, we solve an inverse problem to find strictly concave utility functions, not necessarily quadratic or logarithmic, for which certainty equivalence is fulfilled. We show a variety of models that satisfy the certainty equivalence principle in a non cooperative game of a productive asset.

Efficiency of the price formation process in presence of high frequency participants: a mean field game analysis

This paper deals with a stochastic order-driven market model with waiting costs, for orderbooks with heterogenous traders. Offer and demand of liquidity drives price formation and traders anticipate future evolutions of the orderbook. The natural framework we use is mean field game theory, a class of stochastic differential games with a continuum of anonymous players. Several sources of heterogeneity are considered including the mean size of orders. Thus we are able to consider the coexistence of Institutional Investors and high frequency traders (HFT). We provide both analytical solutions and numerical experiments. Implications on classical quantities are explored: orderbook size, prices, and effective bid/ask spread. According to the model, in markets with Institutional Investors only we show the existence of inefficient liquidity imbalances in equilibrium, with two symmetrical situations corresponding to what we call liquidity calls for liquidity. During these situations the transaction price significantly moves away from the fair price. However this macro phenomenon disappears in markets with both Institutional Investors and HFT, although a more precise study shows that the benefits of the new situation go to HFT only, leaving Institutional Investors even with higher trading costs.

The Derivation of Ergodic Mean Field Game Equations for Several Populations of Players

This note contains a detailed derivation of the equations of the recent mean field games theory (abbr. MFG), developed by M. Huang, P.E. Caines, and R.P. Malhame on one hand and by J.-M. Lasry and P.-L. Lions on the other, associated with a class of stochastic differential games, where the players belong to several populations, each of which consisting of a large number of similar and indistinguishable individuals, in the context of periodic diffusions and long-time-average (or ergodic) costs. After introducing a system of N Hamilton–Jacobi–Bellman (abbr. HJB) and N Kolmogorov–Fokker–Planck (abbr. KFP) equations for an N-player game belonging to such a class of games, the system of MFG equations (consisting of as many HJB equations, and of as many KFP equations as is the number of populations) is derived by letting the number of the members of each population go to infinity. For the sake of clarity and for reader’s convenience, the case of a single population of players, as formulated in the work of J.-M. Lasry and P.-L. Lions, is presented first. The note slightly improves the results in this case too, by dealing with more general dynamics and costs.

Probabilistic Analysis of Mean-Field Games

The purpose of this paper is to provide a complete probabilistic analysis of a large class of stochastic differential games with mean field interactions. We implement the Mean-Field Game strategy developed analytically by Lasry and Lions in a purely probabilistic framework, relying on tailor-made forms of the stochastic maximum principle. While we assume that the state dynamics are affine in the states and the controls, and the costs are convex, our assumptions on the nature of the dependence of all the coefficients upon the statistical distribution of the states of the individual players remains of a rather general nature. Our probabilistic approach calls for the solution of systems of forward-backward stochastic differential equations of a McKean--Vlasov type for which no existence result is known, and for which we prove existence and regularity of the corresponding value function. Finally, we prove that a solution of the Mean-Field Game problem as formulated by Lasry and Lions, does indeed provide approximate Nash equilibriums for games with a large number of players, and we quantify the nature of the approximation.

MEAN FIELD GAMES EQUATIONS WITH QUADRATIC HAMILTONIAN: A SPECIFIC APPROACH

Mean field games models describing the limit of a large class of stochastic differential games, as the number of players goes to +∞, have been introduced by J.-M. Lasry and P.-L. Lions in Refs. 10–12. We use a change of variables to transform the mean field games (MFG) equations into a system of simpler coupled partial differential equations, in the case of a quadratic Hamiltonian. This system is then used to exhibit a monotonic scheme to build solutions of the MFG equations. Effective numerical methods based on this constructive scheme are presented and numerical experiments are carried out.

Saddle points of discrete Markov zero-sum game with stopping

We study the sufficient conditions for the existence of a saddle point of a time-dependent discrete Markov zero-sum game up to a given stopping time. The stopping time is allowed to take either a finite or an infinite non-negative random variable with its associated objective function being well-defined. The result enables us to show the existence of the saddle points of discrete games constructed by Markov chain approximation of a class of stochastic differential games.

Nash and Stackelberg differential games

A large class of stochastic differential games for several players is considered in this paper. The class includes Nash differential games as well as Stackelberg differential games. A mix is possible. The existence of feedback strategies under general conditions is proved. The limitations concern the functionals in which the state and the controls appear separately. This is also true for the state equations. The controls appear in a quadratic form for the payoff and linearly in the state equation. The most serious restriction is the dimension of the state equation, which cannot exceed 2. The reason comes from PDE (partial differential equations) techniques used in studying the system of Bellman equations obtained by Dynamic Programming arguments. In the authors’ previous work in 2002, there is not such a restriction, but there are serious restrictions on the structure of the Hamiltonians, which are violated in the applications dealt with in this article.

Existence and compactness for weak solutions to Bellman systems with critical growth

We deal with nonlinear elliptic and parabolic systems that are the Bellman systems associated to stochastic differential games as a main motivation. We establish the existence of weak solutions in any dimension for an arbitrary number of equations (players). The method is based on using a renormalized sub- and super-solution technique . The main novelty consists in the new structure conditions on the critical growth terms with allow us to show weak solvability for Bellman systems to certain classes of stochastic differential games.

Nonzero Sum Stochastic Differential Games with Discounted Payoff Criterion: An Approximating Markov Chain Approach

We develop a new constructive method for proving the existence of Nash equilibrium for a class of nonzero sum stochastic differential games. Under certain usual assumptions, we prove the existence of Nash equilibrium for discounted payoff criteria. A novel feature of our method is that it allows us to compute Nash equilibrium for a large class of stochastic differential games.

Numerical Approximations for Stochastic Differential Games: The Ergodic Case

The Markov chain approximation method is a widely used, relatively easy to use, and efficient family of methods for the bulk of stochastic control problems in continuous time for reflected-jump-diffusion-type models. It has been shown to converge under broad conditions, and there are good algorithms for solving the numerical problems if the dimension is not too high. We consider a class of stochastic differential games with a reflected diffusion system model and ergodic cost criterion and where the controls for the two players are separated in the dynamics and cost function. It is shown that the value of the game exists and that the numerical method converges to this value as the discretization parameter goes to zero. The actual numerical method solves a stochastic game for a finite state Markov chain and ergodic cost criterion. The essential conditions are nondegeneracy and that a weak local consistency condition hold "almost everywhere" for the numerical approximations, just as for the control problem.

Smooth Solutions of systems of quasilinear parabolic equations

We consider in this article diagonal parabolic systems arising in the context of stochastic differential games. We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear Hamiltonian seems to be the adequate one to achieve the regularity property. A key step in the theory is to prove the existence of Hölder solution.