Maximum principle for partial information non-zero sum stochastic differential games with mixed delays

In this paper, an open-loop two-person non-zero sum stochastic differential game is investigated for fully coupled forward-backward stochastic system driven by a Brownian motion and a Poisson random measure.A variational formula for the cost functionals is obtained directly in terms of the Hamiltonian and the associated adjoint system which is a linear FBSDEs and neither the variational systems nor the corresponding Taylor type expansions of the state process and the cost functional will be considered. As an application, one necessary condition (a stochastic maximum principle) and one sufficient condition (a verification theorem) for the existence of open-loop Nash equilibrium points are proved by the variation formula obtained in a unified way. The control domain need to be convex and the admissible controls for both players are allowed to appear in both the drift and diffusion of the state equations.

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.

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.

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.

A nonzero sum stochastic differential game is considered, with two payoffs, where i ≠ j, w (t) is one dimensional Brownian motion and τ1, τ2 are stopping times≤T. The existence of a saddle point follows from the existence of a “strong” solution to a system of two quasi-variational inequalities. The proof of existence of a “weak” solution is outlined.KeywordsVariational InequalitySaddle PointFree BoundaryParabolic SystemFree Boundary ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper is concerned with a maximum principle for a new class of non-zero sum stochastic differential games. Compared with the existing literature, the gamesystems in this paper are forward-backward systems in which the control variables consist of two components: the continuous controls and the impulse controls.Necessary optimality conditions and sufficient optimality conditions in the form of maximum principle are obtained respectively for open-loop Nash equilibriumpoint of the foregoing games. A fund management problem is used to shed light on the application of the theoretical results, and the optimal investmentportfolio and optimal impulse consumption strategy are obtained explicitly.

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

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.

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.

This technical note is concerned with a maximum principle for a new class of non-zero sum stochastic differential games. The most distinguishing feature, compared with the existing literature, is that the game systems are described by backward stochastic differential equations (BSDEs). This kind of games are motivated by some interesting phenomena arising from financial markets and can be used to characterize the players with different levels of utilities. We establish a necessary condition and a sufficient condition in the form of maximum principle for open-loop equilibrium point of the foregoing games respectively. To explain the theoretical results, we use them to study a financial problem.

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.

This chapter offers a crash course on the theory of nonzero sum stochastic differential games. Its goal is to introduce the jargon and the notation of this class of game models. As the main focus of the text is the probabilistic approach to the solution of stochastic games, we review the strategy based on the stochastic Pontryagin maximum principle and show how BSDEs and FBSDEs can be brought to bear in the search for Nash equilibria. We emphasize the differences between open loop and closed loop or Markov equilibria and we illustrate the results of the chapter with detailed analyses of some of the models introduced in Chapter 1

In this article, we study a class of partially observed non-zero sum stochastic differential game based on forward and backward stochastic differential equations (FBSDEs). It is required that each player has his own observation equation, and the corresponding Nash equilibrium control is required to be adapted to the filtration generated by the observation process. To find the Nash equilibrium point, we establish the maximum principle as a necessary condition and derive the verification theorem as a sufficient condition. Applying the theoretical results and stochastic filtering theory, we obtain the explicit investment strategy of a partial information financial problem.

We study a stochastic differential game problem between two insurers, who invest in a financial market and adopt reinsurance to manage their claim risks. Supposing that their reinsurance premium rates are calculated according to the generalized mean-variance principle, we consider the competition between the two insurers as a non-zero sum stochastic differential game. Using dynamic programming technique, we derive a system of coupled Hamilton–Jacobi–Bellman equations and show the existence of equilibrium strategies. For an exponential utility maximizing game and a probability maximizing game, we obtain semi-explicit solutions for the equilibrium strategies and the equilibrium value functions, respectively. Finally, we provide some detailed comparative-static analyses on the equilibrium strategies and illustrate some economic insights.

Demand for longevity securities under relative performance concerns: Stochastic differential games with cointegration