Stochastic Differential Game Problem Research Articles

Nash Equilibrium of Stochastic Partial Differential Game with Partial Information via Malliavin Calculus

In this article, we consider the Nash equilibrium of stochastic differential game where the state process is governed by a controlled stochastic partial differential equation and the information available to the controllers is possibly less than the general information. All the system coefficients and the objective performance functionals are assumed to be random. We find an explicit strong solution of the linear stochastic partial differential equation with a generalized probabilistic representation for this solution with the benefit of Kunita’s stochastic flow theory. We use Malliavin calculus to derive a stochastic maximum principle for the optimal control and obtain the Nash equilibrium of this type of stochastic differential game problem.

Malliavin Calculus and Its Application to Robust Optimal Investment for an Insider

In the theory of portfolio selection, there are few methods that effectively address the combined challenge of insider information and model uncertainty, despite numerous methods proposed for each individually. This paper studies the problem of the robust optimal investment for an insider under model uncertainty. To address this, we extend the Itô formula for forward integrals by Malliavin calculus, and use it to establish an implicit anticipating stochastic differential game model for the robust optimal investment. Since traditional stochastic control theory proves inadequate for solving anticipating control problems, we introduce a new approach. First, we employ the variational method to convert the original problem into a nonanticipative stochastic differential game problem. Then we use the stochastic maximum principle to derive the Hamiltonian system governing the robust optimal investment. In cases where the insider information filtration is of the initial enlargement type, we derive the closed-form expression for the investment by using the white noise theory when the insider is ’small’. When the insider is ’large’, we articulate a quadratic backward stochastic differential equation characterization of the investment. We present the numerical result and conduct an economic analysis of the optimal strategy across various scenarios.

Reinforcement learning for exploratory linear-quadratic two-person zero-sum stochastic differential games

In this paper, we study an entropy-regularized continuous-time linear-quadratic two-person zero-sum stochastic differential game problem from the perspective of reinforcement learning (RL). By the solvability of a discounted algebraic Riccati equation, we construct a Gaussian closed-loop optimal control pair for the problem, which achieves the best tradeoff between exploration and exploitation. Then, in this exploratory framework, we propose an RL algorithm that relies on only partial system information to solve a stochastic H∞ control problem. The corresponding convergence analysis and simulation examples are also provided to verify the efficiency of the proposed algorithm.

Pareto-based Stackelberg differential game for stochastic systems with multi-followers

This paper is concerned about a Pareto-based Stackelberg stochastic differential game with multi-followers, in which followers are cooperative relations. First of all, necessary and sufficient conditions for the existence of Pareto-based Stackelberg equilibrium are given for the general Pareto-based Stackelberg stochastic differential game problem. Next, we study the above game problem in the linear-quadratic (LQ) case and obtain a feedback form of Pareto-based Stackelberg equilibrium. The solvability of Riccati equations is guaranteed under some assumptions. Finally, the theoretical results are used to solve a national debt management problem.

A representation theorem approach to probabilistic interpretation for viscosity solutions of Isaacs equations

The main objective of this paper is to show a certain representation theorem for generators of generalized backward stochastic differential equations (GBSDEs) and its application to probabilistic interpretation for viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations with nonlinear Neumann boundary value problems. In the representation theorem a time change method is adopted to treat the difficulty brought from the random measure contained in GBSDEs. By means of the representation theorem the viscosity solution of Isaacs equations with nonlinear Neumann problems is proved to be the value function of a two-player zero-sum stochastic differential game problem with state constraints and recursive cost functionals.

Stochastic Recursive Zero-sum Differential Games under Model Uncertainty

In this paper, we study stochastic recursive zero-sum differential game problem where the payoff function is described by the solutions of a class of backward stochastic differential equations with uncertainty parameter , which are used to represent different market conditions. In such case, the existence of saddle points for stochastic differential game problems above-mentioned is proved.

Linear quadratic nonzero‐sum stochastic differential game of a partially observed Markov jump linear systems

AbstractIn this study, a partially observed linear quadratic (LQ) non‐cooperative stochastic differential game of Markov jump linear system (MJLS) is considered. First, a new proof of filtering theory is given, using the separation principle and filtering technology, partial observations can be transformed into complete observation. Second, nonzero‐sum stochastic differential game problem for MJLS are studied using the dynamic programming principle and completion square method. Then, the sufficient and necessary conditions for the Nash equilibrium are obtained. Finally, a numerical simulation is performed.

Risk-sensitive zero-sum stochastic differential game for jump–diffusions

This article is concerned with the infinite horizon risk-sensitive zero-sum stochastic differential game problem for a class of jump–diffusions controlled through the drift, and driven by a compensated Poisson process and a Wiener process. Under certain geometric stability assumption on the dynamics, we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. We obtain our result by exploiting the stochastic representation of the principal eigenfunction of the associated Hamilton–Jacobi–Isaac (HJI) equation, which is a semilinear integro-partial differential equation.

The Principle-Agent Conflict Problem in a Continuous-Time Delegated Asset Management Model

This paper considers the principle-agent conflict problem in a continuous-time delegated asset management model when the investor and the fund manager are all risk-averse with risk sensitivity coefficients γ f and γ m , respectively. Suppose that the investor entrusts his money to the fund manager. The return of the investment is determined by the manager’s effort level and incentive strategy, but the benefit belongs to the investor. In order to encourage the manager to work hard, the investor will determine the manager’s salary according to the terminal income. This is a stochastic differential game problem, and the distribution of income between the manager and the investor is a key point to be solved in the custody model. The uncertain form of the incentive strategy implies that the problem is different from the classical stochastic optimal control problem. In this paper, we first express the investor’s incentive strategy in term of two auxiliary processes and turn this problem into a classical one. Then, we employ the dynamic programming principle to solve the problem.

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.

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.

On a class of non-zero-sum stochastic differential dividend games with regime switching

This paper investigates a class of non-zero-sum stochastic differential game problems between two insurance companies. The surplus process of each company is modeled by a Brownian motion where drift and volatility depend on the continuous-time Markov regime switching process. Both companies have the option of paying dividends. The objective is to maximize the expected discount utility of surplus relative to a reference point for each insurer, the gains brought by the insurer’s own dividend payout, and the losses incurred by the dividend payment of the competitor. The gains and losses are proportional to the amount of corresponding dividend payment. To find the optimal dividend policy, we relate this singular control problem to a stopping game. Further, we prove the link based on the verification theorem and show that value functions of the non-zero-sum stochastic differential problem can be derived by integrating value functions of a stopping game. Finally, we apply our results to the case with two regimes and the case without regime switching. Numerical examples are also provided.

Robust optimal investment and reinsurance for an insurer with inside information

This paper studies a robust optimal investment–reinsurance problem for an insurer who possesses inside information on the financial market and the insurance business under model uncertainty. The insurer’s surplus process and the risky asset process in the financial market are assumed to be correlated jump diffusion processes with random coefficients. The inside information is modelled by a general random variable related to the future realizations of the surplus process and the risky asset process. Under the criterion of expected utility maximization, the optimal investment–reinsurance problem with inside information is transformed into an adapted stochastic control problem. By incorporating Knightian uncertainty into the model, we establish an adapted stochastic differential game problem with a nonstandard performance functional, which aims to select the robust optimal investment–reinsurance strategy for the insurer with inside information. The stochastic maximum principles are adopted to derive a characterization of the optimal strategy. Some interesting special cases are discussed and the corresponding optimal strategies are derived.

Partial Information Stochastic Differential Games for Backward Stochastic Systems Driven by Lévy Processes

In this paper, we consider a partial information two-person zero-sum stochastic differential game problem, where the system is governed by a backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. A sufficient condition and a necessary one for the existence of the saddle point for the game are proved. As an application, a linear quadratic stochastic differential game problem is discussed.

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.

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.

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.

Continuous-Time Portfolio Choice Under Monotone Mean-Variance Preferences—Stochastic Factor Case

We consider an incomplete market with a nontradable stochastic factor and a continuous-time investment problem with an optimality criterion based on monotone mean-variance preferences. We formulate it as a stochastic differential game problem and use Hamilton–Jacobi–Bellman–Isaacs equations to find an optimal investment strategy and the value function. What is more, we show that our solution is also optimal for the classical Markowitz problem, and every optimal solution for the classical Markowitz problem is optimal also for the monotone mean-variance preferences. These results are interesting because the original Markowitz functional is not monotone, and it was observed that in the case of a static one-period optimization problem, the solutions for those two functionals (monotone mean variance and classical Markowitz) are different. In addition, we determine explicit Markowitz strategies in the square root factor models.

A kind of stochastic recursive Zero-Sum differential game problem with double obstacles constraint

This paper concerns a class of stochastic recursive zero-sum differential game problem with recursive utility related to a backward stochastic differential equation (BSDE) with double obstacles. A sufficient condition is provided to obtain the saddle-point strategy under some assumptions. In virtue of the corresponding relationship of doubly reflected BSDE and mixed game problem, a stochastic linear recursive mixed differential game problem is studied to apply our theoretical result, and here the explicit saddle-point strategy as well as the saddle-point stopping time for the mixed game problem are obtained. Besides, a numeral example is also given to demonstrate the result by virtue of partial differential equations (PDEs) computation method.

Optimal Control and Zero-Sum Stochastic Differential Game Problems of Mean-Field Type

We establish existence of nearly-optimal controls, conditions for existence of an optimal control and a saddle-point for respectively a control problem and zero-sum differential game associated with payoff functionals of mean-field type, under dynamics driven by weak solutions of stochastic differential equations of mean-field type.