Two-player Zero-sum Stochastic Differential Game Research Articles

Optimal consumption, investment and life insurance selection under robust utilities

We study the problem faced by a wage earner with an uncertain lifetime who has access to a Black–Scholes-type financial market consisting of one risk-free security and one risky asset. His preferences relative to consumption, investment and life insurance purchase are described by a robust expected utility. We rewrite this problem in terms of a two-player zero-sum stochastic differential game and we derive the wage earner optimal strategies for a general class of utility functions, studying the case of discounted constant relative risk aversion utility functions with more detail.

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.

Gradient Hölder regularity for parabolic normalized p(x,t)-Laplace equation

We consider the interior Hölder regularity of spatial gradient of viscosity solution to the parabolic normalized p ( x , t ) -Laplace equation u t = ( δ i j + ( p ( x , t ) − 2 ) u i u j | D u | 2 ) u i j with some suitable assumptions on p ( x , t ) , which arises naturally from a two-player zero-sum stochastic differential game with probabilities depending on space and time.

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 BSDE Approach to Stochastic Differential Games with Regime Switching

In this paper, we study a two-player zero-sum stochastic differential game with regime switching in the framework of forward-backward stochastic differential equations on a finite time horizon. By means of backward stochastic differential equation methods, in particular that of the notion from stochastic backward semigroups, we prove a dynamic programming principle for both the upper and the lower value functions of the game. Based on the dynamic programming principle, the upper and the lower value functions are shown to be the unique viscosity solutions of the associated upper and lower Hamilton–Jacobi–Bellman–Isaacs equations.

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.

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.

Two-player zero-sum stochastic differential games with random horizon

AbstractWe consider a two-player zero-sum stochastic differential game with a random planning horizon and diffusive state variable dynamics. The random planning horizon is a function of a non-negative continuous random variable, which is assumed to be independent of the Brownian motion driving the state variable dynamics. We study this game using a combination of dynamic programming and viscosity solution techniques. Under some mild assumptions, we prove that the value of the game exists and is the unique viscosity solution of a certain nonlinear partial differential equation of Hamilton–Jacobi–Bellman–Isaacs type.

Zero-Sum Stochastic Differential Game in Finite Horizon Involving Impulse Controls

This paper considers the problem of two-player zero-sum stochastic differential game with both players adopting impulse controls in finite horizon under rather weak assumptions on the cost functions ($c$ and $\chi$ not decreasing in time). We use the dynamic programming principle and viscosity solutions approach to show existence and uniqueness of a solution for the Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equation (PDE) of the game. We prove that the upper and lower value functions coincide.

Continuous-Time Markov Games with Asymmetric Information

We study a two-player zero-sum stochastic differential game with asymmetric information where the payoff depends on a controlled continuous-time Markov chain X with finite state space which is only observed by player 1. This model was already studied in Cardaliaguet et al (Math Oper Res 41(1):49–71, 2016) through an approximating sequence of discrete-time games. Our first contribution is the proof of the existence of the value in the continuous-time model based on duality techniques. This value is shown to be the unique solution of the same Hamilton–Jacobi equation with convexity constraints which characterized the limit value obtained in Cardaliaguet et al. (2016). Our second main contribution is to provide a simpler equivalent formulation for this Hamilton–Jacobi equation using directional derivatives and exposed points, which we think is interesting for its own sake as the associated comparison principle has a very simple proof which avoids all the technical machinery of viscosity solutions.

The Existence of Game Value for Path-dependent Stochastic Differential Game

In this paper we study a two-player zero-sum stochastic differential game for a path-dependent stochastic system under a recursive path-dependent cost functional. Due to the typical non-Markovian structure, the game value is a random field. Dividing the time horizontal into small intervals, we approximate the path-dependent game by a series of state-dependent games. We utilize the state-dependent viscosity solution theory to prove that under Isaacs' condition the game value exists. In our model, coefficients of diffusion of the system contain control and strategy, and could be degenerate The dimension of the state space is high. The existence of approximating Nash equilibrium ($\varepsilon$-saddle point) is given under the formula about nonanticipative strategy with delay.

Stochastic Differential Games Involving Impulse Controls and Double-Obstacle Quasi-variational Inequalities

We study a two-player zero-sum stochastic differential game with both players adopting impulse controls, on a finite time horizon. The Hamilton--Jacobi--Bellman--Isaacs (HJBI) partial differential equation (PDE) of the game turns out to be a double-obstacle quasi-variational inequality; therefore the two obstacles are implicitly given. We prove that the upper and lower value functions coincide; indeed we show, by means of the dynamic programming principle for the stochastic differential game, that they are the unique viscosity solution to the HJBI equation, therefore proving that the game admits a value.

A Probabilistic-Numerical Approximation for an Obstacle Problem Arising in Game Theory

We investigate a two-player zero-sum stochastic differential game in which one of the players has more information on the game than his opponent. We show how to construct numerical schemes for the value function of this game, which is given by the solution of a quasilinear partial differential equation with obstacle.

A BSDE approach to stochastic differential games with incomplete information

We consider a two-player zero-sum stochastic differential game in which one of the players has a private information on the game. Both players observe each other, so that the non-informed player can try to guess his missing information. Our aim is to quantify the amount of information the informed player has to reveal in order to play optimally: to do so, we show that the value function of this zero-sum game can be rewritten as a minimization problem over some martingale measures with a payoff given by the solution of a backward stochastic differential equation.

A stochastic differential game for the inhomogeneous ∞-Laplace equation

Given a bounded $\mathcaligr{C}^2$ domain $G\subset{\mathbb{R}}^m$, functions $g\in\mathcaligr{C}(\partial G,{\mathbb{R}})$ and $h\in\mathcaligr {C}(\bar{G},{\mathbb{R}}\setminus\{0\})$, let $u$ denote the unique viscosity solution to the equation $-2\Delta_{\infty}u=h$ in $G$ with boundary data $g$. We provide a representation for $u$ as the value of a two-player zero-sum stochastic differential game.

ROBUST DYNAMIC PRICING OVER INFINITE HORIZON IN THE PRESENCE OF MODEL UNCERTAINTY

This paper studies a problem of dynamic pricing faced by a retailer with limited inventory, uncertain about the demand rate model, aiming to maximize expected discounted revenue over an infinite time horizon. The retailer doubts his demand model which is generated by historical data and views it as an approximation. Uncertainty in the demand rate model is represented by a notion of generalized relative entropy process, and the robust pricing problem is formulated as a two-player zero-sum stochastic differential game. The pricing policy is obtained through the Hamilton-Jacobi-Isaacs (HJI) equation. The existence and uniqueness of the solution of the HJI equation is shown and a verification theorem is proved to show that the solution of the HJI equation is indeed the value function of the pricing problem. The results are illustrated by an example with exponential nominal demand rate.

Stochastic Differential Games with Asymmetric Information

We investigate a two-player zero-sum stochastic differential game in which the players have an asymmetric information on the random payoff. We prove that the game has a value and characterize this value in terms of dual viscosity solutions of some second order Hamilton-Jacobi equation.