Stochastic differential games with reflection and related obstacle problems for Isaacs equations

In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1997), which turns out to be very useful because it allows us to estimate the L ^p -distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.     

Multi-dimensional backward stochastic differential equations with one reflecting lower barrier of Itô diffusion type

This paper investigates a class of multi-dimensional stochastic differential equations with one reflecting lower barrier (RBSDEs in short), where the random obstacle is described as an Itô diffusion type of stochastic differential equation. The existence and uniqueness results for adapted solutions to such RBSDEs are established based on a penalization scheme and some higher moment estimates for solutions to penalized BSDEs under the Lipschitz condition and a higher moment condition on the coefficients. Finally, two examples are given to illustrate our theory and their applications.     

Reflected backward stochastic differential equations with two barriers and Dynkin games under Knightian uncertainty

This paper is concerned with a class of reflected backward stochastic differential equations (RBSDEs in short) with two barriers. The first purpose of the paper is to establish existence and uniqueness results of adapted solutions for such RBSDEs. Most of existing results on adapted solutions for RBSDEs with two barriers are heavily based on either the Mokobodski condition or other restrictive regularity conditions. In this paper, the two barriers are modeled by stochastic differential equations with coefficients satisfying the local Lipschitz condition and the linear growth condition, which enables us to weaken the regularity conditions on the boundary processes. Existence is proved by a penalization scheme together with a comparison theorem under the Lipschitz condition on the coefficients of RBSDEs. As an application, it is proved that the initial value of an RBSDE with two barriers coincides with the value function of a certain Dynkin game under Knightian uncertainty.     

On the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains and the Isaacs equations

We prove the dynamic programming principle for uniformly nondegenerate stochastic differential games in the framework of time-homogeneous diffusion processes considered up to the first exit time from a domain. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. There is no assumption about solvability of the the Isaacs equation in any sense (classical or viscosity). The zeroth-order “coefficient” and the “free” term are only assumed to be measurable in the space variable. We also prove that value functions are uniquely determined by the functions defining the corresponding Isaacs equations and thus stochastic games with the same Isaacs equation have the same value functions.     

Characterization of Barriers of Differential Games

In pursuit-evasion games, when a barrier occurs, splitting the state space into capture and evasion areas, in order to characterize this manifold, the study of the minimum time function requires discontinuous generalized solutions of the Isaacs equation. Thanks to the minimal oriented distance from the target, we obtain a characterization by approximation with continuous functions. The barrier is characterized by the largest upper semicontinuous viscosity subsolution of a variational inequality. This result extends the Isaacs semipermeability property.     

Reflected Backward Stochastic Differential Equations,Convex Risk Measures and American Options

We use convex risk measures to assess unhedged risks for American-style contingent claims in a continuous-time non-Markovian economy using reflected backward stochastic differential equations (RBSDEs). A two-stage approach is adopted to evaluate the risk. We formulate the evaluation problem as an optimal stopping-control problem and discuss the problem using reflected BSDEs. The convex risk measures are represented as solutions of RBSDEs. In the Markov case, we relate the RBSDE solutions to the unique viscosity solutions of related obstacle problems for parabolic partial differential equations.     

����Ԫ���������������ķ��䵹�����΢�ַ�������Ԫ�ı�ʾ����

In this paper, we establish a local representation theorem for generators of reflected backward stochastic differential equations (RBSDEs), whose generators are continuous with linear growth. It generalizes some known representation theorems for generators of backward stochastic differential equations (BSDEs). As some applications, a general converse comparison theorem for RBSDEs is obtained and some properties of RBSDEs are discussed.     

关于反射倒向随机微分方程的解的一些性质(英文)

研究了关于反射倒向随机微分方程的解的一些性质.同时在适当的条件下建立了关于反射倒向随机微分方程生成元的一个唯一性定理和一个逆比较定理.     

Dynamic mean-variance problem with constrained risk control for the insurers

In this paper, we study optimal reinsurance/new business and investment (no-shorting) strategy for the mean-variance problem in two risk models: a classical risk model and a diffusion model. The problem is firstly reduced to a stochastic linear-quadratic (LQ) control problem with constraints. Then, the efficient frontiers and efficient strategies are derived explicitly by a verification theorem with the viscosity solutions of Hamilton–Jacobi–Bellman (HJB) equations, which is different from that given in Zhou et al. (SIAM J Control Optim 35:243–253, 1997). Furthermore, by comparisons, we find that they are identical under the two risk models. This work was supported by National Basic Research Program of China (973 Program) 2007CB814905 and National Natural Science Foundation of China (10571092).     

关于反射倒向随机微分方程的解的一些性质

研究了关于反射倒向随机微分方程的解的一些性质．同时在适当的条件下建立了关于反射倒向随机微分方程生成元的一个唯一性定理和一个逆比较定理．     

Equivalence between Nonlinear H ∞ Control Problems and Existence of Viscosity Solutions of Hamilton—Jacobi—Isaacs Equations rid="*" id="*" This work was written while the author was visiting the University of California at Santa Barbara. He was partially supported by Consiglio Nazionale delle Ricerche of Italy.

In this paper we extend to completely general nonlinear systems the result stating that the suboptimal control problem is solved if and only if the corresponding Hamilton—Jacobi—Isaacs (HJI) equation has a nonnegative (super)solution. This is well known for linear systems, using the Riccati equation instead of the HJI equation. We do this using the theory of differential games and viscosity solutions. Accepted 14 February 1997     

Risk minimizing portfolios and HJBI equations for stochastic differential games

In this paper, we consider the problem to find a market portfolio that minimizes the convex risk measure of the terminal wealth in a jump diffusion market. We formulate the problem as a two player (zero-sum) stochastic differential game. To help us find a solution, we prove a theorem giving the Hamilton–Jacobi–Bellman–Isaacs (HJBI) conditions for a general zero-sum stochastic differential game in a jump diffusion setting. We then use the theorem to study particular risk minimization problems. Finally, we extend our approach to cover general stochastic differential games (not necessarily zero-sum), and we obtain similar HJBI equations for the Nash equilibria of such games.     

Reflected and Doubly Reflected BSDEs for Levy Processes： Solutions and Comparison

In this paper we study reflected and doubly reflected backward stochastic differential equations （BSDEs, for short） driven by Teugels martingales associated with L~vy process satisfying some moment condi- tions and by an independent Brownian motion. For BSDEs with one reflecting barrier, we obtain a comparison theorem using the Tanaka-Meyer formula. For BSDEs with two reflecting barriers, we first prove the existence and uniqueness of the solutions under the Mokobodski＇s condition by using the Snell envelope theory and then we obtain a comparison result.     

Continuous dependence estimates for viscosity solutions of integro-PDEs

We present a general framework for deriving continuous dependence estimates for, possibly polynomially growing, viscosity solutions of fully nonlinear degenerate parabolic integro-PDEs. We use this framework to provide explicit estimates for the continuous dependence on the coefficients and the “Lévy measure” in the Bellman/Isaacs integro-PDEs arising in stochastic control/differential games. Moreover, these explicit estimates are used to prove regularity results and rates of convergence for some singular perturbation problems. Finally, we illustrate our results on some integro-PDEs arising when attempting to price European/American options in an incomplete stock market driven by a geometric Lévy process. Many of the results obtained herein are new even in the convex case where stochastic control theory provides an alternative to our pure PDE methods.     

Infinite horizon noncooperative differential games with nonsmooth costs

For a noncooperative differential game, the value functions of the various players satisfy a system of Hamilton-Jacobi equations. In the present paper, we study a class of infinite-horizon scalar games with either piecewise linear or piecewise smooth costs, exponentially discounted in time. By the analysis of the value functions, we find that results about existence and uniqueness of admissible solutions to the HJ system, and therefore of Nash equilibrium solutions in feedback form, can be recovered as in the smooth costs case, provided the costs are globally monotone. On the other hand, we present examples of costs such that the corresponding HJ system has infinitely many admissible solutions or no admissible solutions at all, suggesting that new concepts of equilibria may be needed to study games with general nonlinear costs.     

Stochastic representation for solutions of Isaacs’ type integral–partial differential equations

In this paper we study the integral–partial differential equations of Isaacs’ type by zero-sum two-player stochastic differential games (SDGs) with jump-diffusion. The results of Fleming and Souganidis (1989) [9] and those of Biswas (2009) [3] are extended, we investigate a controlled stochastic system with a Brownian motion and a Poisson random measure, and with nonlinear cost functionals defined by controlled backward stochastic differential equations (BSDEs). Furthermore, unlike the two papers cited above the admissible control processes of the two players are allowed to rely on all events from the past. This quite natural generalization permits the players to consider those earlier information, and it makes more convenient to get the dynamic programming principle (DPP). However, the cost functionals are not deterministic anymore and hence also the upper and the lower value functions become a priori random fields. We use a new method to prove that, indeed, the upper and the lower value functions are deterministic. On the other hand, thanks to BSDE methods (Peng, 1997) [18] we can directly prove a DPP for the upper and the lower value functions, and also that both these functions are the unique viscosity solutions of the upper and the lower integral–partial differential equations of Hamilton–Jacobi–Bellman–Isaacs’ type, respectively. Moreover, the existence of the value of the game is got in this more general setting under Isaacs’ condition.     

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

We develop a viscosity solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the case of stochastic optimal switching and control, we prove via dynamic programming methods that the value function is a viscosity solution of the IPDEs. In our setting the value functions or the solutions of the IPDEs are not smooth, so classical verification theorems do not apply.     

反射倒向随机微分方程的逆比较问题(Ⅰ)

在Briand,Coquet,Hu,Memin,Peng[1],Coquet,Hu,Memin,Peng[2],Chen[3],Jiang [8]等中,研究了倒向随机微分方程的逆比较定理,就是通过比较倒向随机微分方程的解来比较倒向随机微分方程的生成元问题．在文[9]中Li和Tang首次研究了反射倒向随机微分方程的逆比较问题．本文考虑在更一般的条件下,反射倒向随机微分方程的生成元的逆比较问题．     

Robust Optimal Stopping-Time Control for Nonlinear Systems

Abstract. We formulate a robust optimal stopping-time problem for a state-space system and give the connection between various notions of lower value function for the associated games (and storage function for the associated dissipative system) with solutions of the appropriate variational inequality (VI) (the analogue of the Hamilton—Jacobi—Bellman—Isaacs equation for this setting). We show that the stopping-time rule can be obtained by solving the VI in the viscosity sense and a positive definite supersolution of the VI can be used for stability analysis.     

Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games

In this paper we deal with the problem of existence of a smooth solution of the Hamilton–Jacobi–Bellman–Isaacs (HJBI for short) system of equations associated with nonzero-sum stochastic differential games. We consider the problem in unbounded domains either in the case of continuous generators or for discontinuous ones. In each case we show the existence of a smooth solution of the system. As a consequence, we show that the game has smooth Nash payoffs which are given by means of the solution of the HJBI system and the stochastic process which governs the dynamic of the controlled system.