Viscosity Solutions of Second-Order Equations, Stochastic Control and Stochastic Differential Games
Viscosity Solutions of Second-Order Equations, Stochastic Control and Stochastic Differential Games
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二阶方程粘度解、随机控制和随机微分博弈
DOI：
10.1007/978-1-4613-8762-6_19
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发表时间：
1988
期刊：
影响因子：
--
通讯作者：
P. Souganidis
中科院分区：
文献类型：
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作者：
P. Lions;P. Souganidis
在这篇文章中，我们回顾、解释和详细说明了关于一般最优随机控制和随机微分对策的各种值函数与相关的Hamilton-Jacobi-Bellman HJB和Bellman-Isaacs BI方程的粘度解之间可能的关系的一些最新结果。众所周知，这些方程的推导是启发式的，只有当值函数足够光滑时才有理由（W.H. Fleming和R. Richel[15]）。另一方面，方程是完全非线性的，二阶的，椭圆的，但可能是退化的。一般情况下，光滑解不存在，而非光滑解（如确定性情况下的Lipschitz连续解）是非唯一的。(举一些简单的例子，我们参考p - l。狮子[24])。对于一阶Hamilton-Jacobi方程，要克服这些典型的困难以及与之相关的数值逼近、渐近问题等。M.G.克兰德尔和p.l。Lions[8]引入了粘度解的概念，并证明了一般唯一性结果。M.G.克兰德尔、L.C.埃文斯和p.l l对这一概念的几个等价公式进行了系统的探索，并对典型的唯一性结果进行了简单易读的解释。狮子[6]。在p - l中也观察到。确定控制问题的Bellman方程的经典推导可以很容易地适用于得出以下一般事实：确定控制问题的值函数总是相关Hamilton-Jacobi-Bellman方程的粘度解。粘度解的唯一性和上述事实意味着值函数的完整表征。这一观察结果随后被E.N. Barron、L.C. Evans和r.j ensen （bb2）、P.E. Souganidis （bb1）和L.C. Evans和P.E. Souganidis （bb0）扩展到不同的游戏中。
In this note we review, explain and detail some recent results concerning the possible relations between various value functions of general optimal stochastic control and stochastic differential games and the viscosity solutions of the associated Hamilton-Jacobi-Bellman HJB and Bellman-Isaacs BI equations. It is well-known that the derivation of these equations is heuristic and it is justified only when the value functions are smooth enough (W.H. Fleming and R. Richel [15]). On the other hand, the equations are fully nonlinear, second-order, elliptic but possibly degenerate. Smooth solutions do not exist in general and nonsmooth solutions (like Lipschitz continuous solutions in the deterministic case) are highly nonunique. (For some simple examples we refer to P.-L. Lions [24]). As far as the first-order Hamilton-Jacobi equations are concerned, to overcome these typical difficulties and related ones like numerical approximations, asymptotic problems etc. M.G. Crandall and P.-L. Lions [8] introduced the notion of viscosity solutions and proved general uniqueness results. A systematic exploration of several equivalent formulations of this notion and an easy and readable account of the typical uniqueness results may be found in M.G. Crandall, L.C. Evans and P.-L. Lions [6]. It was also observed in P.-L. Lions [24] that the classical derivation of the Bellman equation for deterministic control problems can be easily adapted to yield the following general fact: Value functions of deterministic control problems are always viscosity solutions of the associated Hamilton-Jacobi-Bellman equations. The uniqueness of viscosity solutions and the above fact imply then a complete characterization of the value functions. This observation was then extended to differential games by E.N. Barron, L.C. Evans and R. Jensen [3], P.E. Souganidis [36] and L.C. Evans and P.E. Souganidis [14].