Viscosity Solutions of SecondOrder Equations, Stochastic Control and Stochastic Differential Games
 P.L. Lions,
 P. E. Souganidis
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Abstract
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 HamiltonJacobiBellman HJB and BellmanIsaacs BI equations. It is wellknown 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, secondorder, 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 firstorder HamiltonJacobi 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 HamiltonJacobiBellman 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].
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 Title
 Viscosity Solutions of SecondOrder Equations, Stochastic Control and Stochastic Differential Games
 Book Title
 Stochastic Differential Systems, Stochastic Control Theory and Applications
 Pages
 pp 293309
 Copyright
 1988
 DOI
 10.1007/9781461387626_19
 Print ISBN
 9781461387640
 Online ISBN
 9781461387626
 Series Title
 The IMA Volumes in Mathematics and Its Applications
 Series Volume
 10
 Series ISSN
 09406573
 Publisher
 Springer New York
 Copyright Holder
 SpringerVerlag New York
 Additional Links
 Topics
 Industry Sectors
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 Editors

 Wendell Fleming ^{(1)}
 PierreLouis Lions ^{(2)}
 Editor Affiliations

 1. Division of Applied Mathematics, Brown University
 2. Ceremade, Universite ParisDauphine
 Authors

 P.L. Lions ^{(3)}
 P. E. Souganidis ^{(4)}
 Author Affiliations

 3. Ceremade, Universite ParisDauphine, Place de Lattre de Tassigny, 75775, Paris Cedex 16, France
 4. Lefschetz Center for Dynamical Systems Division of Applied Mathematics, Brown University, Providence, Rhode Islands, 02912, USA
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