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On the Hamilton-Jacobi-Bellman equations. (English) Zbl 0594.93069
The author considers optimal stochastic control problems and the associated Hamilton-Jacobi-Bellman equations. The heuristic argument showing that the minimal cost function satisfies the H-J-B equation is given. Then the author shows that the minimal cost function, u, is the maximum element of the set of all sub-solutions v satisfying: \(A_{\alpha}v\leq f_{\alpha}\) in \(R^ n\), for all \(\alpha\in A\) \((A_{\alpha}\) elliptic operator). Continuity and regularity results are obtained for u, which shows u satisfies the H-J-B in a \(W^{2,\infty}\)- sense. Viscosity solutions for general second order PDE are defined.
Reviewer: S.M.Lenhart

MSC:
93E20 Optimal stochastic control
35J65 Nonlinear boundary value problems for linear elliptic equations
60J60 Diffusion processes
35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
49L20 Dynamic programming in optimal control and differential games
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