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A strong comparison result for the Bellman equation arising in stochastic exit time control problems and its applications. (English) Zbl 0919.35009
For the fully nonlinear Hamilton-Jacobi-Bellman equation with Dirichlet boundary conditions arising in stochastic optimal control with exit time is considered. The main result concerns a “Strong Comparison Result” which allows to compare discontinuous viscosity sub- and supersolutions of the mentioned HJB equation. To prove that result some rather complicated conditions for the control problems near the boundary are required. The authors discuss difficulties connected with Dirichlet boundary conditions and show how these conditions can be treated. Some possible extentions of the results of the paper are pointed out.

MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B45 A priori estimates in context of PDEs
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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