Controlled Markov Processes and Viscosity SolutionsThis book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. |
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Page 13
Wendell Helms Fleming, H. Mete Soner. In Theorem 5.1 , x * ( · ) denotes the solution to ( 3.2 ) with u ( · ) = u * ( · ) , x * ( t ) = x . Theorem 5.1 is called a Verification Theorem . Note that , by the definition ( 5.4 ) of H , ( 5.7 ) ...
Wendell Helms Fleming, H. Mete Soner. In Theorem 5.1 , x * ( · ) denotes the solution to ( 3.2 ) with u ( · ) = u * ( · ) , x * ( t ) = x . Theorem 5.1 is called a Verification Theorem . Note that , by the definition ( 5.4 ) of H , ( 5.7 ) ...
Page 16
... ( Theorem II.10.2 ) . Boundary conditions are discussed further in Section II ... 5.1 . In ( 5.8 ) the integral is now from t to the exit time 7 , and W ( 7 ... Theorem 5.1 . Remark 5.2 . An entirely similar Verification Theorem is true for ...
... ( Theorem II.10.2 ) . Boundary conditions are discussed further in Section II ... 5.1 . In ( 5.8 ) the integral is now from t to the exit time 7 , and W ( 7 ... Theorem 5.1 . Remark 5.2 . An entirely similar Verification Theorem is true for ...
Page 333
... ( 5.1 ) can be proved as in Section IV.7 when O = IR " and as in Section V.2 when O is bounded . In the latter case we need to assume I ( 3.11 ) and V ( 2.3 ) . Theorem 5.1 . Assume V Є Cp ( O ) and ( 5.1 ) . Then V is a viscosity ...
... ( 5.1 ) can be proved as in Section IV.7 when O = IR " and as in Section V.2 when O is bounded . In the latter case we need to assume I ( 3.11 ) and V ( 2.3 ) . Theorem 5.1 . Assume V Є Cp ( O ) and ( 5.1 ) . Then V is a viscosity ...
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Controlled Markov Processes and Viscosity Solutions Wendell H. Fleming,Halil Mete Soner Limited preview - 2006 |
Controlled Markov Processes and Viscosity Solutions Wendell H. Fleming,Halil Mete Soner No preview available - 2006 |
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admissible control assume assumptions boundary condition boundary data bounded c₁ Cą(Q calculus of variations Chapter classical solution consider constant continuous on Q convergence convex Corollary cylindrical region defined definition denote dynamic programming equation dynamic programming principle Dynkin formula Example exists exit finite first-order formulation Hamilton-Jacobi equation Hence HJB equation holds implies inequality initial data lateral boundary Lemma lim sup linear Lipschitz continuous Markov chain Markov control policy Markov processes maximum principle minimizing Moreover nonlinear obtain optimal control optimal control problem partial derivatives partial differential equation proof of Theorem prove result satisfies second-order Section semigroup stochastic differential equation Suppose t₁ test function Theorem 5.1 uniformly continuous unique value function variations problem Verification Theorem viscosity solution viscosity subsolution viscosity supersolution yields