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 68
... solution of ( 2.1 ) . However , we claim that for k > 1 W is not a viscosity supersolution of ( 2.1 ) , and consequently not a viscosity solution of ( 2.1 ) . Indeed , let ... Viscosity Solutions 5 Dynamic programming and viscosity property.
... solution of ( 2.1 ) . However , we claim that for k > 1 W is not a viscosity supersolution of ( 2.1 ) , and consequently not a viscosity solution of ( 2.1 ) . Indeed , let ... Viscosity Solutions 5 Dynamic programming and viscosity property.
Page 69
... viscosity solution of ( 3.12 ) in Q. Proof . Let w E D and ( t , x ) EQ be a maximizer of the difference Vw on Q satisfying V ( t , x ) w ( t , x ) . Then , w > V. Using ( 3.2b ) with = = w ( r ,. ) and s = t1 , we obtain , for every r ...
... viscosity solution of ( 3.12 ) in Q. Proof . Let w E D and ( t , x ) EQ be a maximizer of the difference Vw on Q satisfying V ( t , x ) w ( t , x ) . Then , w > V. Using ( 3.2b ) with = = w ( r ,. ) and s = t1 , we obtain , for every r ...
Page 107
... solution of ( 8.1 ) in Q and ( 9.3a ) if it is both a viscosity subsolution and a viscosity supersolution of ( 8.1 ) in Q and ( 9.3a ) . Notice that ( 13.1 ) implies ( 13.4 ) . Also we have formally argued that if ... Viscosity Solutions 107.
... solution of ( 8.1 ) in Q and ( 9.3a ) if it is both a viscosity subsolution and a viscosity supersolution of ( 8.1 ) in Q and ( 9.3a ) . Notice that ( 13.1 ) implies ( 13.4 ) . Also we have formally argued that if ... Viscosity Solutions 107.
Other editions - View all
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 |
Common terms and phrases
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