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 199
... compact set BC Qo . Lemma 10.1 . Let Yn Є Loc ( Qo ) be such that , for every compact set BC Qo , Yn ( t , x ) | ≤ KB for all ( t , x ) E B and n = 1 , 2 , ... ... ... .. Then : nk ( a ) There is a subsequence In which tends to a limit ...
... compact set BC Qo . Lemma 10.1 . Let Yn Є Loc ( Qo ) be such that , for every compact set BC Qo , Yn ( t , x ) | ≤ KB for all ( t , x ) E B and n = 1 , 2 , ... ... ... .. Then : nk ( a ) There is a subsequence In which tends to a limit ...
Page 200
... compact set B C Qo , Lemma 10.1 implies that AV tends weakly in Loc ( Qo ) to a limit y " . Since y satisfies ( 10.5 ) ... compact sets to f , o , L as n → ∞o ; ( b ) the partial derivatives of fn , on with respect to t , xi , xixj , i ...
... compact set B C Qo , Lemma 10.1 implies that AV tends weakly in Loc ( Qo ) to a limit y " . Since y satisfies ( 10.5 ) ... compact sets to f , o , L as n → ∞o ; ( b ) the partial derivatives of fn , on with respect to t , xi , xixj , i ...
Page 202
... compact set BC Qo since Aon = An – fn · D2 and AnVn , fn · DxVn are uniformly bounded on compact sets . The proof is then the same as for Lemma 10.3 . ᄆ Definition . Let W be locally Lipschitz on Qo . We call W a generalized solution ...
... compact set BC Qo since Aon = An – fn · D2 and AnVn , fn · DxVn are uniformly bounded on compact sets . The proof is then the same as for Lemma 10.3 . ᄆ Definition . Let W be locally Lipschitz on Qo . We call W a generalized solution ...
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 |
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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