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 173
... Moreover , assume either that ẞ > 0 or that T < ∞ with probability 1 for every admissible progressively measurable control process u ( · ) . Then W ( x ) = J ( x ; u * ) = Vpm ( x ) . Minimum mean exit time from O. Let us apply ...
... Moreover , assume either that ẞ > 0 or that T < ∞ with probability 1 for every admissible progressively measurable control process u ( · ) . Then W ( x ) = J ( x ; u * ) = Vpm ( x ) . Minimum mean exit time from O. Let us apply ...
Page 200
... Moreover , V is a generalized subsolution of the HJB equation . Proof . The same kind of approximations used in the proof of Theorem 6.1 provide fn , on , Ln with the following properties : ( a ) fn , On , Ln converge uniformly on ...
... Moreover , V is a generalized subsolution of the HJB equation . Proof . The same kind of approximations used in the proof of Theorem 6.1 provide fn , on , Ln with the following properties : ( a ) fn , On , Ln converge uniformly on ...
Page 306
... Moreover , although ặ¤ is not continuous at { t1 } x 80 , ge is continuous and ge = 1 on { t1 } x 80. We now obtain ( 10.11 ) by using the maximum principle . Since pe≤ 1 , Vε > 0. Hence by ( 10.8 ) , Ve ( t , x ) is uniformly bounded ...
... Moreover , although ặ¤ is not continuous at { t1 } x 80 , ge is continuous and ge = 1 on { t1 } x 80. We now obtain ( 10.11 ) by using the maximum principle . Since pe≤ 1 , Vε > 0. Hence by ( 10.8 ) , Ve ( t , x ) is uniformly bounded ...
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