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 66
... maximum principle . Hence ( 4.3ii ) is equivalent to the maximum principle property of F. We will now give the definition of a viscosity subsolution and a superso- lution of nonlinear partial differential equations . Definition 4.2 ...
... maximum principle . Hence ( 4.3ii ) is equivalent to the maximum principle property of F. We will now give the definition of a viscosity subsolution and a superso- lution of nonlinear partial differential equations . Definition 4.2 ...
Page 72
... maximum of W - w with W ( t , x ) = w ( t , x ) . In view of Lemma 6.1 , we may assume that ( t , x ) is a strict maximum . Since DC C1 , 2 ( Q ) , w € C1 , 2 ( Q ) . Therefore there is an open set Q * Q such that wЄ C1,2 ( Q * ) . As ...
... maximum of W - w with W ( t , x ) = w ( t , x ) . In view of Lemma 6.1 , we may assume that ( t , x ) is a strict maximum . Since DC C1 , 2 ( Q ) , w € C1 , 2 ( Q ) . Therefore there is an open set Q * Q such that wЄ C1,2 ( Q * ) . As ...
Page 234
... maximum principle with G = Q. This corollary is a generalization of the following version of the classical maximum principle : if W , V E C1,2 ( Q ) and W - V has an interior maximum ( t , x ) , then at ( t , x ) Wt = Vt , D2W = D2V and ...
... maximum principle with G = Q. This corollary is a generalization of the following version of the classical maximum principle : if W , V E C1,2 ( Q ) and W - V has an interior maximum ( t , x ) , then at ( t , x ) Wt = Vt , D2W = D2V and ...
Contents
Viscosity Solutions | 53 |
Controlled Markov Diffusions in R | 157 |
SecondOrder Case | 213 |
Copyright | |
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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 |
Common terms and phrases
admissible control assume assumptions boundary condition boundary data bounded c₁ C¹(Q calculus of variations Chapter classical solution consider constant convergence convex Corollary cylindrical region D₂V defined definition denote dynamic programming equation dynamic programming principle Dynkin formula Example exists exit finite first-order formulation given Hamilton-Jacobi equation Hence HJB equation holds implies inequality initial condition initial data lateral boundary Lebesgue left endpoint Lemma linear Lipschitz continuous Markov chain Markov control policy Markov processes maximum principle minimizing Moreover nonlinear obtain optimal control optimal control problems partial derivatives partial differential equation proof of Theorem prove R₁ reference probability system result satisfies second-order Section stochastic control stochastic differential equation Suppose t₁ Theorem 5.1 tion unique value function variations problem Verification Theorem viscosity solution viscosity subsolution viscosity supersolution yields