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. |
From inside the book
Results 1-3 of 53
Page 65
... maximum principle : we say that Gt obeys the maximum principle if for every te [ to , til , and , in the domain of Gt , we have ( Gtø ) ( T ) ≤ ( Gt ¥ ) ( X ) , whenever I Є arg min { ( ø – ¥ ) ( x ) | x € Σ } ~ Σ ′ with p ( x ) = ( x ) ...
... maximum principle : we say that Gt obeys the maximum principle if for every te [ to , til , and , in the domain of Gt , we have ( Gtø ) ( T ) ≤ ( Gt ¥ ) ( X ) , whenever I Є arg min { ( ø – ¥ ) ( x ) | x € Σ } ~ Σ ′ with p ( x ) = ( x ) ...
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 ...
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 C¹(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