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 118
... constant K3 such that ( 16.11 ) 0 ≤ V ( t , x ) ≤ K3 distance ( x , 0 ) , ( t , x ) EQ . Indeed fix ( t , x ) EQ and choose z Є 00 satisfying | xz | = distance ( x , d0 ) . - u Let 8 > 0 be as in ( 10.6 ) . Set = 8 ( x - x ) / | z ...
... constant K3 such that ( 16.11 ) 0 ≤ V ( t , x ) ≤ K3 distance ( x , 0 ) , ( t , x ) EQ . Indeed fix ( t , x ) EQ and choose z Є 00 satisfying | xz | = distance ( x , d0 ) . - u Let 8 > 0 be as in ( 10.6 ) . Set = 8 ( x - x ) / | z ...
Page 155
... constant con- trols . If t1t Mh , M = 1 , 2 , ... , let Vh ( t , x ) = Th_tv ( x ) . - Then Vh ( t , x ) turns out to be the value function obtained by requiring that u ( s ) is constant on each interval [ t + mh , t + ( m + 1 ) h ] , m ...
... constant con- trols . If t1t Mh , M = 1 , 2 , ... , let Vh ( t , x ) = Th_tv ( x ) . - Then Vh ( t , x ) turns out to be the value function obtained by requiring that u ( s ) is constant on each interval [ t + mh , t + ( m + 1 ) h ] , m ...
Page 312
... constant K , depending on the Lipschitz constant of b . Using the Gronwall inequality and the fact that § ( t ) = 0 , we have E ( s ) ≤ eK ( s − t ) [ \ u ( z ) P2 dz , 8 € ( [ t , 7 ] . The condition ( 10.23 ) implies that for x E O ...
... constant K , depending on the Lipschitz constant of b . Using the Gronwall inequality and the fact that § ( t ) = 0 , we have E ( s ) ≤ eK ( s − t ) [ \ u ( z ) P2 dz , 8 € ( [ t , 7 ] . The condition ( 10.23 ) implies that for x E O ...
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