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 36
... smooth ( for example , g Є C4 ( Qo ) ) , then g ( T , x ( T ) ) = g ( t , x ) + [ * [ 9 : ( 8 , x ( 8 ) ) + Dz9 ( 3 , x ( 3 ) ) · Ż ( 8 ) ] ds by the Fundamental Theorem of Calculus . We then rewrite J in ( 8.11 ) as ( 8.11 ' ) J = Ĵ + ...
... smooth ( for example , g Є C4 ( Qo ) ) , then g ( T , x ( T ) ) = g ( t , x ) + [ * [ 9 : ( 8 , x ( 8 ) ) + Dz9 ( 3 , x ( 3 ) ) · Ż ( 8 ) ] ds by the Fundamental Theorem of Calculus . We then rewrite J in ( 8.11 ) as ( 8.11 ' ) J = Ĵ + ...
Page 61
... smooth . " So we assume that there exist Σ ' CΣ , DC C ( [ to , t1 ) × Σ ' ) and a one - parameter family of nonlinear operators { G } telto , t ] of functions of Σ , satisfying the following conditions with Q = [ to , t1 ) x Σ ...
... smooth . " So we assume that there exist Σ ' CΣ , DC C ( [ to , t1 ) × Σ ' ) and a one - parameter family of nonlinear operators { G } telto , t ] of functions of Σ , satisfying the following conditions with Q = [ to , t1 ) x Σ ...
Page 384
... smooth subsolution of the HJB equation with the boundary conditions ( 5.1 ) if ( 5.16 ) -9t + H ( x , Dzg , D2g ) ≤ 0 , ( t , x ) EQ , g ( t , x ) ≤ ¥ ( t , x ) , ( t , x ) € 8 * Q . A smooth subsolution is of course also a viscosity ...
... smooth subsolution of the HJB equation with the boundary conditions ( 5.1 ) if ( 5.16 ) -9t + H ( x , Dzg , D2g ) ≤ 0 , ( t , x ) EQ , g ( t , x ) ≤ ¥ ( t , x ) , ( t , x ) € 8 * Q . A smooth subsolution is of course also a viscosity ...
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