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 92
Page 68
... solution of ( 2.1 ) . However , we claim that for k > 1 Wk is not a viscosity supersolution of ( 2.1 ) , and consequently not a viscosity solution of ( 2.1 ) . Indeed , let ... Viscosity Solutions 5 Dynamic programming and viscosity property.
... solution of ( 2.1 ) . However , we claim that for k > 1 Wk is not a viscosity supersolution of ( 2.1 ) , and consequently not a viscosity solution of ( 2.1 ) . Indeed , let ... Viscosity Solutions 5 Dynamic programming and viscosity property.
Page 69
... viscosity solution of ( 3.12 ) in Q. Proof . Let wЄ D and ( t , x ) EQ be a maximizer of the difference V - w on Q satisfying V ( t , x ) = w ( t , x ) . Then , w ≥ V. Using ( 3.2b ) with $ = w ( r ,. ) and s = t1 , we obtain , for ...
... viscosity solution of ( 3.12 ) in Q. Proof . Let wЄ D and ( t , x ) EQ be a maximizer of the difference V - w on Q satisfying V ( t , x ) = w ( t , x ) . Then , w ≥ V. Using ( 3.2b ) with $ = w ( r ,. ) and s = t1 , we obtain , for ...
Page 107
... solution of ( 8.1 ) in Q and ( 9.3a ) if it is both a viscosity subsolution and a viscosity supersolution of ( 8.1 ) in Q and ( 9.3a ) . Notice that ( 13.1 ) implies ( 13.4 ) . Also we have formally argued that if ... Viscosity Solutions 107.
... solution of ( 8.1 ) in Q and ( 9.3a ) if it is both a viscosity subsolution and a viscosity supersolution of ( 8.1 ) in Q and ( 9.3a ) . Notice that ( 13.1 ) implies ( 13.4 ) . Also we have formally argued that if ... Viscosity Solutions 107.
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 brownian motion C₁ Cą(Q calculus of variations Chapter classical solution consider continuous on Q controlled Markov diffusion convergence convex Corollary D₂V defined definition denote deterministic dynamic programming equation dynamic programming principle Dynkin formula Example exists exit finite formula Hence HJB equation holds implies inequality initial data lateral boundary Lebesgue Lemma lim sup linear Lipschitz continuous Markov chain Markov control policy Markov processes maximum principle minimizing Moreover nonlinear obtain optimal control optimal Markov control partial derivatives partial differential equation proof of Theorem prove reference probability system Remark result satisfies second-order Section semiconvex stochastic control stochastic differential equation Suppose t₁ Theorem 5.1 uniformly continuous unique value function Verification Theorem viscosity solution viscosity subsolution viscosity supersolution yields