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 168
... derivatives 9t , 9 , 9xx , are continuous on Q × U , i , j = 1 , ... , n ; ( d ) ve C3 ( Qo ) . Theorem 4.1 . If assumptions ( 3.5 ) , ( 4.1 ) hold , then ( 3.3 ) – ( 3.4 ) has a unique solution W € C1 , 2 ( Q ) ΜC ( Q ) . This theorem ...
... derivatives 9t , 9 , 9xx , are continuous on Q × U , i , j = 1 , ... , n ; ( d ) ve C3 ( Qo ) . Theorem 4.1 . If assumptions ( 3.5 ) , ( 4.1 ) hold , then ( 3.3 ) – ( 3.4 ) has a unique solution W € C1 , 2 ( Q ) ΜC ( Q ) . This theorem ...
Page 198
... derivatives of V are defined as follows [ Zi ] . Suppose that there exists Vi Є Loc ( Qo ) such that ( 10.1 ) V1Þdxdt = - Jao Vox , dxdt for all Є Co ° ( Qo ) . Then ; is called a generalized first - order partial derivative of V with ...
... derivatives of V are defined as follows [ Zi ] . Suppose that there exists Vi Є Loc ( Qo ) such that ( 10.1 ) V1Þdxdt = - Jao Vox , dxdt for all Є Co ° ( Qo ) . Then ; is called a generalized first - order partial derivative of V with ...
Page 203
... derivative Var if and only if V1 ( t ,. ) is absolutely continuous on each finite subinterval of R1 . Similarly , for n > 1 the semiconcavity of V ( t ,. ) implies that the Schwartz distribution second derivatives V. , are measures ...
... derivative Var if and only if V1 ( t ,. ) is absolutely continuous on each finite subinterval of R1 . Similarly , for n > 1 the semiconcavity of V ( t ,. ) implies that the Schwartz distribution second derivatives V. , are measures ...
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