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 198
... derivatives of V are defined as follows [ Zi ] . Suppose that there exists i E Loc ( Qo ) such that ( 10.1 ) E Voz.drdt for all € Co ° ( Qo ) . V1odxdt = - LV42 , Qo Then ; is called a generalized first - order partial derivative of V ...
... derivatives of V are defined as follows [ Zi ] . Suppose that there exists i E Loc ( Qo ) such that ( 10.1 ) E Voz.drdt for all € Co ° ( Qo ) . V1odxdt = - LV42 , Qo Then ; is called a generalized first - order partial derivative of V ...
Page 199
... derivatives Vxx , in ( 10.3 ) exist . We say that a sequence In converges to weakly * in Lo ( Qo ) if loc ( 10.6 ) lim Yn Pdxdt = Vodxdt ∞18 for every Є L1 ( Qo ) with support in some compact set B C Qo . Lemma 10.1 . Let Yn E Loc ( Qo ) ...
... derivatives Vxx , in ( 10.3 ) exist . We say that a sequence In converges to weakly * in Lo ( Qo ) if loc ( 10.6 ) lim Yn Pdxdt = Vodxdt ∞18 for every Є L1 ( Qo ) with support in some compact set B C Qo . Lemma 10.1 . Let Yn E Loc ( Qo ) ...
Page 203
... derivative Var if and only if Va ( 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 Va ( 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 ...
Contents
Viscosity Solutions | 53 |
Controlled Markov Diffusions in R | 157 |
SecondOrder Case | 213 |
Copyright | |
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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 convergence convex Corollary cylindrical region D₂V defined definition denote dynamic programming equation dynamic programming principle Dynkin formula Example exists exit finite first-order formulation given Hamilton-Jacobi equation Hence HJB equation holds implies inequality initial condition initial data lateral boundary Lebesgue left endpoint Lemma linear Lipschitz continuous Markov chain Markov control policy Markov processes maximum principle minimizing Moreover nonlinear obtain optimal control optimal control problems partial derivatives partial differential equation proof of Theorem prove R₁ reference probability system result satisfies second-order Section stochastic control stochastic differential equation Suppose t₁ Theorem 5.1 tion unique value function variations problem Verification Theorem viscosity solution viscosity subsolution viscosity supersolution yields