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 157
... depend on the theory of parabolic partial differential equations , and are quoted without proof in Section 4 . In Sections 6-10 the assumption of uniform parabolicity is abandoned . Hence , the value function need not be a classical ...
... depend on the theory of parabolic partial differential equations , and are quoted without proof in Section 4 . In Sections 6-10 the assumption of uniform parabolicity is abandoned . Hence , the value function need not be a classical ...
Page 196
... depend only on C1 , C2 , ... as stated in Lemma 9.1 . From Lemma 9.1 , let us obtain a one sided bound for A2V : ( 9.7 ) A2V ( t , x ) ≤ M3 ( 1 + | x | 2 + | x | TM ) . To obtain ( 9.7 ) , given 8 > 0 choose u ( · ) such that Then 8h2 ...
... depend only on C1 , C2 , ... as stated in Lemma 9.1 . From Lemma 9.1 , let us obtain a one sided bound for A2V : ( 9.7 ) A2V ( t , x ) ≤ M3 ( 1 + | x | 2 + | x | TM ) . To obtain ( 9.7 ) , given 8 > 0 choose u ( · ) such that Then 8h2 ...
Page 208
... depend on p and R ) . Moreover , F , L ,, and the eigenvalues of L ,, ( t , x , v ) are bounded below by > 0 , for ( x , v ) E B × UR . Then ( 3.12ii ) follows from Lemma 11.1 , and ( 3.12iii ) is automatic since UR is compact . = Our ...
... depend on p and R ) . Moreover , F , L ,, and the eigenvalues of L ,, ( t , x , v ) are bounded below by > 0 , for ( x , v ) E B × UR . Then ( 3.12ii ) follows from Lemma 11.1 , and ( 3.12iii ) is automatic since UR is compact . = Our ...
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