Controlled Markov Processes and Viscosity SolutionsThis book is an introduction to optimal stochastic control for continuous time Markov processes and the theory of viscosity solutions. It covers dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. New chapters in this second edition introduce the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets and two-controller, zero-sum differential games. |
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Results 1-5 of 32
Page 11
... corollary of the above computations is the following . Corollary 4.1 . An admissible control u ( · ) € U ( t , x ) is 8 - optimal at ( t , x ) if any only if it is 8 - optimal at every ( r , x ( r ) ) with r Є [ t , T ] . I.5 Dynamic ...
... corollary of the above computations is the following . Corollary 4.1 . An admissible control u ( · ) € U ( t , x ) is 8 - optimal at ( t , x ) if any only if it is 8 - optimal at every ( r , x ( r ) ) with r Є [ t , T ] . I.5 Dynamic ...
Page 16
... Corollary 5.1 . A control u * ( ) satisfies. v EU We may now restate ( 5.7 ' ) as u * ( s ) € v * ( s , x * ( s ) ) . Substituting this into the state dynamics yields ( 5.22 ) x * ( s ) = F * ( s , x * ( s ) ) , 8 € [ t , T * ] . Thus ...
... Corollary 5.1 . A control u * ( ) satisfies. v EU We may now restate ( 5.7 ' ) as u * ( s ) € v * ( s , x * ( s ) ) . Substituting this into the state dynamics yields ( 5.22 ) x * ( s ) = F * ( s , x * ( s ) ) , 8 € [ t , T * ] . Thus ...
Page 17
Wendell H. Fleming, Halil Mete Soner. Corollary 5.1 . A control u * ( ) satisfies the optimality condition ( 5.7 ) if x * ( ) is a solution to the differential inclusion ( 5.22 ) . Feedback controls ( Markov control policies ) . Corollary ...
Wendell H. Fleming, Halil Mete Soner. Corollary 5.1 . A control u * ( ) satisfies the optimality condition ( 5.7 ) if x * ( ) is a solution to the differential inclusion ( 5.22 ) . Feedback controls ( Markov control policies ) . Corollary ...
Page 19
... Corollary II.8.1 . ) . By the dynamic programming principle ( 4.3 ) , we have ( see ( 5.1 ) ) for small h > 0 inf 1 t + h 1 L ( s , x ( s ) , u ( s ) ) ds + h — [ V ( t + h , x ( t + h ) ) − V ( t , x ) ] } = 0 . u ( · ) ЄU ° ( t ) h ...
... Corollary II.8.1 . ) . By the dynamic programming principle ( 4.3 ) , we have ( see ( 5.1 ) ) for small h > 0 inf 1 t + h 1 L ( s , x ( s ) , u ( s ) ) ds + h — [ V ( t + h , x ( t + h ) ) − V ( t , x ) ] } = 0 . u ( · ) ЄU ° ( t ) h ...
Page 20
... Corollary 6.1 . Let U be compact and U ( t , x ) = U ° ( t ) . If V is locally Lipschitz on Q , then V is a generalized solution of the dynamic programming equation ( 5.3 ) . Later we will prove two theorems which give sufficient ...
... Corollary 6.1 . Let U be compact and U ( t , x ) = U ° ( t ) . If V is locally Lipschitz on Q , then V is a generalized solution of the dynamic programming equation ( 5.3 ) . Later we will prove two theorems which give sufficient ...
Contents
1 | |
Viscosity Solutions | 57 |
Classical Solutions119 | 118 |
Controlled Markov Diffusions in R | 151 |
SecondOrder Case | 199 |
Logarithmic Transformations and Risk Sensitivity | 227 |
Singular Perturbations 261 | 260 |
Singular Stochastic Control | 293 |
Finite Difference Numerical Approximations | 321 |
Differential Games | 375 |
A Duality Relationships 397 | 396 |
References | 409 |
Index 425 | 424 |
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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 C¹(Q calculus of variations Chapter classical solution consider constant constraint controlled Markov diffusion convergence convex Corollary defined definition denote differential games dynamic programming equation dynamic programming principle Dynkin formula Example exists exit finite formulation given Hence HJB equation holds implies inequality initial data Ishii Lemma linear Lipschitz continuous Markov chain Markov control policy Markov processes minimizing Moreover nonlinear obtain optimal control optimal control problem partial derivatives partial differential equation progressively measurable proof of Theorem prove reference probability system Remark result risk sensitive satisfies Section semigroup Soner stochastic control stochastic control problem stochastic differential equations subset Suppose t₁ Theorem 9.1 uniformly continuous unique value function Verification Theorem viscosity solution viscosity subsolution viscosity supersolution