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 84
Page 16
... exists a control u ( · ) such that J ( t , x ; u ( · ) ) < g ( t , x ) . See Example II.2.3 . At such points , strict inequality holds in ( 5.19 ) . If ( 3.10 ) is assumed , in addition to ( 3.11 ) , then V ( t , x ) > 0. Since g = 0 ...
... exists a control u ( · ) such that J ( t , x ; u ( · ) ) < g ( t , x ) . See Example II.2.3 . At such points , strict inequality holds in ( 5.19 ) . If ( 3.10 ) is assumed , in addition to ( 3.11 ) , then V ( t , x ) > 0. Since g = 0 ...
Page 17
... exists . We shall not undertake to deal with it here , but will only indicate some of the difficulties . In order to implement the pro- cedure just outlined above , one needs first a smooth solution W to the dy- namic programming ...
... exists . We shall not undertake to deal with it here , but will only indicate some of the difficulties . In order to implement the pro- cedure just outlined above , one needs first a smooth solution W to the dy- namic programming ...
Page 18
... exists an optimal control u * ( · ) € U ( t , x ) such that u * ( s ) tends to a limit v * as s ↓ t , then Vt ( t , x ) + L ( t , x , v * ) + f ( t , x , v * ) · DxV ( t , x ) = 0 . Hence the dynamic programming equation ( 5.3 ) holds ...
... exists an optimal control u * ( · ) € U ( t , x ) such that u * ( s ) tends to a limit v * as s ↓ t , then Vt ( t , x ) + L ( t , x , v * ) + f ( t , x , v * ) · DxV ( t , x ) = 0 . Hence the dynamic programming equation ( 5.3 ) holds ...
Page 19
... exists . In Section 9 we will show that there is a continuous optimal control , for a special class of control problems of calculus of variations type . For further results about existence and continuity properties of optimal controls ...
... exists . In Section 9 we will show that there is a continuous optimal control , for a special class of control problems of calculus of variations type . For further results about existence and continuity properties of optimal controls ...
Page 26
... exists v ( § ) € U such that ƒ ( § , v ( § ) ) · n ( § ) > 0 . Here n ( ) is the exterior unit normal at § . The value function V is defined by ( 7.8 ) V ( x ) = inf J ( x ; u ) , xɛŌ . Их We will only consider problems in which V ( x ) ...
... exists v ( § ) € U such that ƒ ( § , v ( § ) ) · n ( § ) > 0 . Here n ( ) is the exterior unit normal at § . The value function V is defined by ( 7.8 ) V ( x ) = inf J ( x ; u ) , xɛŌ . Их We will only consider problems in which V ( x ) ...
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