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 6-10 of 85
Page 20
... assume that the partial derivatives fx , La . , 9x , Va , exist and are continuous for i = 1 , ... · · , n . As above , let u * ( · ) denote an optimal control and x * ( · ) the corresponding solution to ( 3.2 ) with x * ( t ) = x . Let ...
... assume that the partial derivatives fx , La . , 9x , Va , exist and are continuous for i = 1 , ... · · , n . As above , let u * ( · ) denote an optimal control and x * ( · ) the corresponding solution to ( 3.2 ) with x * ( t ) = x . Let ...
Page 25
... assume that all the given data are time inde- pendent . Thus L , ğ and ƒ , are independent of t . With an abuse of notation we use L ( x , v ) and g ( x ) to denote L ( x , v ) and g ( x ) respectively . Then ( 7.1 ) becomes • T ( 7.1 ...
... assume that all the given data are time inde- pendent . Thus L , ğ and ƒ , are independent of t . With an abuse of notation we use L ( x , v ) and g ( x ) to denote L ( x , v ) and g ( x ) respectively . Then ( 7.1 ) becomes • T ( 7.1 ...
Page 26
... assume that ( 7.6 ) Ux is nonempty for all x Є Ō . We also assume the analogue of ( 3.11 ) : ( 7.7 ) For every do there exists v ( § ) € U such that ƒ ( § , v ( § ) ) · n ( § ) > 0 . Here n ( ) is the exterior unit normal at § . The ...
... assume that ( 7.6 ) Ux is nonempty for all x Є Ō . We also assume the analogue of ( 3.11 ) : ( 7.7 ) For every do there exists v ( § ) € U such that ƒ ( § , v ( § ) ) · n ( § ) > 0 . Here n ( ) is the exterior unit normal at § . The ...
Page 30
... assume that We C1 ( O ) n C ( O ) rather than W = C1 ( O ) . See Remark 5.2 for the corresponding finite time horizon problem . Remark 7.2 will be used in the next example , in which O is the interval ( 0 , ∞ ) . Example 7.3 . This ...
... assume that We C1 ( O ) n C ( O ) rather than W = C1 ( O ) . See Remark 5.2 for the corresponding finite time horizon problem . Remark 7.2 will be used in the next example , in which O is the interval ( 0 , ∞ ) . Example 7.3 . This ...
Page 32
... assume that the running cost is given by h ( x ) = α | x1 | + | x2 | , x = ( x1 , x2 ) IR2 with some a > 0. Let U be as in ( 2.3 ) with c1d1 + c2d2 < 1. Then the production planning is to minimize the total discounted holding and ...
... assume that the running cost is given by h ( x ) = α | x1 | + | x2 | , x = ( x1 , x2 ) IR2 with some a > 0. Let U be as in ( 2.3 ) with c1d1 + c2d2 < 1. Then the production planning is to minimize the total discounted holding and ...
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 |
Other editions - View all
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