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 51
Page 16
... boundary for which there exists a control u ( · ) such that J ( t , x ; u ... condition V ( t , x ) = 0 holds for all ( t , x ) = [ to , t1 ) × dO , if ... Boundary conditions are discussed further in Section II.13 . Theorem 5.2 . Let WE ...
... boundary for which there exists a control u ( · ) such that J ( t , x ; u ... condition V ( t , x ) = 0 holds for all ( t , x ) = [ to , t1 ) × dO , if ... Boundary conditions are discussed further in Section II.13 . Theorem 5.2 . Let WE ...
Page 17
... condition ( 5.7 ) if x * ( ) is a solution to the differential inclusion ... conditions ( t , x ) . In particular , suppose that u * is a feedback control policy ... boundary conditions . If L ( s , y , v ) + f ( s , y , v ) . DxW ( s , y ) ...
... condition ( 5.7 ) if x * ( ) is a solution to the differential inclusion ... conditions ( t , x ) . In particular , suppose that u * is a feedback control policy ... boundary conditions . If L ( s , y , v ) + f ( s , y , v ) . DxW ( s , y ) ...
Page 20
... conditions for the value function V to be Lipschitz on Q. See Theorem 9.3 , Theorem II.10.2 . A local Lipschitz condition ... boundary data can have infinitely many generalized solutions . See Example II.2.2 below . This difficulty is ...
... conditions for the value function V to be Lipschitz on Q. See Theorem 9.3 , Theorem II.10.2 . A local Lipschitz condition ... boundary data can have infinitely many generalized solutions . See Example II.2.2 below . This difficulty is ...
Page 23
... condition . 7 * Again let P ( s ) be as as in ( 6.4 ) . If * = t1 and x * ( t1 ) € O , then the transversality condition ... boundary is as follows : there exists a scalar such that ( 6.11a ) P ( 7 * ) = Dxg ( 7 * , x * ( 7 * ) ) + \ n ( x ...
... condition . 7 * Again let P ( s ) be as as in ( 6.4 ) . If * = t1 and x * ( t1 ) € O , then the transversality condition ... boundary is as follows : there exists a scalar such that ( 6.11a ) P ( 7 * ) = Dxg ( 7 * , x * ( 7 * ) ) + \ n ( x ...
Page 27
... condition will be imposed to exclude solutions to ( 7.10 ) which " grow too rapidly " as | x | → ∞ . See ( 7.14 ) ... boundary conditions ( 7.11 ) . As in the proof of Theorem 5.1 , using the state dynamics and ( 7.10 ) we calculate that ...
... condition will be imposed to exclude solutions to ( 7.10 ) which " grow too rapidly " as | x | → ∞ . See ( 7.14 ) ... boundary conditions ( 7.11 ) . As in the proof of Theorem 5.1 , using the state dynamics and ( 7.10 ) we calculate that ...
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