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 77
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... Definition ... 62 67 II.5 Dynamic programming and viscosity property 72 II.6 Properties of viscosity solutions . . . . 73 II.7 Deterministic optimal control and viscosity solutions 78 II.8 Viscosity solutions : first order case 83 II.9 ...
... Definition ... 62 67 II.5 Dynamic programming and viscosity property 72 II.6 Properties of viscosity solutions . . . . 73 II.7 Deterministic optimal control and viscosity solutions 78 II.8 Viscosity solutions : first order case 83 II.9 ...
Page 13
... definition ( 5.4 ) of H , ( 5.7 ) is equivalent to ( 5.7 ' ) u * ( s ) = arg min { ƒ ( s , x * ( s ) , v ) · DxW ( s , x * ( s ) ) + L ( s , x * ( s ) , v ) } . VEU Proof of Theorem 5.1 . Consider any u ( · ) € U ° ( t ) . Using ...
... definition ( 5.4 ) of H , ( 5.7 ) is equivalent to ( 5.7 ' ) u * ( s ) = arg min { ƒ ( s , x * ( s ) , v ) · DxW ( s , x * ( s ) ) + L ( s , x * ( s ) , v ) } . VEU Proof of Theorem 5.1 . Consider any u ( · ) € U ° ( t ) . Using ...
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... Definition . W is a generalized solution to the dynamic programming equation in Q if W is locally Lipschitz and satisfies ( 5.3 ) for almost all ( t , x ) EQ . Corollary 6.1 . Let U be compact and U ( t , x ) = U ° ( t ) . If V is ...
... Definition . W is a generalized solution to the dynamic programming equation in Q if W is locally Lipschitz and satisfies ( 5.3 ) for almost all ( t , x ) EQ . Corollary 6.1 . Let U be compact and U ( t , x ) = U ° ( t ) . If V is ...
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... definition of approximately continuous function , see [ EG ] or [ McS , p . 224 ] . ) Given v € U and 0 < 8 < t1 − s , let us ( r ) = u * ( r ) if r ¢ [ s , s + 8 ] ข if rЄ [ s , s +8 ] , and let xs ( r ) be the solution to I ...
... definition of approximately continuous function , see [ EG ] or [ McS , p . 224 ] . ) Given v € U and 0 < 8 < t1 − s , let us ( r ) = u * ( r ) if r ¢ [ s , s + 8 ] ข if rЄ [ s , s +8 ] , and let xs ( r ) be the solution to I ...
Page 26
... defined for all u ( · ) Є Ux . We 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 ...
... defined for all u ( · ) Є Ux . We 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 ...
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