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. |
From inside the book
Results 1-5 of 76
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... remarks 117 III Optimal Control of Markov Processes : Classical Solutions119 III.1 Introduction . 119 III.2 Markov processes and their evolution operators . 120 III.3 Autonomous ( time - homogeneous ) Markov processes 123 III.4 Classes ...
... remarks 117 III Optimal Control of Markov Processes : Classical Solutions119 III.1 Introduction . 119 III.2 Markov processes and their evolution operators . 120 III.3 Autonomous ( time - homogeneous ) Markov processes 123 III.4 Classes ...
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... remarks Singular Perturbations VII.1 Introduction VII.2 Examples ... VII.3 Barles and Perthame procedure . VII.4 Discontinuous viscosity solutions . VII.5 Terminal condition . . ix . 227 227 228 230 235 238 239 245 250 255 . 259 . 261 ...
... remarks Singular Perturbations VII.1 Introduction VII.2 Examples ... VII.3 Barles and Perthame procedure . VII.4 Discontinuous viscosity solutions . VII.5 Terminal condition . . ix . 227 227 228 230 235 238 239 245 250 255 . 259 . 261 ...
Page 13
... Remark 5.1 . Condition ( 5.7 ) is necessary as well as sufficient . Indeed , from the proof of Theorem 5.1 and the definition ( 5.4 ) of H it is immediate that ( 5.7 ) holds for almost all s if u * ( ) is optimal . We illustrate the use ...
... Remark 5.1 . Condition ( 5.7 ) is necessary as well as sufficient . Indeed , from the proof of Theorem 5.1 and the definition ( 5.4 ) of H it is immediate that ( 5.7 ) holds for almost all s if u * ( ) is optimal . We illustrate the use ...
Page 16
... Remark 5.2 . An entirely similar Verification Theorem is true for the problem of control until the time 7 ' of exit from Q ( rather from Q. ) In fact , since ( s , x ( s ) ) Є Q for t < s < 7 ' , the proof of Theorem 5.2 shows that it ...
... Remark 5.2 . An entirely similar Verification Theorem is true for the problem of control until the time 7 ' of exit from Q ( rather from Q. ) In fact , since ( s , x ( s ) ) Є Q for t < s < 7 ' , the proof of Theorem 5.2 shows that it ...
Page 28
... Remark 7.1 . A slight generalization of Theorem 7.1 can be proved in which ( 7.14 ) is replaced by assumptions like those in Theorem III.9.1 and Theorem IV.5.1 . Those results concern stochastic control problems . In the deterministic ...
... Remark 7.1 . A slight generalization of Theorem 7.1 can be proved in which ( 7.14 ) is replaced by assumptions like those in Theorem III.9.1 and Theorem IV.5.1 . Those results concern stochastic control problems . In the deterministic ...
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