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 51
Page
... condition . . ix . 227 227 228 230 235 238 239 245 250 255 . 259 . 261 261 . 263 . 265 266 . 269 VII.6 Boundary condition VII.7 Convergence VII.8 Comparison 271 . 272 273 VII.9 Vanishing viscosity VII.10 Large deviations for exit ...
... condition . . ix . 227 227 228 230 235 238 239 245 250 255 . 259 . 261 261 . 263 . 265 266 . 269 VII.6 Boundary condition VII.7 Convergence VII.8 Comparison 271 . 272 273 VII.9 Vanishing viscosity VII.10 Large deviations for exit ...
Page 6
... condition , but this dependence is suppressed in our notation . Let U ° ( t ) denote the set of all controls u ... boundary and terminal boundary , respectively , of Q. Given initial data ( t , x ) Є Q , let 7 denote the exit time of ( s ...
... condition , but this dependence is suppressed in our notation . Let U ° ( t ) denote the set of all controls u ... boundary and terminal boundary , respectively , of Q. Given initial data ( t , x ) Є Q , let 7 denote the exit time of ( s ...
Page 7
... boundary cost function , and is assumed continuous . B ' . Control until exit from Q. Let ( t , x ) E Q , and let 7 ... condition that U ( t , x ) is nonempty is called a reachability condition . See Sontag [ Sg ] . If U = IR TM , it is ...
... boundary cost function , and is assumed continuous . B ' . Control until exit from Q. Let ( t , x ) E Q , and let 7 ... condition that U ( t , x ) is nonempty is called a reachability condition . See Sontag [ Sg ] . If U = IR TM , it is ...
Page 9
... conditions ( t , x ) . Consider the minimum value of the payoff function as a function of this initial point . Thus define a ... boundary cost ( see ( 3.5 ) ) . Lemma 4.1 . For every initial condition ( t , x ) Є Q , admissible control u ...
... conditions ( t , x ) . Consider the minimum value of the payoff function as a function of this initial point . Thus define a ... boundary cost ( see ( 3.5 ) ) . Lemma 4.1 . For every initial condition ( t , x ) Є Q , admissible control u ...
Page 15
... condition x * ( t ) = x . Thus there is a unique control u ... boundary conditions for the dynamic pro- gramming equation ( 5.3 ) . Then we ... boundary [ to , t1 ) × 20. This implies that , for ( t , x ) E [ to , t1 ) x 00 , one choice ...
... condition x * ( t ) = x . Thus there is a unique control u ... boundary conditions for the dynamic pro- gramming equation ( 5.3 ) . Then we ... boundary [ to , t1 ) × 20. This implies that , for ( t , x ) E [ to , t1 ) x 00 , one choice ...
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