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 82
Page 13
... Theorem 5.1 is called a Verification Theorem . Note that , by the 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 ...
... Theorem 5.1 is called a Verification Theorem . Note that , by the 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 ...
Page 15
... Theorem 5.1 to show that V ( t , x ) tmin < tt1 , and to find an explicit formula for the optimal u * ( s ) . In ... proof of a Verification Theorem ( Theorem 5.2 ) similar to Theorem 5.1 . When t = t1 , we have as in ( 5.5 ) : ( 5.18 ) ...
... Theorem 5.1 to show that V ( t , x ) tmin < tt1 , and to find an explicit formula for the optimal u * ( s ) . In ... proof of a Verification Theorem ( Theorem 5.2 ) similar to Theorem 5.1 . When t = t1 , we have as in ( 5.5 ) : ( 5.18 ) ...
Page 16
... proof of Theorem 5.2 is almost the same as for Theorem 5.1 . In ( 5.8 ) the integral is now from t to the exit time 7 , and W ( 7 , x ( 7 ) ) is on the left side . By ( 5.18 ) and ( 5.19 ) , W ( t , x ( 7 ) ) ≤ ( 7 , x ( 7 ) ) with V ...
... proof of Theorem 5.2 is almost the same as for Theorem 5.1 . In ( 5.8 ) the integral is now from t to the exit time 7 , and W ( 7 , x ( 7 ) ) is on the left side . By ( 5.18 ) and ( 5.19 ) , W ( t , x ( 7 ) ) ≤ ( 7 , x ( 7 ) ) with V ...
Page 19
... proof of Theorem 6.1 ' . Another proof of this result is also given by using the theory of viscosity solutions ( See Corollary II.8.1 . ) . By the dynamic programming principle ( 4.3 ) , we have ( see ( 5.1 ) ) for small h > 0 inf 1 t + ...
... proof of Theorem 6.1 ' . Another proof of this result is also given by using the theory of viscosity solutions ( See Corollary II.8.1 . ) . By the dynamic programming principle ( 4.3 ) , we have ( see ( 5.1 ) ) for small h > 0 inf 1 t + ...
Page 20
... prove two theorems which give sufficient conditions for the value function V to be Lipschitz on Q. See Theorem 9.3 , Theorem II.10.2 . A local Lipschitz condition for V follows from the estimates in Section IV.8 . Unfortunately , when ...
... prove two theorems which give sufficient conditions for the value function V to be Lipschitz on Q. See Theorem 9.3 , Theorem II.10.2 . A local Lipschitz condition for V follows from the estimates in Section IV.8 . Unfortunately , when ...
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