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 82
Page 5
... implies that , given any control u ( · ) , the differ- ential equation ( 3.2 ) d x ( s ) = f ( s , x ( s ) , u ( s ) ) , t ≤ s ≤ ti ds with initial condition ( 3.3 ) x ( t ) I. Deterministic Optimal Control 5 Finite time horizon problems.
... implies that , given any control u ( · ) , the differ- ential equation ( 3.2 ) d x ( s ) = f ( s , x ( s ) , u ( s ) ) , t ≤ s ≤ ti ds with initial condition ( 3.3 ) x ( t ) I. Deterministic Optimal Control 5 Finite time horizon problems.
Page 8
... implies , in particular , that an admissible control always stays admissible . Indeed , simply take in ( 3.7 ) r = 7 and ũç ( · ) = u ( · ) . The control problem is as follows : given initial data ( t , x ) Q , find u * ( · ) € U ( t ...
... implies , in particular , that an admissible control always stays admissible . Indeed , simply take in ( 3.7 ) r = 7 and ũç ( · ) = u ( · ) . The control problem is as follows : given initial data ( t , x ) Q , find u * ( · ) € U ( t ...
Page 15
... implies that u * ( ) is optimal at ( t , x ) . Notice that the optimal control u * ( s ) in ( 5.17 ) is a linear function of the state x * ( s ) . The matrix N - 1 ( s ) B ' ( s ) P ( s ) can be precomputed by solving the Riccati ...
... implies that u * ( ) is optimal at ( t , x ) . Notice that the optimal control u * ( s ) in ( 5.17 ) is a linear function of the state x * ( s ) . The matrix N - 1 ( s ) B ' ( s ) P ( s ) can be precomputed by solving the Riccati ...
Page 16
... implies that the lateral boundary condition V ( t , x ) = 0 holds for all ( t , x ) = [ to , t1 ) × dO , if both ( 3.10 ) and ( 3.11 ) hold . Note that we have not yet proved that the value function V is continuous on Q. However , such ...
... implies that the lateral boundary condition V ( t , x ) = 0 holds for all ( t , x ) = [ to , t1 ) × dO , if both ( 3.10 ) and ( 3.11 ) hold . Note that we have not yet proved that the value function V is continuous on Q. However , such ...
Page 23
... implies that D & V - Dag at ( 7 * , x * ( * ) ) is a scalar multiple A of the exterior unit normal n ( x * ( * ) ) . Thus , P ( * ) - Dxg ( T * , x * ( 7 * ) ) = Dx ( V − 9 ) ( T * , x * ( T * ) ) = λn ( x * ( 7 * ) ) , which is ( 6.11 ...
... implies that D & V - Dag at ( 7 * , x * ( * ) ) is a scalar multiple A of the exterior unit normal n ( x * ( * ) ) . Thus , P ( * ) - Dxg ( T * , x * ( 7 * ) ) = Dx ( V − 9 ) ( T * , x * ( T * ) ) = λn ( x * ( 7 * ) ) , which is ( 6.11 ...
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