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 6-10 of 49
Page 6
... formulation of an optimal control problem , we must specify for each initial data ( t , x ) a set U ( t , x ) CU ° ( t ) of admissible controls and a payoff functional J ( t , x ; u ) to be minimized . Let us first for- mulate some ...
... formulation of an optimal control problem , we must specify for each initial data ( t , x ) a set U ( t , x ) CU ° ( t ) of admissible controls and a payoff functional J ( t , x ; u ) to be minimized . Let us first for- mulate some ...
Page 7
... formulation . Let us now formulate a general class of control problems , which includes each of the classes A through D above . Let OCR be open , with either : ( i ) O IR " , or ( ii ) do a compact manifold of class C2 . Let Q = [ to ...
... formulation . Let us now formulate a general class of control problems , which includes each of the classes A through D above . Let OCR be open , with either : ( i ) O IR " , or ( ii ) do a compact manifold of class C2 . Let Q = [ to ...
Page 15
... formulated in Example 2.3 , the matrices M ( s ) and D are nonnegative definite and N ( s ) is positive definite . This implies that P ( t ) is nonnegative definite and that tmin = -∞ . To see this , for tmin < t < t1 , 0 ≤ V ( t , x ) ...
... formulated in Example 2.3 , the matrices M ( s ) and D are nonnegative definite and N ( s ) is positive definite . This implies that P ( t ) is nonnegative definite and that tmin = -∞ . To see this , for tmin < t < t1 , 0 ≤ V ( t , x ) ...
Page 20
... formulated a " maximum principle " which provides a general set of necessary conditions for an extremum in an optimal control problem . A statement and proof of Pontryagin's principle in its full generality is rather lengthy , and will ...
... formulated a " maximum principle " which provides a general set of necessary conditions for an extremum in an optimal control problem . A statement and proof of Pontryagin's principle in its full generality is rather lengthy , and will ...
Page 26
... formulation in Section 3. In particular , we do not consider the infinite horizon problem with a state constraint x ( s ) = O for all s ≥ 0. However such problems can be analyzed exactly as below . Equation ( 7.2 ) suggests that ( 7.9 ) ...
... formulation in Section 3. In particular , we do not consider the infinite horizon problem with a state constraint x ( s ) = O for all s ≥ 0. However such problems can be analyzed exactly as below . Equation ( 7.2 ) suggests that ( 7.9 ) ...
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