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 81
Page 19
... Section 9 we will show that there is a continuous optimal control , for a special class of control problems of calculus of variations type . For further results about existence and continuity properties of optimal controls see [ FR ...
... Section 9 we will show that there is a continuous optimal control , for a special class of control problems of calculus of variations type . For further results about existence and continuity properties of optimal controls see [ FR ...
Page 25
... section we study a class of problems with infinite time horizon ( tı ∞ ) . With the notation of Section 3 , the payoff functional is T ( 7.1 ) J ( t , x ; u ) = [ [ " L ( s , x ( s ) , u ( s ) ) ds + 9 ( 7 , x ( 7 ) ) Xr < ∞ , = t ...
... section we study a class of problems with infinite time horizon ( tı ∞ ) . With the notation of Section 3 , the payoff functional is T ( 7.1 ) J ( t , x ; u ) = [ [ " L ( s , x ( s ) , u ( s ) ) ds + 9 ( 7 , x ( 7 ) ) Xr < ∞ , = t ...
Page 26
... section only the problem of control until exit from O , rather than the more general 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 ...
... section only the problem of control until exit from O , rather than the more general 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 ...
Page 28
... ( Section III.9 ) or progressively measurable control processes ( Section IV.5 ) . The following is one among many conditions sufficient for ( 7.14 ) . Lemma 7.1 . Suppose that ẞ > 0 , that ( 7.16 ) sup { f ( x , v ) x x € Ō , v € U } ...
... ( Section III.9 ) or progressively measurable control processes ( Section IV.5 ) . The following is one among many conditions sufficient for ( 7.14 ) . Lemma 7.1 . Suppose that ẞ > 0 , that ( 7.16 ) sup { f ( x , v ) x x € Ō , v € U } ...
Page 33
... Section 9. See ( 9.2 ) . It will be shown in Section 9 , for the special case M IR " ( no constraint on x ( t ) ) , that there exists x * ( - ) which minimizes J. Moreover , x * ( - ) C2 ( [ t , t1 ] ) . See Theorem 9.2 . = In a ...
... Section 9. See ( 9.2 ) . It will be shown in Section 9 , for the special case M IR " ( no constraint on x ( t ) ) , that there exists x * ( - ) which minimizes J. Moreover , x * ( - ) C2 ( [ t , t1 ] ) . See Theorem 9.2 . = In a ...
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