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 87
Page 19
... holds if a continuous optimal control u * ( · ) exists . In 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 ...
... holds if a continuous optimal control u * ( · ) exists . In 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 ...
Page 23
... holds true . However , a weaker notion of gradient is needed . See Section II.15 in this book and [ Cle1,2 ] . Results similar to Theorem 6.2 are known for the problem of control up to the time of exit from the closure Q of a ...
... holds true . However , a weaker notion of gradient is needed . See Section II.15 in this book and [ Cle1,2 ] . Results similar to Theorem 6.2 are known for the problem of control up to the time of exit from the closure Q of a ...
Page 28
... holds . Proof . We calculate that ( 7.17 ) d ( | x ( s ) | 2 ) = 2ƒ ( x ( s ) , u ( s ) ) · x ( s ) ds for any control u ( · ) . By ( 7.16 ) | x ( s ) | 2 ≤ | x | 2 + Ks for some K. Then , for suitable C1 | W ( x ( s ) ) | ≤ C1 ( 1+ ...
... holds . Proof . We calculate that ( 7.17 ) d ( | x ( s ) | 2 ) = 2ƒ ( x ( s ) , u ( s ) ) · x ( s ) ds for any control u ( · ) . By ( 7.16 ) | x ( s ) | 2 ≤ | x | 2 + Ks for some K. Then , for suitable C1 | W ( x ( s ) ) | ≤ C1 ( 1+ ...
Page 29
... holds for D the positive root and b = (2D)−1. Then the linear equation (7.20) is of 2D2 + 3D solved for x≥ − b, 12 with = 0 W(b) = Db2; and we set W(−x) = W(x). This solution W to (7.18) is of class C1(IR1) and grows quadratically ...
... holds for D the positive root and b = (2D)−1. Then the linear equation (7.20) is of 2D2 + 3D solved for x≥ − b, 12 with = 0 W(b) = Db2; and we set W(−x) = W(x). This solution W to (7.18) is of class C1(IR1) and grows quadratically ...
Page 31
... holds and use Theorem 7.1 with Remark 7.2 . Since c ( s ) > 0 , ( 7.23 ) implies that ( 7.28 ) x ( s ) ≤ xers , x = x ( 0 ) . Then ( 7.23 ) and ( 7.28 ) imply an upper bound on the total consumption up to time s : S - [ c ( p ) dp ...
... holds and use Theorem 7.1 with Remark 7.2 . Since c ( s ) > 0 , ( 7.23 ) implies that ( 7.28 ) x ( s ) ≤ xers , x = x ( 0 ) . Then ( 7.23 ) and ( 7.28 ) imply an upper bound on the total consumption up to time s : S - [ c ( p ) dp ...
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