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 79
Page 7
... Suppose that in case A , the additional restriction x ( t ) EM is imposed , where M is a given closed subset of IR " . In particular , if M = { 1 } consists of a single point , then both endpoints ( t , x ) and ( t1 , x1 ) of the curve ...
... Suppose that in case A , the additional restriction x ( t ) EM is imposed , where M is a given closed subset of IR " . In particular , if M = { 1 } consists of a single point , then both endpoints ( t , x ) and ( t1 , x1 ) of the curve ...
Page 9
... Suppose that < r < t1 . Then ( r Ʌ 7 , x ( r ^ T ) ) = g ( t , x ( T ) ) , and ( 4.2 ) follows from the definition of the value function . Now suppose that r ≤ 7. For any 8 > 0 , choose I. Deterministic Optimal Control 9 Dynamic ...
... Suppose that < r < t1 . Then ( r Ʌ 7 , x ( r ^ T ) ) = g ( t , x ( T ) ) , and ( 4.2 ) follows from the definition of the value function . Now suppose that r ≤ 7. For any 8 > 0 , choose I. Deterministic Optimal Control 9 Dynamic ...
Page 16
... suppose that there exists u * ( · ) € Uo ( t ) such that ( 5.7 ) holds for almost all s Є [ t , 7 * ] and W ( 7 * , x * ( 7 * ) ) = g ( 7 * , x * ( 7 * ) ) in case 7 * < t1 . Then u * ( · ) is optimal for initial data ( t , x ) and W ...
... suppose that there exists u * ( · ) € Uo ( t ) such that ( 5.7 ) holds for almost all s Є [ t , 7 * ] and W ( 7 * , x * ( 7 * ) ) = g ( 7 * , x * ( 7 * ) ) in case 7 * < t1 . Then u * ( · ) is optimal for initial data ( t , x ) and W ...
Page 17
... suppose that u * is a feedback control policy such that u * ( s , y ) € v * ( s , y ) for all ( s , y ) = Q. If u * is admissible for initial conditions ( t , x ) , then we let x * ( s ) be the corresponding solution of ( 5.23 ) with u ...
... suppose that u * is a feedback control policy such that u * ( s , y ) € v * ( s , y ) for all ( s , y ) = Q. If u * is admissible for initial conditions ( t , x ) , then we let x * ( s ) be the corresponding solution of ( 5.23 ) with u ...
Page 23
... Suppose that * < t1 . Then the transversality condition at the point ( 7 * , x * ( 7 * ) ) of the lateral boundary is as follows : there exists a scalar such that ( 6.11a ) P ( 7 * ) = Dxg ( 7 * , x * ( 7 * ) ) + \ n ( x * ( 7 ...
... Suppose that * < t1 . Then the transversality condition at the point ( 7 * , x * ( 7 * ) ) of the lateral boundary is as follows : there exists a scalar such that ( 6.11a ) P ( 7 * ) = Dxg ( 7 * , x * ( 7 * ) ) + \ n ( x * ( 7 ...
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