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 87
Page 12
... holds . For instance , we may take u ( s ) = v . For the state constrained problem ( Case D , Section 3 ) , ( 5.2 ) holds provided U ( r , ) is not empty for every ( r , c ) Q ( See Theorem II.12.1 . ) . Note that we assume ( 5.2 ) for ...
... holds . For instance , we may take u ( s ) = v . For the state constrained problem ( Case D , Section 3 ) , ( 5.2 ) holds provided U ( r , ) is not empty for every ( r , c ) Q ( See Theorem II.12.1 . ) . Note that we assume ( 5.2 ) for ...
Page 13
... holds for almost all s if u * ( ) is optimal . We illustrate the use of the Verification Theorem 5.1 in an example . Example 5.1 . Consider the linear quadratic regulator problem ( LQRP ) described in Example 2.3 . In this example , O ...
... holds for almost all s if u * ( ) is optimal . We illustrate the use of the Verification Theorem 5.1 in an example . Example 5.1 . Consider the linear quadratic regulator problem ( LQRP ) described in Example 2.3 . In this example , O ...
Page 15
... holds at any s € [ t , t1 ] if and only if ( 5.17 ) 1 u * ( s ) = — — N − 1 ( s ) B ′ ( s ) D ̧W ( s , x * ( s ) ) = -N - 1 ( s ) B ' ( s ) P ( s ) x * ( s ) . = Now substitute ( 5.17 ) back into the state equation ( 2.6 ) to obtain d ...
... holds at any s € [ t , t1 ] if and only if ( 5.17 ) 1 u * ( s ) = — — N − 1 ( s ) B ′ ( s ) D ̧W ( s , x * ( s ) ) = -N - 1 ( s ) B ' ( s ) P ( s ) x * ( s ) . = Now substitute ( 5.17 ) back into the state equation ( 2.6 ) to obtain d ...
Page 16
... holds in ( 5.19 ) . If ( 3.10 ) is assumed , in addition to ( 3.11 ) , then V ( t , x ) > 0. Since g = 0 when ( 3.10 ) holds , ( 5.19 ) implies that the lateral boundary condition V ( t , x ) = 0 holds for all ( t , x ) = [ to , t1 ) ...
... holds in ( 5.19 ) . If ( 3.10 ) is assumed , in addition to ( 3.11 ) , then V ( t , x ) > 0. Since g = 0 when ( 3.10 ) holds , ( 5.19 ) implies that the lateral boundary condition V ( t , x ) = 0 holds for all ( t , x ) = [ to , t1 ) ...
Page 18
... holds at ( t , x ) . Proof . By assumption ( 5.2 ) , for any v Є U there exists a control u ( · ) € U ( t , x ) such that u ( s ) tends to v as st . By Lemma 4.1 , if t + h < 7 , then t + h ( 6.1 ) V ( t , x ) < L ( s , x ( s ) , u ( s ) ...
... holds at ( t , x ) . Proof . By assumption ( 5.2 ) , for any v Є U there exists a control u ( · ) € U ( t , x ) such that u ( s ) tends to v as st . By Lemma 4.1 , if t + h < 7 , then t + h ( 6.1 ) V ( t , x ) < L ( s , x ( s ) , u ( s ) ...
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 |
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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