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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Page 11
... dynamic programming principle . It is the basis of the solution technique developed by Bellman in the 1950's [ Be ] . An interesting observation is that an optimal control u * ( · ) € U ( t , x ) minimizes ( 4.3 ) at every r . Hence to ...
... dynamic programming principle . It is the basis of the solution technique developed by Bellman in the 1950's [ Be ] . An interesting observation is that an optimal control u * ( · ) € U ( t , x ) minimizes ( 4.3 ) at every r . Hence to ...
Page 17
... Dynamic programming and Pontryagin's principle In this section we. If ( 5.23 ) with the initial data x ( t ) = x has a unique solution x ( · ) , and if ( 5.24 ) u ( s ) = u ( s , x ( s ) ) belongs to U ° ( t ) , then we call u an ...
... Dynamic programming and Pontryagin's principle In this section we. If ( 5.23 ) with the initial data x ( t ) = x has a unique solution x ( · ) , and if ( 5.24 ) u ( s ) = u ( s , x ( s ) ) belongs to U ° ( t ) , then we call u an ...
Page 18
Wendell H. Fleming, Halil Mete Soner. 1.6 Dynamic programming and Pontryagin's principle In this section we first give a sufficient condition that the value function V satisfies the dynamic programming equation ( 5.3 ) at a point ( t , x ) ...
Wendell H. Fleming, Halil Mete Soner. 1.6 Dynamic programming and Pontryagin's principle In this section we first give a sufficient condition that the value function V satisfies the dynamic programming equation ( 5.3 ) at a point ( t , x ) ...
Page 19
... dynamic programming principle ( 4.3 ) , we have ( see ( 5.1 ) ) for small h > 0 inf 1 t + h 1 L ( s , x ( s ) , u ( s ) ) ds + h — [ V ( t + h , x ( t + h ) ) − V ( t , x ) ] } = 0 . u ( · ) ЄU ° ( t ) h Since U is compact , it can be ...
... dynamic programming principle ( 4.3 ) , we have ( see ( 5.1 ) ) for small h > 0 inf 1 t + h 1 L ( s , x ( s ) , u ( s ) ) ds + h — [ V ( t + h , x ( t + h ) ) − V ( t , x ) ] } = 0 . u ( · ) ЄU ° ( t ) h Since U is compact , it can be ...
Page 20
... dynamic programming equation in Q if W is locally Lipschitz and satisfies ( 5.3 ) for almost all ( t , x ) EQ ... Principle . During the 1950's Pontryagin formulated a " maximum principle " which provides a general set of necessary ...
... dynamic programming equation in Q if W is locally Lipschitz and satisfies ( 5.3 ) for almost all ( t , x ) EQ ... Principle . During the 1950's Pontryagin formulated a " maximum principle " which provides a general set of necessary ...
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