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 80
Page 2
... prove two special existence theorems which are used elsewhere in this book. The proofs rely on lower semicontinuity of the cost function in the control problem. The reader should refer to Section 3 for notations and assumptions used in ...
... prove two special existence theorems which are used elsewhere in this book. The proofs rely on lower semicontinuity of the cost function in the control problem. The reader should refer to Section 3 for notations and assumptions used in ...
Page 11
... proved the following . Lemma 4.2 . For any initial condition ( t , x ) = Q and r € [ t , t1 ] , ( 4.3 ) CrɅT V ( t , x ) ... prove two Veri- fication Theorems ( Theorems 5.1 and 5.2 ) which give sufficient conditions for a solution to the ...
... proved the following . Lemma 4.2 . For any initial condition ( t , x ) = Q and r € [ t , t1 ] , ( 4.3 ) CrɅT V ( t , x ) ... prove two Veri- fication Theorems ( Theorems 5.1 and 5.2 ) which give sufficient conditions for a solution to the ...
Page 13
... prove the second assertion of the theorem , let u * ( - ) € U ° ( t ) satisfy ( 5.7 ) . We redo the calculation above with u * ( · ) . This yields ( 5.8 ) with an equality . Hence ( 5.9 ) W ( t , x ) = J ( t , x ; u * ) . By combining ...
... prove the second assertion of the theorem , let u * ( - ) € U ° ( t ) satisfy ( 5.7 ) . We redo the calculation above with u * ( · ) . This yields ( 5.8 ) with an equality . Hence ( 5.9 ) W ( t , x ) = J ( t , x ; u * ) . By combining ...
Page 16
... proved that the value function V is continuous on Q. However , such a result will be proved later ( Theorem II.10.2 . ) . Boundary conditions are discussed further in Section II.13 . Theorem 5.2 . Let WE C1 ( Q ) satisfy ( 5.3 ) ...
... proved that the value function V is continuous on Q. However , such a result will be proved later ( Theorem II.10.2 . ) . Boundary conditions are discussed further in Section II.13 . Theorem 5.2 . Let WE C1 ( Q ) satisfy ( 5.3 ) ...
Page 18
... proves ( a ) . To prove ( b ) , we use the same argument , observing that equality holds in ( 6.1 ) when u ( · ) = u * ( · ) and x ( · ) = x * ( · ) is the corresponding solution of ( 3-2 ) - ( 3.3 ) . In particular , assumption ( b ) ...
... proves ( a ) . To prove ( b ) , we use the same argument , observing that equality holds in ( 6.1 ) when u ( · ) = u * ( · ) and x ( · ) = x * ( · ) is the corresponding solution of ( 3-2 ) - ( 3.3 ) . In particular , assumption ( b ) ...
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