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 82
Page 18
... value function V satisfies the dynamic programming equation ( 5.3 ) at a point ( t , x ) . Then we show how dynamic programming is related to Pontryagin's principle , which gives necessary conditions for u * ( - ) to minimize J ( t , x ...
... value function V satisfies the dynamic programming equation ( 5.3 ) at a point ( t , x ) . Then we show how dynamic programming is related to Pontryagin's principle , which gives necessary conditions for u * ( - ) to minimize J ( t , x ...
Page 19
... value function V fails to be differentiable at some points ( t , x ) . Thus , V may not satisfy the dynamic programming equation ( 5.3 ) everywhere in Q. In such cases , we wish to interpret V as a solution in some extended sense . One ...
... value function V fails to be differentiable at some points ( t , x ) . Thus , V may not satisfy the dynamic programming equation ( 5.3 ) everywhere in Q. In such cases , we wish to interpret V as a solution in some extended sense . One ...
Page 20
... value function V to be Lipschitz on Q. See Theorem 9.3 , Theorem II.10.2 . A local Lipschitz condition for V follows from the estimates in Section IV.8 . Unfortunately , when generalized solutions are considered instead of “ clas- sical ...
... value function V to be Lipschitz on Q. See Theorem 9.3 , Theorem II.10.2 . A local Lipschitz condition for V follows from the estimates in Section IV.8 . Unfortunately , when generalized solutions are considered instead of “ clas- sical ...
Page 23
... value function V or other restrictive assumptions in Theorem 6.2 . Theorem 6.3 . Let U ( t , x ) = U ° ( t ) and O = R " . Let u * ( · ) be an optimal control , and let P ( s ) be the solution to ( 6.2 ) for t≤ st1 with P ( t1 ) = Dy ...
... value function V or other restrictive assumptions in Theorem 6.2 . Theorem 6.3 . Let U ( t , x ) = U ° ( t ) and O = R " . Let u * ( · ) be an optimal control , and let P ( s ) be the solution to ( 6.2 ) for t≤ st1 with P ( t1 ) = Dy ...
Page 25
... value function . A standard procedure is to introduce a discount factor ẞ ≥ 0 such that - L ( t , x , v ) = e ̄ßt Ĩ ( t , x , v ) g ( t , x ) = e - tğ ( t , x ) . If ß > 0 and Ĩ , ğ are bounded , then the value function V is always ...
... value function . A standard procedure is to introduce a discount factor ẞ ≥ 0 such that - L ( t , x , v ) = e ̄ßt Ĩ ( t , x , v ) g ( t , x ) = e - tğ ( t , x ) . If ß > 0 and Ĩ , ğ are bounded , then the value function V is always ...
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