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 56
Page 2
... initial data. See Section 4, and also Section 7 for infinite time horizon problems. The value function V for a deterministic optimal control problem satisfies, at least for- mally, a first order nonlinear partial differential equation ...
... initial data. See Section 4, and also Section 7 for infinite time horizon problems. The value function V for a deterministic optimal control problem satisfies, at least for- mally, a first order nonlinear partial differential equation ...
Page 6
... initial data ( t , x ) a set U ( t , x ) CU ° ( t ) of admissible controls and a payoff functional J ( t , x ; u ) to be minimized . Let us first for- mulate some particular classes of problems ( A through D below ) . Then we subsume ...
... initial data ( t , x ) a set U ( t , x ) CU ° ( t ) of admissible controls and a payoff functional J ( t , x ; u ) to be minimized . Let us first for- mulate some particular classes of problems ( A through D below ) . Then we subsume ...
Page 8
... initial condition x ( t ) = x . Then we assume that ( 3.9 ) ũs ( · ) EU ( s ... data ( t , x ) Q , find u * ( · ) € U ( t , x ) such that J ( t , x ; u ... first term on the right side by J ' ( t , x ; u ) . J ' is the payoff for the ...
... initial condition x ( t ) = x . Then we assume that ( 3.9 ) ũs ( · ) EU ( s ... data ( t , x ) Q , find u * ( · ) € U ( t , x ) such that J ( t , x ; u ... first term on the right side by J ' ( t , x ; u ) . J ' is the payoff for the ...
Page 12
... data are ( 5.5 ) V ( t1 , x ) = ( x ) , x = R " . We now state a theorem which connects the dynamic programming ... initial data ( t , x ) and W ( t , x ) = V ( t , x ) . v EU We may now restate ( 5.7 ' ) 12 I. Deterministic Optimal Control.
... data are ( 5.5 ) V ( t1 , x ) = ( x ) , x = R " . We now state a theorem which connects the dynamic programming ... initial data ( t , x ) and W ( t , x ) = V ( t , x ) . v EU We may now restate ( 5.7 ' ) 12 I. Deterministic Optimal Control.
Page 15
... data ( 5.16 ) , without reference to the initial conditions for x ( s ) . This is one of the important aspects of the LQRP . In the LQRP as formulated in Example 2.3 , the matrices M ( s ) and D are nonnegative definite and N ( s ) is ...
... data ( 5.16 ) , without reference to the initial conditions for x ( s ) . This is one of the important aspects of the LQRP . In the LQRP as formulated in Example 2.3 , the matrices M ( s ) and D are nonnegative definite and N ( s ) is ...
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