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 56
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... unique vis- cosity solution of the HJB equation with appropriate boundary conditions. In addition, the viscosity solution framework is well suited to proving continuous dependence of solutions on problem data. The book begins with an ...
... unique vis- cosity solution of the HJB equation with appropriate boundary conditions. In addition, the viscosity solution framework is well suited to proving continuous dependence of solutions on problem data. The book begins with an ...
Page 3
... unique minimum at x = 0. A typical example of h is n h ( x ) = Σ [ αi ( xi ) + + Vi ( Xi ) ̄ ] , i = 1 where ai , i are positive constants interpreted respectively as a unit holding cost and a unit shortage cost . Here , a + = max { a ...
... unique minimum at x = 0. A typical example of h is n h ( x ) = Σ [ αi ( xi ) + + Vi ( Xi ) ̄ ] , i = 1 where ai , i are positive constants interpreted respectively as a unit holding cost and a unit shortage cost . Here , a + = max { a ...
Page 6
... unique solution . The solution x ( s ) of ( 3.2 ) and ( 3.3 ) is called the state of the system at time s . Clearly the state depends on the control u ( · ) and the initial condition , but this dependence is suppressed in our notation ...
... unique solution . The solution x ( s ) of ( 3.2 ) and ( 3.3 ) is called the state of the system at time s . Clearly the state depends on the control u ( · ) and the initial condition , but this dependence is suppressed in our notation ...
Page 14
... unique maximizer of (5.12) is (5.13) 1 v∗ = - N −1(t)B (t)p. 2 To use the Verification Theorem 5.1, first we have to solve (5.11) with the terminal condition (5.14) V (t1 ,x) = x · Dx, x ∈ Rn. We guess that the solution of (5.11) and ...
... unique maximizer of (5.12) is (5.13) 1 v∗ = - N −1(t)B (t)p. 2 To use the Verification Theorem 5.1, first we have to solve (5.11) with the terminal condition (5.14) V (t1 ,x) = x · Dx, x ∈ Rn. We guess that the solution of (5.11) and ...
Page 15
... unique solution satisfying the initial condition x * ( t ) = x . Thus there is a unique control u * ( · ) satisfying ( 5.17 ) . Theorem 5.1 then implies that u * ( ) is optimal at ( t , x ) . Notice that the optimal control u * ( s ) in ...
... unique solution satisfying the initial condition x * ( t ) = x . Thus there is a unique control u * ( · ) satisfying ( 5.17 ) . Theorem 5.1 then implies that u * ( ) is optimal at ( t , x ) . Notice that the optimal control u * ( s ) in ...
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