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 82
Page
... assumptions the value function is the 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 ...
... assumptions the value function is the 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 ...
Page 2
... assumptions used in this chapter, for finite-time horizon deterministic optimal control problems. For infinite-time horizon problems, these are summarized in Section 7. I.2. Examples. We start our discussion by giving some examples. In ...
... assumptions used in this chapter, for finite-time horizon deterministic optimal control problems. For infinite-time horizon problems, these are summarized in Section 7. I.2. Examples. We start our discussion by giving some examples. In ...
Page 5
... assumptions in the problem formulation become slightly more complicated . ) The objective is to minimize some payoff ... Assumption ( 3.1 ) implies that , given any control u ( · ) , the differ- ential equation ( 3.2 ) d x ( s ) = f ( s ...
... assumptions in the problem formulation become slightly more complicated . ) The objective is to minimize some payoff ... Assumption ( 3.1 ) implies that , given any control u ( · ) , the differ- ential equation ( 3.2 ) d x ( s ) = f ( s ...
Page 8
... assumption g = 0 , J ( t , x ; u ) = Lds + J Lds + v ( x ( t ) ) XT = t1 Let us denote the first term on the right side by J ' ( t , x ; u ) . J ' is the payoff for the problem of control up to time 7 ' , in case 7 ' < t1 . Since L ≥ 0 ...
... assumption g = 0 , J ( t , x ; u ) = Lds + J Lds + v ( x ( t ) ) XT = t1 Let us denote the first term on the right side by J ' ( t , x ; u ) . J ' is the payoff for the problem of control up to time 7 ' , in case 7 ' < t1 . Since L ≥ 0 ...
Page 16
... assumption on W is convenient will arise in Example 7.3 . In Example 5.1 we constructed an admissible control by using the value function . To generalize the procedure , let W be as in Theorem 5.2 ( or as in Theorem 5.1 in case Q = Qo ...
... assumption on W is convenient will arise in Example 7.3 . In Example 5.1 we constructed an admissible control by using the value function . To generalize the procedure , let W be as in Theorem 5.2 ( or as in Theorem 5.1 in case Q = Qo ...
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