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
Page 3
... satisfy certain constraints related to the physical capabilities of the factory and the workforce . These capacity con- straints translate into upper bounds for the production rates . We assume that these take the form c1u1 + ··· + cnun ...
... satisfy certain constraints related to the physical capabilities of the factory and the workforce . These capacity con- straints translate into upper bounds for the production rates . We assume that these take the form c1u1 + ··· + cnun ...
Page 13
... 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 this equality with ( 5.6 ) , we conclude that u * ( · ) is optimal at ...
... 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 this equality with ( 5.6 ) , we conclude that u * ( · ) is optimal at ...
Page 15
... 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 ( 5.17 ) is a ...
... 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 ( 5.17 ) is a ...
Page 19
... 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 such interpretation is as a generalized solution , a concept which we shall now define ...
... 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 such interpretation is as a generalized solution , a concept which we shall now define ...
Page 21
... satisfying ( 6.5 ) . So the task in front of us is to show that P ( s ) is indeed equal to P ( s ) . For t ≤ r < t1 , the restriction u * ( · ) of u * ( · ) to [ r , t1 ] is admissible . Hence , for any yЄIR " ( 6.6 ) V ( r , y ) ≤ J ...
... satisfying ( 6.5 ) . So the task in front of us is to show that P ( s ) is indeed equal to P ( s ) . For t ≤ r < t1 , the restriction u * ( · ) of u * ( · ) to [ r , t1 ] is admissible . Hence , for any yЄIR " ( 6.6 ) V ( r , y ) ≤ J ...
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