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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Page 11
... Hence to determine the optimal control u * ( t ) , it suffices to analyze ( 4.3 ) with r arbitrarily close to t . Intuitively this yields a simple optimization problem that is minimized by u * ( t ) . However , as we shall see in later ...
... Hence to determine the optimal control u * ( t ) , it suffices to analyze ( 4.3 ) with r arbitrarily close to t . Intuitively this yields a simple optimization problem that is minimized by u * ( t ) . However , as we shall see in later ...
Page 13
... Hence W ( t , x ) J ( t , x ; u ) . We get ( 5.6 ) by taking the infimum over u ( · ) . To prove the second assertion of the theorem , let u * ( - ) € U ° ( t ) satisfy ( 5.7 ) . We redo the calculation above with u * ( · ) . This ...
... Hence W ( t , x ) J ( t , x ; u ) . We get ( 5.6 ) by taking the infimum over u ( · ) . To prove the second assertion of the theorem , let u * ( - ) € U ° ( t ) satisfy ( 5.7 ) . We redo the calculation above with u * ( · ) . This ...
Page 18
... Hence the dynamic programming equation ( 5.3 ) holds at ( t , x ) . Proof . By assumption ( 5.2 ) , for any v Є U there exists a control u ( · ) € U ( t , x ) such that u ( s ) tends to v as st . By Lemma 4.1 , if t + h < 7 , then t + h ...
... Hence the dynamic programming equation ( 5.3 ) holds at ( t , x ) . Proof . By assumption ( 5.2 ) , for any v Є U there exists a control u ( · ) € U ( t , x ) such that u ( s ) tends to v as st . By Lemma 4.1 , if t + h < 7 , then t + h ...
Page 21
... hence one can take Po = 1 . In addition to ( 6.2 ) and ( 6.3 ) , the adjoint variable satisfies conditions at the ... Hence it has a unique solution P ( s ) satisfying ( 6.5 ) . So the task in front of us is to show that P ( s ) is ...
... hence one can take Po = 1 . In addition to ( 6.2 ) and ( 6.3 ) , the adjoint variable satisfies conditions at the ... Hence it has a unique solution P ( s ) satisfying ( 6.5 ) . So the task in front of us is to show that P ( s ) is ...
Page 25
... Hence , it suffices to consider initial time t = 0. From now on we shall do so , and will write J ( x ; u ) instead of J ( 0 , x ; u ) . Let us now formulate more precisely the class of infinite horizon control problems which we shall ...
... Hence , it suffices to consider initial time t = 0. From now on we shall do so , and will write J ( x ; u ) instead of J ( 0 , x ; u ) . Let us now formulate more precisely the class of infinite horizon control problems which we shall ...
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