## Controlled Markov Processes and Viscosity SolutionsThis book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. |

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Page 69

Then , V is a

Then , V is a

**viscosity solution**of ( 3.12 ) in Q. Proof . Let w E D and ( 1,7 ) E Q be a maximizer of the difference V – w on ở satisfying V ( T , ) = wst , 7 ) . Then , w > V. Using ( 3.2b ) with 0 = w ( r , · ) and s = tı , we obtain ...Page 107

( c ) W is a

( c ) W is a

**viscosity solution**of ( 8.1 ) in Q and ( 9.3a ) if it is both a viscosity subsolution and a viscosity supersolution of ( 8.1 ) in Q and ( 9.3a ) . Notice that ( 13.1 ) implies ( 13.4 ) . Also we have formally argued that if ...Page 177

This case has been studied by Fleming - Zariphopoulou ( FZ ) and Fitzpatrick - Fleming ( FF ] using

This case has been studied by Fleming - Zariphopoulou ( FZ ) and Fitzpatrick - Fleming ( FF ] using

**viscosity solution**techniques and numerical solutions of the dynamic programming equation . Another simplification in Merton's problem ...### What people are saying - Write a review

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### Contents

Viscosity Solutions | 53 |

Controlled Markov Diffusions in R | 157 |

SecondOrder Case | 213 |

Copyright | |

7 other sections not shown

### Other editions - View all

Controlled Markov Processes and Viscosity Solutions Wendell H. Fleming,Halil Mete Soner Limited preview - 2006 |

Controlled Markov Processes and Viscosity Solutions Wendell H. Fleming,Halil Mete Soner No preview available - 2006 |

### Common terms and phrases

admissible apply approximation assume assumptions boundary condition bounded calculus called Chapter compact condition consider constant continuous control problem convergence convex Corollary corresponding cost defined definition denote depend derivatives deterministic difference discussion dynamic programming equation equivalent estimate Example exists exit fact finite fixed formula given gives Hence holds horizon implies inequality lateral Lemma limit linear Lipschitz Markov Markov diffusion Markov processes maximum measurable method minimizing Moreover nonlinear obtain operator optimal control partial differential equation particular positive principle probability proof prove Recall reference Remark replaced require respectively result satisfies Section Similarly smooth space step stochastic control stochastic differential equation subset sufficiently suitable supersolution Suppose term terminal Theorem 5.1 theory tion uniformly unique value function Verification viscosity solution viscosity subsolution yields