## 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

However, under quite general

However, under quite general

**assumptions**the value function is the unique viscosity solution of the HJB equation with appropriate boundary conditions. In addition, the viscosity solution framework is well suited to proving continuous ... Page 2

The reader should refer to Section 3 for notations and

The reader should refer to Section 3 for notations and

**assumptions**used in this chapter, for finite-time horizon deterministic optimal control problems. For infinite-time horizon problems, these are summarized in Section 7. Page 5

(One could equally well take −∞ <t<t 1, but then certain

(One could equally well take −∞ <t<t 1, but then certain

**assumptions**in the problem formulation become slightly more complicated.) The objective is to minimize some payoff functional J, which depends on states x(s) and controls u(s) ... Page 8

... then by (3.5) and the

... then by (3.5) and the

**assumption**g E 0, .](t,ac;u) If Lds+ Lds + 1b($(t1))X.,:,;1 . t T' Let us denote the first term on the right side by J'(t, cc; J' is the payoff for the problem of control up to time T', in case T' < t1. Page 16

... the proof of Theorem 5.2 shows that it suffices in this case to assume W G C1(Q) Fl rather than W G G1 A situation where such a weaker

... the proof of Theorem 5.2 shows that it suffices in this case to assume W G C1(Q) Fl rather than W G G1 A situation where such a weaker

**assumption**on W is convenient will arise in Example 7.3. 1n Example 5.1 we constructed an ...### What people are saying - Write a review

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### Contents

1 | |

Viscosity Solutions | 57 |

Differential Games | 375 |

A Duality Relationships 397 | 396 |

References | 409 |

### 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 calculus of variations Chapter classical solution consider constant controlled Markov diffusion convergence convex Corollary cost function deﬁne deﬁnition denote differential games dynamic programming equation dynamic programming principle Dynkin formula Example exists exit ﬁnite ﬁrst formulation G Q0 Hamilton-Jacobi equation Hence HJB equation holds implies inequality initial data Ishii Lemma linear Lipschitz continuous Markov chain Markov control policy Markov processes maximum principle 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 satisfying Section semigroup Soner stochastic control stochastic control problem stochastic differential equations subset Suppose Theorem 9.1 uniformly continuous unique value function Veriﬁcation Theorem viscosity solution viscosity subsolution viscosity supersolution