## 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

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Page

62 II.4

62 II.4

**Definition**. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 II.5 Dynamic programming and viscosity property . . . . . . . . . . 72 II.6 Properties of viscosity solutions . Page 9

... the minimum value of the payoff function as a function of this initial point. Thus

... the minimum value of the payoff function as a function of this initial point. Thus

**define**a value function by V(t1*/B): ... L'(T)), and (4.2) follows from the**definition**of the value function. Now suppose that r § T. For any 5 > 0, ... Page 13

Wendell H. Fleming, Halil Mete Soner. In Theorem 5.1, denotes the solution to (3.2) with :u*(-), ac* (t) : m. Theorem 5.1 is called a Verification Theorem. Note that, by the

Wendell H. Fleming, Halil Mete Soner. In Theorem 5.1, denotes the solution to (3.2) with :u*(-), ac* (t) : m. Theorem 5.1 is called a Verification Theorem. Note that, by the

**definition**(5.4) of H, (5.7) is equivalent to (5-7') u*($) Q ... Page 20

or [Zi]) every locally Lipschitz function is differentiable at almost all points (t,x)∈Q.

or [Zi]) every locally Lipschitz function is differentiable at almost all points (t,x)∈Q.

**Definition**. W is a generalized solution to the dynamic programming equation in Q if W is locally Lipschitz and satisfies (5.3) for almost all (t ... Page 23

Then (6.3) holds for almost all s G [t, t1.] Proof. Let s G [t,t1) be any point at which is approximately continuous. (For a

Then (6.3) holds for almost all s G [t, t1.] Proof. Let s G [t,t1) be any point at which is approximately continuous. (For a

**definition**of approximately continuous function, see [EG] or [McS, p. 224].) Given v G U and U < 5 < t1 — s, ...### 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