## 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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Typically, the value function is not smooth enough to satisfy the HJB equation in a

Typically, the value function is not smooth enough to satisfy the HJB equation in a

**classical**sense. However, under quite general assumptions the value function is the unique viscosity**solution**of the HJB equation with appropriate ... Page 2

In fact, the value function V often does not have the smoothness properties needed to interpret it as a

In fact, the value function V often does not have the smoothness properties needed to interpret it as a

**solution**to the dynamic programming partial differential equation in the usual (“**classical**”) sense. However, in such cases V can be ... Page 3

models chosen from such diverse applications as inventory theory, control of physical devices, financial economics and

models chosen from such diverse applications as inventory theory, control of physical devices, financial economics and

**classical**mechanics. Example 2.1. Consider the production planning of a factory producing n commodities. Page 4

One can also consider the problem of controlling the

One can also consider the problem of controlling the

**solution**x(s)=(x 1(s), x2 (s)) to (2.4) on an infinite time ... The simplest kind of problem in**classical**calculus of variations is to determine a function x(·) which minimizes a ... Page 12

... {_p ' .f(t:$vv) _ L(tv $1 14)} ' 1n analogy with a quantity occurring in

... {_p ' .f(t:$vv) _ L(tv $1 14)} ' 1n analogy with a quantity occurring in

**classical**mechanics, we call this function the Hamiltonian. The dynamic programming equation (5.3') is sometimes also called a HamiltonGJacobiGBellman PDE.### 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