## 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

Results 1-5 of 38

Page

... value function is the unique viscosity solution of the HJB equation with appropriate

... 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 dependence of solutions on problem**data**. Page 2

... value function V is introduced which is the optimum value of the payoff considered as a function of initial data. ... a classical solution of the dynamic programming partial differential equation with the appropriate

... value function V is introduced which is the optimum value of the payoff considered as a function of initial data. ... a classical solution of the dynamic programming partial differential equation with the appropriate

**boundary data**. Page 6

In order to complete the formulation of an optimal control problem, we must specify for each initial

In order to complete the formulation of an optimal control problem, we must specify for each initial

**data**(t, ... the lateral**boundary**and terminal**boundary**, respectively, of Q. Given initial**data**(t, x) ∈ Q, let τ denote the exit time ... Page 12

**Boundary**conditions for state constrained problems (class D, Section 3) will be described later in Section 11.12. For class A, we have Q : Q0. By (3.4) the terminal (Cauchy)**data**are (5.5) V(t1,:1;) : 1b(:1;), cc G IR". Page 15

... by solving the Riccati differential equation (5.15) with terminal

... by solving the Riccati differential equation (5.15) with terminal

**data**(5.16), without reference to the initial conditions for x(s). ... We first formulate appropriate**boundary**conditions for the dynamic programming equation (5.3).### 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