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 59
Page
... 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. The book begins with an ...
... 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. The book begins with an ...
Page 3
... unique minimum at x = 0. A typical example of h is h(x) = n∑i=1 [ αi (xi )+ + γi (xi )− ] , where αi ,γ i are positive constants interpreted respectively as a unit holding cost and a unit shortage cost. Here, a+ = max{a,0}, a− = max ...
... unique minimum at x = 0. A typical example of h is h(x) = n∑i=1 [ αi (xi )+ + γi (xi )− ] , where αi ,γ i are positive constants interpreted respectively as a unit holding cost and a unit shortage cost. Here, a+ = max{a,0}, a− = max ...
Page 5
... that, given any control u(·), the differential equation (3.2) ddsx(s) = f(s,x(s),u(s)), t ≤ s ≤ t1 with initial condition (3.3) x(t) = x has a unique I. Deterministic Optimal Control 5 Finite time horizon problems.
... that, given any control u(·), the differential equation (3.2) ddsx(s) = f(s,x(s),u(s)), t ≤ s ≤ t1 with initial condition (3.3) x(t) = x has a unique I. Deterministic Optimal Control 5 Finite time horizon problems.
Page 6
... unique solution. The solution x(s) of (3.2) and (3.3) is called the state of the system at time s. Clearly the state depends on the control u(·) and the initial condition, but this dependence is suppressed in our notation. Let U0(t) ...
... unique solution. The solution x(s) of (3.2) and (3.3) is called the state of the system at time s. Clearly the state depends on the control u(·) and the initial condition, but this dependence is suppressed in our notation. Let U0(t) ...
Page 14
... unique maximizer of (5.12) is (5.13) 11* I -%N*1(1s)B'(t)p. To use the Verification Theorem 5.1, first we have to solve (5.11) with the terminal condition (5.14) V(t1,$) : m - Dav, at G R”. We guess that the solution of (5.11) and (5.14) ...
... unique maximizer of (5.12) is (5.13) 11* I -%N*1(1s)B'(t)p. To use the Verification Theorem 5.1, first we have to solve (5.11) with the terminal condition (5.14) V(t1,$) : m - Dav, at G R”. We guess that the solution of (5.11) and (5.14) ...
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