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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Results 1-5 of 84
Page 3
... constants, known to the planner. Let x(s)=(x 1(s),···,x n(s)), u(s)=(u 1(s),···,u n(s)), d = (d1,···,dn). They are, respectively, the inventory and control vectors at time s, and the demand vector. The rate of change of the inventory x ...
... constants, known to the planner. Let x(s)=(x 1(s),···,x n(s)), u(s)=(u 1(s),···,u n(s)), d = (d1,···,dn). They are, respectively, the inventory and control vectors at time s, and the demand vector. The rate of change of the inventory x ...
Page 5
... constant K, since U ⊂ {v : |v| ≤ ρ} for large enough ρ. If f(t,·,v) has a continuous gradient fx, (3.1) is equivalent to the condition |fx(t,x,v)| ≤ Kρ whenever |v| ≤ ρ. A control is a bounded, Lebesgue measurable function u(·) on ...
... constant K, since U ⊂ {v : |v| ≤ ρ} for large enough ρ. If f(t,·,v) has a continuous gradient fx, (3.1) is equivalent to the condition |fx(t,x,v)| ≤ Kρ whenever |v| ≤ ρ. A control is a bounded, Lebesgue measurable function u(·) on ...
Page 9
... constant M Z O.) The method of dynamic programming uses the value function as a tool in the analysis of the optimal control problem. In this section and the following one we study some basic properties of the value function. Then we ...
... constant M Z O.) The method of dynamic programming uses the value function as a tool in the analysis of the optimal control problem. In this section and the following one we study some basic properties of the value function. Then we ...
Page 10
... constant (or nearly constant). Indeed, for a small positive 5, choose a 5-optimal admissible control G Z/l(t, Then for any r G lt, t1] we have 5 + V(t,ac) Z .](t,:L';u) - ;Le»e»uows+@w»v» I\/ H H 3 3 3 N E3? H 3 Q 3 ds + [T L(s, ac(s),u ...
... constant (or nearly constant). Indeed, for a small positive 5, choose a 5-optimal admissible control G Z/l(t, Then for any r G lt, t1] we have 5 + V(t,ac) Z .](t,:L';u) - ;Le»e»uows+@w»v» I\/ H H 3 3 3 N E3? H 3 Q 3 ds + [T L(s, ac(s),u ...
Page 19
... constant M K such that |W(t,x) − W(t,x)| ≤ MK(|t− t| + |x − x|) for all (t, x), (t ,x) ∈ K. If one can choose M = MK which does not depend on K, then W is Lipschitz continuous on Q. By Rademacher's Theorem ([EG] or [Zi]) every ...
... constant M K such that |W(t,x) − W(t,x)| ≤ MK(|t− t| + |x − x|) for all (t, x), (t ,x) ∈ K. If one can choose M = MK which does not depend on K, then W is Lipschitz continuous on Q. By Rademacher's Theorem ([EG] or [Zi]) every ...
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