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 77
Page
... . . . . . . 62 II.4 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 II.5 Dynamic programming and viscosity property . . . . . . . . . . 72 II.6 Properties of viscosity solutions ...
... . . . . . . 62 II.4 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 II.5 Dynamic programming and viscosity property . . . . . . . . . . 72 II.6 Properties of viscosity solutions ...
Page 13
... definition (5.4) of H, (5.7) is equivalent to (5-7') u*($) Q aYg€11,}iH{f($.w*($).v) - D@W($.w*($)) + L($»w*($)»v)}Proof of Theorem 5.1. Consider any G Z/l0(t). Using multivariate calculus and the dynamic programming equation (5.3), we ...
... definition (5.4) of H, (5.7) is equivalent to (5-7') u*($) Q aYg€11,}iH{f($.w*($).v) - D@W($.w*($)) + L($»w*($)»v)}Proof of Theorem 5.1. Consider any G Z/l0(t). Using multivariate calculus and the dynamic programming equation (5.3), we ...
Page 20
... 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,x)∈Q. Corollary 6.1. Let U be compact and U(t,x) = U0(t). If V is locally Lipschitz on 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,x)∈Q. Corollary 6.1. Let U be compact and U(t,x) = U0(t). If V is locally Lipschitz on Q ...
Page 23
... definition of approximately continuous function, see [EG] or [McS, p. 224].) Given v G U and U < 5 < t1 — s, let u*(r)ifr¢[s,s+5] u5(r)I v ifrG[s,s+5], and let :c5(r) be the solution to (3.2) with u(r) I. Deterministic Optimal Control 23.
... definition of approximately continuous function, see [EG] or [McS, p. 224].) Given v G U and U < 5 < t1 — s, let u*(r)ifr¢[s,s+5] u5(r)I v ifrG[s,s+5], and let :c5(r) be the solution to (3.2) with u(r) I. Deterministic Optimal Control 23.
Page 26
... defined for all G Mm. We assume that (7.6) Mm is nonempty for all at G O. We also assume the analogue of (3.11): (7.7) For every 5 G 30 there exists G U such that f(€,v(€)) -17(5) > 0Here is the exterior unit normal at f. The value ...
... defined for all G Mm. We assume that (7.6) Mm is nonempty for all at G O. We also assume the analogue of (3.11): (7.7) For every 5 G 30 there exists G U such that f(€,v(€)) -17(5) > 0Here is the exterior unit normal at f. The value ...
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