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 78
Page 4
... Suppose we are also given M(s),N(s), and D, such that M(s) and D are nonnegative definite, symmetric n × n matrices and N(s) is a symmetric, positive definite m × m matrix. The LQRP is to choose u(s) so that (2.7) ∫ t1t [x(s) · M(s)x(s) ...
... Suppose we are also given M(s),N(s), and D, such that M(s) and D are nonnegative definite, symmetric n × n matrices and N(s) is a symmetric, positive definite m × m matrix. The LQRP is to choose u(s) so that (2.7) ∫ t1t [x(s) · M(s)x(s) ...
Page 7
... Suppose that in case A, the additional restriction a:(t1) G ./\/l is imposed, where ./\/l is a given closed subset of R". In particular, if ./\/l I {ml} consists of a single point, then both endpoints (t, ac) and (t1, $1) of the curve ...
... Suppose that in case A, the additional restriction a:(t1) G ./\/l is imposed, where ./\/l is a given closed subset of R". In particular, if ./\/l I {ml} consists of a single point, then both endpoints (t, ac) and (t1, $1) of the curve ...
Page 9
... Suppose that T < r § t1. Then II/(r /\ T,5c(r /\ T)) : g(T,.'L'(T)), and (4.2) follows from the definition of the value function. Now suppose that r § T. For any 5 > 0, choose an I. Deterministic Optimal Control 9 Dynamic programming ...
... Suppose that T < r § t1. Then II/(r /\ T,5c(r /\ T)) : g(T,.'L'(T)), and (4.2) follows from the definition of the value function. Now suppose that r § T. For any 5 > 0, choose an I. Deterministic Optimal Control 9 Dynamic programming ...
Page 16
... suppose that there exists ( ) such that (5.7) holds for almost all 5 G [t,T*] and W(T*,:c* : g(T*,a'* in case T* < t1. Then is optimal for initial data (t, ac) and W(t,m) : 1/(t, Here T* is the exit time of (s,at* from The proof of ...
... suppose that there exists ( ) such that (5.7) holds for almost all 5 G [t,T*] and W(T*,:c* : g(T*,a'* in case T* < t1. Then is optimal for initial data (t, ac) and W(t,m) : 1/(t, Here T* is the exit time of (s,at* from The proof of ...
Page 17
... suppose that u∗ is a feedback control policy such that u∗(s, y) ∈ v∗(s, y) for all (s, y) ∈ Q. If u∗ is admissible for initial conditions (t, x), then we let x∗(s) be the corresponding solution of (5.23) with u = u∗, and (5.25) ...
... suppose that u∗ is a feedback control policy such that u∗(s, y) ∈ v∗(s, y) for all (s, y) ∈ Q. If u∗ is admissible for initial conditions (t, x), then we let x∗(s) be the corresponding solution of (5.23) with u = u∗, and (5.25) ...
Contents
1 | |
Viscosity Solutions | 57 |
Differential Games | 375 |
A Duality Relationships 397 | 396 |
References | 409 |
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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