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 79
Page 8
... resulting control is still admissible. More precisely, let G Z/{(t,ac) and G Z/{(r,a:(r)) for some r G [t, T]. Define a new control by u(s), t § 5 § r (3.8) 11(5) :{ u'(s), r < s § t1. Let .i'(s) be the solution to (3.2) corresponding ...
... resulting control is still admissible. More precisely, let G Z/{(t,ac) and G Z/{(r,a:(r)) for some r G [t, T]. Define a new control by u(s), t § 5 § r (3.8) 11(5) :{ u'(s), r < s § t1. Let .i'(s) be the solution to (3.2) corresponding ...
Page 16
... result will be proved later (Theorem 11.10.2.). Boundary conditions are discussed further in Section 11.13. Theorem 5.2. Let W G C1(Q) satisfy (5.3), (5.18) and (5.19). Then (5.20) W(t,:L') § V(t,a') for all (t, ac) G (I) Q 0 G4Moreouer ...
... result will be proved later (Theorem 11.10.2.). Boundary conditions are discussed further in Section 11.13. Theorem 5.2. Let W G C1(Q) satisfy (5.3), (5.18) and (5.19). Then (5.20) W(t,:L') § V(t,a') for all (t, ac) G (I) Q 0 G4Moreouer ...
Page 19
... results about existence and continuity properties of optimal controls see [FR, Chap III], Cesari [Ce] and Section 11 ... result is also given by using the theory of viscosity solutions (See Corollary II.8.1.). By the dynamic programming ...
... results about existence and continuity properties of optimal controls see [FR, Chap III], Cesari [Ce] and Section 11 ... result is also given by using the theory of viscosity solutions (See Corollary II.8.1.). By the dynamic programming ...
Page 20
... result (Theorem 6.2) which connects Pontryagin's principle and dynamic programming. For this purpose we assume that the partial derivatives fxi,Lxi,gxi,ψxi exist and are continuous for i = 1,···,n. As above, let u∗(·) denote an optimal ...
... result (Theorem 6.2) which connects Pontryagin's principle and dynamic programming. For this purpose we assume that the partial derivatives fxi,Lxi,gxi,ψxi exist and are continuous for i = 1,···,n. As above, let u∗(·) denote an optimal ...
Page 23
... Results similar to Theorem 6.2 are known for the problem of control up to the time of exit from the closure Q of a ... result here. However, we will derive the transversality condition. Again let P(s) be as as in (6.4). If T* I t1 and ...
... Results similar to Theorem 6.2 are known for the problem of control up to the time of exit from the closure Q of a ... result here. However, we will derive the transversality condition. Again let P(s) be as as in (6.4). If T* I t1 and ...
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