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 87
Page 12
... holds. For instance, we may take u(s) E u. For the state constrained problem (Case D, Section 3), (5.2) holds provided Z/{(r, is not empty for every (r, G Q (See Theorem 11.12.1.). Note that we assume (5.2) for (t, m) G Q, not (t, at) G ...
... holds. For instance, we may take u(s) E u. For the state constrained problem (Case D, Section 3), (5.2) holds provided Z/{(r, is not empty for every (r, G Q (See Theorem 11.12.1.). Note that we assume (5.2) for (t, m) G Q, not (t, at) G ...
Page 13
... holds for almost all s if is optimal. We illustrate the use of the Verification Theorem 5.1 in an example. Example 5.1. Consider the linear quadratic regulator problem (LQRP) described in Example 2.3. In this example, O : R”, U : 1Rm,Z ...
... holds for almost all s if is optimal. We illustrate the use of the Verification Theorem 5.1 in an example. Example 5.1. Consider the linear quadratic regulator problem (LQRP) described in Example 2.3. In this example, O : R”, U : 1Rm,Z ...
Page 15
... In view of (5.13), (5.7) holds at any s ∈ [t, t1] if and only if u∗(s) = −12N−1(s)B(s)DxW(s, x∗(s)) (5.17) = −N−1(s)B(s)P(s)x∗(s). Now substitute (5.17) back into the state equation (2.6) to obtain ddsx∗(s)=[A(s) − B(s)N−1(s) ...
... In view of (5.13), (5.7) holds at any s ∈ [t, t1] if and only if u∗(s) = −12N−1(s)B(s)DxW(s, x∗(s)) (5.17) = −N−1(s)B(s)P(s)x∗(s). Now substitute (5.17) back into the state equation (2.6) to obtain ddsx∗(s)=[A(s) − B(s)N−1(s) ...
Page 16
... holds, (5.19) implies that the lateral boundary condition 1/(t,$) : 0 holds for all (t, cc) G [t0,t1) >< 30, if both (3.10) and (3.11) hold. Note that we have not yet proved that the value function V is continuous on However, such a ...
... holds, (5.19) implies that the lateral boundary condition 1/(t,$) : 0 holds for all (t, cc) G [t0,t1) >< 30, if both (3.10) and (3.11) hold. Note that we have not yet proved that the value function V is continuous on However, such a ...
Page 18
... holds at (t, Proof. By assumption (5.2), for any v G U there exists a control G Z/l(t,a3) such that tends to v as s ... holds in (6.1) when : and : is the corresponding solution of (3-2)-(3.3). U In particular, assumption (b) holds if a ...
... holds at (t, Proof. By assumption (5.2), for any v G U there exists a control G Z/l(t,a3) such that tends to v as s ... holds in (6.1) when : and : is the corresponding solution of (3-2)-(3.3). U In particular, assumption (b) holds if a ...
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