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 81
Page 13
... Theorem 5.1 is called a Verification Theorem. Note that, by the 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*($)»v)}Proof of Theorem 5.1. Consider any G Z/l0(t). Using ...
... Theorem 5.1 is called a Verification Theorem. Note that, by the definition (5.4) of H, (5.7) is equivalent to (5-7') u*($) 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 ...
Page 15
... Theorem 5.1 to show that V(t,x) = W(t,x) for tmin < t ≤ t1, and to find an explicit formula for the optimal u∗(s) ... proof of a Verification Theorem (Theorem 5.2) similar to Theorem 5.1. When t = t1, we have as in (5.5): (5.18) V(t1,x) ...
... Theorem 5.1 to show that V(t,x) = W(t,x) for tmin < t ≤ t1, and to find an explicit formula for the optimal u∗(s) ... proof of a Verification Theorem (Theorem 5.2) similar to Theorem 5.1. When t = t1, we have as in (5.5): (5.18) V(t1,x) ...
Page 16
... proof of Theorem 5.2 is almost the same as for Theorem 5.1. 1n (5.8) the integral is now from t to the exit time T, and W(T,ac(T)) is on the left side. By (5.18) and (5.19), W(T, § Q(T, with W as in (3.6). This gives (5.20). The second ...
... proof of Theorem 5.2 is almost the same as for Theorem 5.1. 1n (5.8) the integral is now from t to the exit time T, and W(T,ac(T)) is on the left side. By (5.18) and (5.19), W(T, § Q(T, with W as in (3.6). This gives (5.20). The second ...
Page 19
... proof of Theorem 6.1. Another proof of this result is also given by using the theory of viscosity solutions (See Corollary II.8.1.). By the dynamic programming principle (4.3), we have (see (5.1)) for small h > 0 infu(·)∈U0(t) { 1 h ...
... proof of Theorem 6.1. Another proof of this result is also given by using the theory of viscosity solutions (See Corollary II.8.1.). By the dynamic programming principle (4.3), we have (see (5.1)) for small h > 0 infu(·)∈U0(t) { 1 h ...
Page 20
... Theorem 9.3, Theorem II.10.2. A local Lipschitz condition for V follows from the estimates in Section IV.8 ... proof of Pontryagin's principle in its full generality is rather lengthy, and will not be given in this book. See for instance [FR, ...
... Theorem 9.3, Theorem II.10.2. A local Lipschitz condition for V follows from the estimates in Section IV.8 ... proof of Pontryagin's principle in its full generality is rather lengthy, and will not be given in this book. See for instance [FR, ...
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