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 86
Page 16
... exists a control u0(-) such that J(t,a3;u(~)) < g(t, See Example 11.2.3. At such points, strict inequality holds in (5.19). 1f (3.10) is assumed, in addition to (3.11), then 1/(t,5c) Z 0. Since g E 0 when (3.10) holds, (5.19) implies ...
... exists a control u0(-) such that J(t,a3;u(~)) < g(t, See Example 11.2.3. At such points, strict inequality holds in (5.19). 1f (3.10) is assumed, in addition to (3.11), then 1/(t,5c) Z 0. Since g E 0 when (3.10) holds, (5.19) implies ...
Page 17
... exists. We shall not undertake to deal with it here, but will only indicate some of the difficulties. In order to implement the procedure just outlined above, one needs first a smooth solution W to the dynamic programming equation (5.3) ...
... exists. We shall not undertake to deal with it here, but will only indicate some of the difficulties. In order to implement the procedure just outlined above, one needs first a smooth solution W to the dynamic programming equation (5.3) ...
Page 18
... exists an optimal control G Z/{(t, £17) such that u* (s) tends to a limit v* as s J, t, then V1(i,w) + L(trI1.v*)+ f(i.w,v*) ~ DJ/(i,w) I 0Hence the dynamic programming equation (5.3) holds at (t, Proof. By assumption (5.2), for any v ...
... exists an optimal control G Z/{(t, £17) such that u* (s) tends to a limit v* as s J, t, then V1(i,w) + L(trI1.v*)+ f(i.w,v*) ~ DJ/(i,w) I 0Hence the dynamic programming equation (5.3) holds at (t, Proof. By assumption (5.2), for any v ...
Page 19
... exists. In Section 9 we will show that there is a continuous optimal control, for a special class of control problems of calculus of variations type. For further results about existence and continuity properties of optimal controls see ...
... exists. In Section 9 we will show that there is a continuous optimal control, for a special class of control problems of calculus of variations type. For further results about existence and continuity properties of optimal controls see ...
Page 23
... exists a scalar A such that (6-11¢) P(T*) I Dm9(T*,fv*(T*))+>"7(flY*(T*)) with g as in (3.6) and the exterior unit normal at 5 G 30. Moreover, (6-115) 9i(T*,flF*(T*)) :H(T*»$*(T*),P(T*))We prove (6.11 a,b) as follows. If T* ...
... exists a scalar A such that (6-11¢) P(T*) I Dm9(T*,fv*(T*))+>"7(flY*(T*)) with g as in (3.6) and the exterior unit normal at 5 G 30. Moreover, (6-115) 9i(T*,flF*(T*)) :H(T*»$*(T*),P(T*))We prove (6.11 a,b) as follows. If T* ...
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