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 47
Page 6
... control problem, we must specify for each initial data (t, x) a set U(t, x) ⊂ U0(t) of admissible controls and a payoff functional J(t, x;u) to be minimized. Let us first formulate some particular classes of problems (A through D below) ...
... control problem, we must specify for each initial data (t, x) a set U(t, x) ⊂ U0(t) of admissible controls and a payoff functional J(t, x;u) to be minimized. Let us first formulate some particular classes of problems (A through D below) ...
Page 8
... admissible control by another admissible one after a certain time, then the 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 ...
... admissible control by another admissible one after a certain time, then the 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 ...
Page 9
... control policy. We start with a simple property of V. Let r /\ T : min(r, T). Recall that g is the boundary cost (see (3.5)). Lemma 4.1. For every initial condition (t, ac) G Q, admissible control G Z/{(t,;v) and t § r § t1, we have <4 ...
... control policy. We start with a simple property of V. Let r /\ T : min(r, T). Recall that g is the boundary cost (see (3.5)). Lemma 4.1. For every initial condition (t, ac) G Q, admissible control G Z/{(t,;v) and t § r § t1, we have <4 ...
Page 10
... control is called 5 - optimal.) As in (3.8) define an admissible control G M (15, 111) by u(s), s § r : { u1(s), 5 > r. Let be the state corresponding to with initial condition (t, av), and T the exit time of (s, from Since r < T, T1 ...
... control is called 5 - optimal.) As in (3.8) define an admissible control G M (15, 111) by u(s), s § r : { u1(s), 5 > r. Let be the state corresponding to with initial condition (t, av), and T the exit time of (s, from Since r < T, T1 ...
Page 11
... admissible control u(·) ∈ U(t,x) is δ-optimal at (t, x) if any only if it is δ-optimal at every (r, x(r)) with r ∈ [t, τ]. I.5. Dynamic. programming. equation. In this section, we assume that the value function is continuously ...
... admissible control u(·) ∈ U(t,x) is δ-optimal at (t, x) if any only if it is δ-optimal at every (r, x(r)) with r ∈ [t, τ]. I.5. Dynamic. programming. equation. In this section, we assume that the value function is continuously ...
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