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 48
Page 4
... minimized. Another possible criterion to be minimized is the time for x(s) to reach a given target. If the target is the point (0,0), then the control function u(·) is to be chosen such that the first time θ when x(θ) = (0,0) is minimized ...
... minimized. Another possible criterion to be minimized is the time for x(s) to reach a given target. If the target is the point (0,0), then the control function u(·) is to be chosen such that the first time θ when x(θ) = (0,0) is minimized ...
Page 6
... minimized. Let us first formulate some particular classes of problems (A through D below). Then we subsume all of ... minimizes (3.4) J(t, x;u) = ∫ t1t L(s, x(s),u(s))ds + ψ(x(t1)), where L ∈ C(Q0 ×U). We call L the running cost ...
... minimized. Let us first formulate some particular classes of problems (A through D below). Then we subsume all of ... minimizes (3.4) J(t, x;u) = ∫ t1t L(s, x(s),u(s))ds + ψ(x(t1)), where L ∈ C(Q0 ×U). We call L the running cost ...
Page 7
... minimizing J in (3.5) subject to an endpoint constraint (T,.'IJ(T)) G S, where S is a given closed subset of 3*Q. D. State constraint. This is the problem of minimizing .](t, at; u) in (3.4) subject to the constraint ac(s) G C. In this ...
... minimizing J in (3.5) subject to an endpoint constraint (T,.'IJ(T)) G S, where S is a given closed subset of 3*Q. D. State constraint. This is the problem of minimizing .](t, at; u) in (3.4) subject to the constraint ac(s) G C. In this ...
Page 11
... minimizes (4.3) at every r. Hence to determine the optimal control u∗(t), it suffices to analyze (4.3) with r arbitrarily close to t. Intuitively this yields a simple optimization problem that is minimized by u∗(t). However, as we ...
... minimizes (4.3) at every r. Hence to determine the optimal control u∗(t), it suffices to analyze (4.3) with r arbitrarily close to t. Intuitively this yields a simple optimization problem that is minimized by u∗(t). However, as we ...
Page 12
... to the control problem of minimizing (3.4). Theorem 5.1. (Q I Q0). Let W e C1(Q0) satisfy (5.3) and (5.5). Then: (5.6) W(t,ac) g 1/(t,:c), V(t,ac) e Q0. Moreover, if there exists u*(-) e Z/l0(t) such that (5.7) L(s,a3*(s),u*(s)) + f(s,m*( ...
... to the control problem of minimizing (3.4). Theorem 5.1. (Q I Q0). Let W e C1(Q0) satisfy (5.3) and (5.5). Then: (5.6) W(t,ac) g 1/(t,:c), V(t,ac) e Q0. Moreover, if there exists u*(-) e Z/l0(t) such that (5.7) L(s,a3*(s),u*(s)) + f(s,m*( ...
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