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 51
Page
... condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 VII.6 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 VII.7 Convergence . . . . . . . . . . . . . . . . . . . . .
... condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 VII.6 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 VII.7 Convergence . . . . . . . . . . . . . . . . . . . . .
Page 6
... condition, but this dependence is suppressed in our notation. Let U0(t) denote the set of all controls u(·). In ... boundary and terminal boundary, respectively, of Q. Given initial data (t, x) ∈ Q, let τ denote the exit time of (s,x(s)) ...
... condition, but this dependence is suppressed in our notation. Let U0(t) denote the set of all controls u(·). In ... boundary and terminal boundary, respectively, of Q. Given initial data (t, x) ∈ Q, let τ denote the exit time of (s,x(s)) ...
Page 7
... boundary cost function, and is assumed continuous. B'. Control until exit from Let (t,ac) G Q, and let T' be the ... condition that Z/{(75, cc) is nonempty is called a reachability condition. See Sontag [Sg]. If U I Rm, it is related to ...
... boundary cost function, and is assumed continuous. B'. Control until exit from Let (t,ac) G Q, and let T' be the ... condition that Z/{(75, cc) is nonempty is called a reachability condition. See Sontag [Sg]. If U I Rm, it is related to ...
Page 9
... conditions (t, Consider the minimum value of the payoff function as a function of this initial point. Thus define a ... boundary cost (see (3.5)). Lemma 4.1. For every initial condition (t, ac) G Q, admissible control G Z/{(t,;v) and t ...
... conditions (t, Consider the minimum value of the payoff function as a function of this initial point. Thus define a ... boundary cost (see (3.5)). Lemma 4.1. For every initial condition (t, ac) G Q, admissible control G Z/{(t,;v) and t ...
Page 15
... condition x∗(t) = x. Thus there is a unique control u∗(·) satisfying (5.17) ... boundary conditions for the dynamic programming equation (5.3). Then we ... boundary [t0,t1) × ∂O. This implies that, for (t, x) ∈ [t0 ,t1) × ∂O, one ...
... condition x∗(t) = x. Thus there is a unique control u∗(·) satisfying (5.17) ... boundary conditions for the dynamic programming equation (5.3). Then we ... boundary [t0,t1) × ∂O. This implies that, for (t, x) ∈ [t0 ,t1) × ∂O, one ...
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