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 11
... 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 shall see in later chapters ...
... 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 shall see in later chapters ...
Page 13
... Hence W(t,5c) § .](t,w;u). We get (5.6) by taking the infimum over To prove the second assertion of the theorem, let G Z/l0(t) satisfy (5.7). We redo the calculation above with This yields (5.8) with an equality. Hence (5.9) W(t, ac) ...
... Hence W(t,5c) § .](t,w;u). We get (5.6) by taking the infimum over To prove the second assertion of the theorem, let G Z/l0(t) satisfy (5.7). We redo the calculation above with This yields (5.8) with an equality. Hence (5.9) W(t, ac) ...
Page 21
... hence one can take P0 = 1. In addition to (6.2) and (6.3), the adjoint variable satisfies conditions at the final ... Hence it has a unique solution ̄P(s) satisfying (6.5). So the task in front of us is to show that P(s) is indeed equal ...
... hence one can take P0 = 1. In addition to (6.2) and (6.3), the adjoint variable satisfies conditions at the final ... Hence it has a unique solution ̄P(s) satisfying (6.5). So the task in front of us is to show that P(s) is indeed equal ...
Page 25
... Hence, it sufiices to consider initial time t : 0. From now on we shall do so, and will write .]($; u) instead of J(0, cc; Let us now formulate more precisely the class of infinite horizon control problems which we shall consider. Let ...
... Hence, it sufiices to consider initial time t : 0. From now on we shall do so, and will write .]($; u) instead of J(0, cc; Let us now formulate more precisely the class of infinite horizon control problems which we shall consider. Let ...
Page 28
... hence (7.16) holds. Consider the problem of minimizing O0 —s 1 2 1 2 J(a3; u) : e —:c(s) + —u(s) ds. 0 2 2 The stationary dynamic programming equation (7.10) is _ 1 1 (7.18) V(at) — [5v2 + vV'(a3)] — 50102 + :cV'(at) : 0. Here = d/dx ...
... hence (7.16) holds. Consider the problem of minimizing O0 —s 1 2 1 2 J(a3; u) : e —:c(s) + —u(s) ds. 0 2 2 The stationary dynamic programming equation (7.10) is _ 1 1 (7.18) V(at) — [5v2 + vV'(a3)] — 50102 + :cV'(at) : 0. Here = d/dx ...
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