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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Page
... value function. For controlled Markov diffusion processes on n - dimensional euclidean space, the dynamic programming equation becomes a nonlinear partial differential equation of second order, called a Hamilton – Jacobi – Bellman (HJB) ...
... value function. For controlled Markov diffusion processes on n - dimensional euclidean space, the dynamic programming equation becomes a nonlinear partial differential equation of second order, called a Hamilton – Jacobi – Bellman (HJB) ...
Page 2
... value function V is introduced which is the optimum value of the payoff considered as a function of initial data. See Section 4, and also Section 7 for infinite time horizon problems. The value function V for a deterministic optimal ...
... value function V is introduced which is the optimum value of the payoff considered as a function of initial data. See Section 4, and also Section 7 for infinite time horizon problems. The value function V for a deterministic optimal ...
Page 9
... value of the payoff function as a function of this initial point. Thus define a value function by V(t1*/B): J(t1$iU')a for all (t, at') G We shall assume that V(t,a3) > —oo. This is always true if the control space U is compact, or if U ...
... value of the payoff function as a function of this initial point. Thus define a value function by V(t1*/B): J(t1$iU')a for all (t, at') G We shall assume that V(t,a3) > —oo. This is always true if the control space U is compact, or if U ...
Page 11
... value function is continuously differentiable and proceed formally to obtain a nonlinear partial differential equation satisfied by the value function. In general however, the value function is not differentiable. In that case a notion ...
... value function is continuously differentiable and proceed formally to obtain a nonlinear partial differential equation satisfied by the value function. In general however, the value function is not differentiable. In that case a notion ...
Page 16
... value function. To generalize the procedure, let W be as in Theorem 5.2 (or as in Theorem 5.1 in case Q : Q0.) For (t, ac) G Q define a set-valued map F* (t, m) by F*<r.r> - {f<r.r.r> = r e r*<r.r>} where 12* (t, ac) is another set-valued ...
... value function. To generalize the procedure, let W be as in Theorem 5.2 (or as in Theorem 5.1 in case Q : Q0.) For (t, ac) G Q define a set-valued map F* (t, m) by F*<r.r> - {f<r.r.r> = r e r*<r.r>} where 12* (t, ac) is another set-valued ...
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