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 76
Page 2
... prove two special existence theorems which are used elsewhere in this book. The proofs rely on lower semicontinuity of the cost function in the control problem. The reader should refer to Section 3 for notations and assumptions used in ...
... prove two special existence theorems which are used elsewhere in this book. The proofs rely on lower semicontinuity of the cost function in the control problem. The reader should refer to Section 3 for notations and assumptions used in ...
Page 11
... proved the following. Lemma 4.2. For any initial condition (t, x) ∈ Q and r ∈ [t, t1], (4.3) V(t, x) = infu(·)∈U(t,x) ... prove two Verification Theorems (Theorems 5.1 and 5.2) which give sufficient conditions for a solution to the ...
... proved the following. Lemma 4.2. For any initial condition (t, x) ∈ Q and r ∈ [t, t1], (4.3) V(t, x) = infu(·)∈U(t,x) ... prove two Verification Theorems (Theorems 5.1 and 5.2) which give sufficient conditions for a solution to the ...
Page 13
... 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) : .](t, ac; By combining this equality with (5.6), we conclude that is ...
... 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) : .](t, ac; By combining this equality with (5.6), we conclude that is ...
Page 16
... proved that the value function V is continuous on However, such a result will be proved later (Theorem 11.10.2.). Boundary conditions are discussed further in Section 11.13. Theorem 5.2. Let W G C1(Q) satisfy (5.3), (5.18) and (5.19) ...
... proved that the value function V is continuous on However, such a result will be proved later (Theorem 11.10.2.). Boundary conditions are discussed further in Section 11.13. Theorem 5.2. Let W G C1(Q) satisfy (5.3), (5.18) and (5.19) ...
Page 18
... proves (a). To prove (b), we use the same argument, observing that equality holds in (6.1) when : and : is the corresponding solution of (3-2)-(3.3). U In particular, assumption (b) holds if a continuous optimal control 18 I ...
... proves (a). To prove (b), we use the same argument, observing that equality holds in (6.1) when : and : is the corresponding solution of (3-2)-(3.3). U In particular, assumption (b) holds if a continuous optimal control 18 I ...
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