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 8
... Define a new control by u(s), t § 5 § r (3.8) 11(5) :{ u'(s), r < s § t1. Let .i'(s) be the solution to (3.2) corresponding to control and initial condition :Z(t) : at. Then we assume that (3-9) 11s(-) E 7/{(8, H2 /3 in §/ ?/ PlI/\ CI ...
... Define a new control by u(s), t § 5 § r (3.8) 11(5) :{ u'(s), r < s § t1. Let .i'(s) be the solution to (3.2) corresponding to control and initial condition :Z(t) : at. Then we assume that (3-9) 11s(-) E 7/{(8, H2 /3 in §/ ?/ PlI/\ CI ...
Page 9
... define a value function by V(t1*/B): J(t1$iU')a for all (t, at') G We shall assume that V(t,a3) > —oo. This is ... definition of the value function. Now suppose that r § T. For any 5 > 0, choose an I. Deterministic Optimal Control 9 ...
... define a value function by V(t1*/B): J(t1$iU')a for all (t, at') G We shall assume that V(t,a3) > —oo. This is ... definition of the value function. Now suppose that r § T. For any 5 > 0, choose an I. Deterministic Optimal Control 9 ...
Page 10
... define an admissible control G M (15, 111) by u(s), s § r : { u1(s), 5 > r. Let be the state corresponding to with initial condition (t, av), and T the exit time of (s, from Since r < T, T1 : T and we have V0. 111) S J(i»w;fl) [le@@@e»w ...
... define an admissible control G M (15, 111) by u(s), s § r : { u1(s), 5 > r. Let be the state corresponding to with initial condition (t, av), and T the exit time of (s, from Since r < T, T1 : T and we have V0. 111) S J(i»w;fl) [le@@@e»w ...
Page 13
... definition (5.4) of H, (5.7) is equivalent to (5-7') u*($) Q aYg€11,}iH{f($.w*($).v) - D@W($.w*($)) + L($»w*($)»v)}Proof of Theorem 5.1. Consider any G Z/l0(t). Using multivariate calculus and the dynamic programming equation (5.3), we ...
... definition (5.4) of H, (5.7) is equivalent to (5-7') u*($) Q aYg€11,}iH{f($.w*($).v) - D@W($.w*($)) + L($»w*($)»v)}Proof of Theorem 5.1. Consider any G Z/l0(t). Using multivariate calculus and the dynamic programming equation (5.3), we ...
Page 16
... 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 map (5.21) 'u*(t, ac) : argengin [f(t, av, v) -D1,W(t,a3) + L(t, av, 12)]. We may now restate (5.7') as u* (s) ...
... 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 map (5.21) 'u*(t, ac) : argengin [f(t, av, v) -D1,W(t,a3) + L(t, av, 12)]. We may now restate (5.7') as u* (s) ...
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