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 78
Page 5
... Moreover, for suitable Kρ: (3.1) |f(t,x,v)−f(t,y,v)|≤ Kρ|x−y| for all t ∈ [t0,t1], x,y ∈ IRn and v ∈ U such that |v| ≤ ρ. If the control space U is compact, we can replace Kρ by a constant K, since U ⊂ {v : |v| ≤ ρ} for large ...
... Moreover, for suitable Kρ: (3.1) |f(t,x,v)−f(t,y,v)|≤ Kρ|x−y| for all t ∈ [t0,t1], x,y ∈ IRn and v ∈ U such that |v| ≤ ρ. If the control space U is compact, we can replace Kρ by a constant K, since U ⊂ {v : |v| ≤ ρ} for large ...
Page 12
... ac) g 1/(t,:c), V(t,ac) e Q0. Moreover, if there exists u*(-) e Z/l0(t) such that (5.7) L(s,a3*(s),u*(s)) + f(s,m*(s),u*(s)) -D@W(s,w*(s)) : —H(s, 010* (s), D@W(s, ac* for almost all 5 G [t,t1], then is optimal for initial data (t, ...
... ac) g 1/(t,:c), V(t,ac) e Q0. Moreover, if there exists u*(-) e Z/l0(t) such that (5.7) L(s,a3*(s),u*(s)) + f(s,m*(s),u*(s)) -D@W(s,w*(s)) : —H(s, 010* (s), D@W(s, ac* for almost all 5 G [t,t1], then is optimal for initial data (t, ...
Page 18
... Moreover, ling [ttTh L(s,x(s),u(s))ds : L(t, £17, v). This 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 ...
... Moreover, ling [ttTh L(s,x(s),u(s))ds : L(t, £17, v). This 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 ...
Page 23
... Moreover, (6-115) 9i(T*,flF*(T*)) :H(T*»$*(T*),P(T*))We prove (6.11 a,b) as follows. If T* < t1, then V(T*,§) § g(T*,§) for all E G 30 by (5.19). Equality holds when § I a:* Hence, the derivative of V(T*, — g(T*, at $* ( ...
... Moreover, (6-115) 9i(T*,flF*(T*)) :H(T*»$*(T*),P(T*))We prove (6.11 a,b) as follows. If T* < t1, then V(T*,§) § g(T*,§) for all E G 30 by (5.19). Equality holds when § I a:* Hence, the derivative of V(T*, — g(T*, at $* ( ...
Page 25
... Moreover, for suitable K P, (7-3) |f(w»v) — f(y.v)| 3 Kplw — 1/I for all ac,y G IR” and v G U such that |v| § p. By a control we mean any Uvalued, Lebesgue measurable function on [0, oo) such that is bounded on [0, t] for any t < oo ...
... Moreover, for suitable K P, (7-3) |f(w»v) — f(y.v)| 3 Kplw — 1/I for all ac,y G IR” and v G U such that |v| § p. By a control we mean any Uvalued, Lebesgue measurable function on [0, oo) such that is bounded on [0, t] for any t < oo ...
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