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. |
From inside the book
Results 1-5 of 85
Page 3
... assume that the demand rates d2 are fixed constants , known to the planner . Let x ( s ) = ( x1 ( s ) , ··· , xn ( s ) ) , u ( s ) = ( u1 ( s ) , ··· , Un ( s ) ) , d = ( d1 , ··· , dn ) . They are , respectively , the inventory and ...
... assume that the demand rates d2 are fixed constants , known to the planner . Let x ( s ) = ( x1 ( s ) , ··· , xn ( s ) ) , u ( s ) = ( u1 ( s ) , ··· , Un ( s ) ) , d = ( d1 , ··· , dn ) . They are , respectively , the inventory and ...
Page 8
... assume that ( 3.9 ) ũs ( · ) EU ( s , x ( s ) ) , t ≤ s ≤ ĩ , where u ( · ) denotes the restriction to [ s , t1 ] of ũ ( · ) and ĩ is the exit time from Qof ( s , ( s ) ) . Note that ( 3.9 ) implies , in particular , that an ...
... assume that ( 3.9 ) ũs ( · ) EU ( s , x ( s ) ) , t ≤ s ≤ ĩ , where u ( · ) denotes the restriction to [ s , t1 ] of ũ ( · ) and ĩ is the exit time from Qof ( s , ( s ) ) . Note that ( 3.9 ) implies , in particular , that an ...
Page 11
... assume that the value function is continuously differen- tiable and proceed formally to obtain a nonlinear partial differential equation satisfied by the value function . In general however , the value function is not differentiable ...
... assume that the value function is continuously differen- tiable and proceed formally to obtain a nonlinear partial differential equation satisfied by the value function . In general however , the value function is not differentiable ...
Page 15
... assume that ( 3.11 ) holds on the lateral boundary [ to , t1 ) × 20. This implies that , for ( t , x ) E [ to , t1 ) x 00 , one choice is to exit immediately from Q ( thus , T = t ) . Therefore , V ( t , x ) ≤ g ( t I. Deterministic ...
... assume that ( 3.11 ) holds on the lateral boundary [ to , t1 ) × 20. This implies that , for ( t , x ) E [ to , t1 ) x 00 , one choice is to exit immediately from Q ( thus , T = t ) . Therefore , V ( t , x ) ≤ g ( t I. Deterministic ...
Page 16
... assumed , in addition to ( 3.11 ) , then V ( t , x ) > 0. Since g = 0 when ( 3.10 ) holds , ( 5.19 ) implies that the lateral ... assume W € C1 ( Q ) ~ C ( Q ) rather than W € C1 ( Q ) . A situation where such a weaker assumption on W is ...
... assumed , in addition to ( 3.11 ) , then V ( t , x ) > 0. Since g = 0 when ( 3.10 ) holds , ( 5.19 ) implies that the lateral ... assume W € C1 ( Q ) ~ C ( Q ) rather than W € C1 ( Q ) . A situation where such a weaker assumption on W is ...
Contents
1 | |
Viscosity Solutions | 57 |
Classical Solutions119 | 118 |
Controlled Markov Diffusions in R | 151 |
SecondOrder Case | 199 |
Logarithmic Transformations and Risk Sensitivity | 227 |
Singular Perturbations 261 | 260 |
Singular Stochastic Control | 293 |
Finite Difference Numerical Approximations | 321 |
Differential Games | 375 |
A Duality Relationships 397 | 396 |
References | 409 |
Index 425 | 424 |
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 C₁ C¹(Q calculus of variations Chapter classical solution consider constant constraint controlled Markov diffusion convergence convex Corollary defined definition denote differential games dynamic programming equation dynamic programming principle Dynkin formula Example exists exit finite formulation given Hence HJB equation holds implies inequality initial data Ishii Lemma linear Lipschitz continuous Markov chain Markov control policy Markov processes 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 Section semigroup Soner stochastic control stochastic control problem stochastic differential equations subset Suppose t₁ Theorem 9.1 uniformly continuous unique value function Verification Theorem viscosity solution viscosity subsolution viscosity supersolution