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 6-10 of 83
Page 6
... denote the set of all controls u ( · ) . In notation which we shall use later ( Section 9 ) U ° ( t ) = L∞ ( [ t , t1 ] ; U ) . This is the space of all bounded , Lebesgue measurable , U - valued functions on [ t , to ] . In order to ...
... denote the set of all controls u ( · ) . In notation which we shall use later ( Section 9 ) U ° ( t ) = L∞ ( [ t , t1 ] ; U ) . This is the space of all bounded , Lebesgue measurable , U - valued functions on [ t , to ] . In order to ...
Page 7
... denotes an indicator function . Thus , for real numbers a , b , Xa < b = 1 if a < b O if a > b , and Xa < b is defined similarly . The function g is called a boundary cost function , and is assumed continuous . B ' . Control until exit ...
... denotes an indicator function . Thus , for real numbers a , b , Xa < b = 1 if a < b O if a > b , and Xa < b is defined similarly . The function g is called a boundary cost function , and is assumed continuous . B ' . Control until exit ...
Page 8
... denotes the restriction to [ s , t1 ] of ũ ( · ) and ĩ is the exit time from Qof ( s , ( s ) ) . Note that ( 3.9 ) ... denote the first term on the right side by J ' ( t , x ; u ) . J ' is the payoff for the problem of control up to ...
... denotes the restriction to [ s , t1 ] of ũ ( · ) and ĩ is the exit time from Qof ( s , ( s ) ) . Note that ( 3.9 ) ... denote the first term on the right side by J ' ( t , x ; u ) . J ' is the payoff for the problem of control up to ...
Page 12
... denotes the gradient of V ( t ,. ) . It is notationally convenient to rewrite ( 5.3 ) as ( 5.3 ' ) მ Ət · V ( t , x ) ... denotes. In analogy with a quantity occurring in classical mechanics , we call this func- tion the Hamiltonian . The ...
... denotes the gradient of V ( t ,. ) . It is notationally convenient to rewrite ( 5.3 ) as ( 5.3 ' ) მ Ət · V ( t , x ) ... denotes. In analogy with a quantity occurring in classical mechanics , we call this func- tion the Hamiltonian . The ...
Page 13
... denotes the solution to ( 3.2 ) with u ( · ) = u * ( · ) , x * ( t ) x . Theorem 5.1 is called a Verification Theorem . Note that , by the definition ( 5.4 ) of H , ( 5.7 ) is equivalent to ( 5.7 ' ) u * ( s ) = arg min { ƒ ( s , x ...
... denotes the solution to ( 3.2 ) with u ( · ) = u * ( · ) , x * ( t ) x . Theorem 5.1 is called a Verification Theorem . Note that , by the definition ( 5.4 ) of H , ( 5.7 ) is equivalent to ( 5.7 ' ) u * ( s ) = arg min { ƒ ( s , x ...
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