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 84
Page 6
... Consider the following payoff functional J , which depends on states x ( s ) and controls u ( s ) for times s Є [ t , 7 ) , where 7 is the smaller of t1 and the exit time of x ( s ) from the closure O of an open set O CIR " . We let Q ...
... Consider the following payoff functional J , which depends on states x ( s ) and controls u ( s ) for times s Є [ t , 7 ) , where 7 is the smaller of t1 and the exit time of x ( s ) from the closure O of an open set O CIR " . We let Q ...
Page 9
... consider a family of optimization problems with different initial conditions ( t , x ) . Consider the minimum value of the payoff function as a function of this initial point . Thus define a value function by ( 4.1 ) V ( t , x ) = inf u ...
... consider a family of optimization problems with different initial conditions ( t , x ) . Consider the minimum value of the payoff function as a function of this initial point . Thus define a value function by ( 4.1 ) V ( t , x ) = inf u ...
Page 13
... Consider any u ( · ) € U ° ( t ) . Using multivariate calculus and the dynamic programming equation ( 5.3 ) , we obtain მ ( 5.8 ) W ( t1 , x ( t ) ) = W ( t , x ) + " & W ( 3 , x ( 8 ) ) + à ( 8 ) · D2W ( s , x ( s ) ] ds rti a t [ ~ W ...
... Consider any u ( · ) € U ° ( t ) . Using multivariate calculus and the dynamic programming equation ( 5.3 ) , we obtain მ ( 5.8 ) W ( t1 , x ( t ) ) = W ( t , x ) + " & W ( 3 , x ( 8 ) ) + à ( 8 ) · D2W ( s , x ( s ) ] ds rti a t [ ~ W ...
Page 15
... consider the problem of control until the time of exit from a closed cylindrical region Q ( class B , Section 3. ) We first formulate appropriate boundary conditions for the dynamic pro- gramming equation ( 5.3 ) . Then we outline a ...
... consider the problem of control until the time of exit from a closed cylindrical region Q ( class B , Section 3. ) We first formulate appropriate boundary conditions for the dynamic pro- gramming equation ( 5.3 ) . Then we outline a ...
Page 17
... Consider the differential equation ( 5.23 ) d ds -x ( s ) = f ( s , x ( s ) , u ( s , x ( s ) ) ) , s Є [ t , t1 ] . 1.6 Dynamic programming and Pontryagin's principle In this section we. If ( 5.23 ) with the initial data x ( t ) = x has ...
... Consider the differential equation ( 5.23 ) d ds -x ( s ) = f ( s , x ( s ) , u ( s , x ( s ) ) ) , s Є [ t , t1 ] . 1.6 Dynamic programming and Pontryagin's principle In this section we. If ( 5.23 ) with the initial data x ( t ) = x has ...
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 |
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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 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