Controlled Markov Processes and Viscosity SolutionsThis book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. |
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Page 4
... given . We seek to minimize a quadratic crite- rion of the form ( 2.5 ) [ m1x1 ( s ) 2 + m2x2 ( 8 ) 2 + u ( s ) 2 ] ds + d1xı ( tı ) 2 + d2x2 ( t2 ) 2 , where m1 , m2 , dı , d2 are nonnegative constants . If there is no constraint on ...
... given . We seek to minimize a quadratic crite- rion of the form ( 2.5 ) [ m1x1 ( s ) 2 + m2x2 ( 8 ) 2 + u ( s ) 2 ] ds + d1xı ( tı ) 2 + d2x2 ( t2 ) 2 , where m1 , m2 , dı , d2 are nonnegative constants . If there is no constraint on ...
Page 5
... given closed subset of R " . If M = { 1 } consists of a single point , then the right endpoint ( t1 , 1 ) is fixed . At the opposite extreme , there is no restriction on x ( t ) if M = R " . - We will discuss calculus of variations ...
... given closed subset of R " . If M = { 1 } consists of a single point , then the right endpoint ( t1 , 1 ) is fixed . At the opposite extreme , there is no restriction on x ( t ) if M = R " . - We will discuss calculus of variations ...
Page 108
... given by ( 2.4 ) is a viscosity solution of ( 2.5 ) in Q = [ 0 , 1 ) × ( −1 , 1 ) and ( 13.7ii ) is satisfied . Moreover , for every t Є [ 0,1 ) , V ( t , 1 ) = ( −1 ) v ( 1 − a + at ) < 1 . Also except at t = [ ( a − 2 ) / a ] v 0 ...
... given by ( 2.4 ) is a viscosity solution of ( 2.5 ) in Q = [ 0 , 1 ) × ( −1 , 1 ) and ( 13.7ii ) is satisfied . Moreover , for every t Є [ 0,1 ) , V ( t , 1 ) = ( −1 ) v ( 1 − a + at ) < 1 . Also except at t = [ ( a − 2 ) / a ] v 0 ...
Other editions - View all
Controlled Markov Processes and Viscosity Solutions Wendell H. Fleming,Halil Mete Soner Limited preview - 2006 |
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 c₁ Cą(Q calculus of variations Chapter classical solution consider constant continuous on Q convergence convex Corollary cylindrical region defined definition denote dynamic programming equation dynamic programming principle Dynkin formula Example exists exit finite first-order formulation Hamilton-Jacobi equation Hence HJB equation holds implies inequality initial data lateral boundary Lemma lim sup 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 proof of Theorem prove result satisfies second-order Section semigroup stochastic differential equation Suppose t₁ test function Theorem 5.1 uniformly continuous unique value function variations problem Verification Theorem viscosity solution viscosity subsolution viscosity supersolution yields