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
Wendell Helms Fleming, H. Mete Soner. We require that u ( s ) € U , where U is a closed interval . For instance , if ... requiring x ( t ) = x where x Є IR " is given . For the right endpoint , let us fix t1 and require that 4 I ...
Wendell Helms Fleming, H. Mete Soner. We require that u ( s ) € U , where U is a closed interval . For instance , if ... requiring x ( t ) = x where x Є IR " is given . For the right endpoint , let us fix t1 and require that 4 I ...
Page 100
... require W = w at the extrema . In the above definition , however , we do not require this equality , but instead we use ẞW ( T ) both in ( 11.4 ) and ( 11.5 ) . This modification is discussed in detail in Remark 4.2 . Also , in the ...
... require W = w at the extrema . In the above definition , however , we do not require this equality , but instead we use ẞW ( T ) both in ( 11.4 ) and ( 11.5 ) . This modification is discussed in detail in Remark 4.2 . Also , in the ...
Page 103
... require neither V nor the boundary 00 to be differentiable . Definition 12.1 . We say that We C ( Q ) is a viscosity supersolution of I ( 5.3 ′ ) on [ to , t1 ) × Ō if , for each w € C∞ ( Q ) , ( 12.5 ) - მ Ət · w ( t , x ) + H ( t ...
... require neither V nor the boundary 00 to be differentiable . Definition 12.1 . We say that We C ( Q ) is a viscosity supersolution of I ( 5.3 ′ ) on [ to , t1 ) × Ō if , for each w € C∞ ( Q ) , ( 12.5 ) - მ Ət · w ( t , x ) + H ( t ...
Contents
Viscosity Solutions | 53 |
Controlled Markov Diffusions in R | 157 |
SecondOrder Case | 213 |
Copyright | |
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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 C¹(Q calculus of variations Chapter classical solution consider constant convergence convex Corollary cylindrical region D₂V defined definition denote dynamic programming equation dynamic programming principle Dynkin formula Example exists exit finite first-order formulation given Hamilton-Jacobi equation Hence HJB equation holds implies inequality initial condition initial data lateral boundary Lebesgue left endpoint Lemma linear Lipschitz continuous Markov chain Markov control policy Markov processes maximum principle minimizing Moreover nonlinear obtain optimal control optimal control problems partial derivatives partial differential equation proof of Theorem prove R₁ reference probability system result satisfies second-order Section stochastic control stochastic differential equation Suppose t₁ Theorem 5.1 tion unique value function variations problem Verification Theorem viscosity solution viscosity subsolution viscosity supersolution yields