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 193
... Similarly , ( 8.14b ) follows from ( 8.10 ) . IV.9 Estimates for second - order difference quotients = We will now obtain a bound for second difference quotients A2J △ J ( t ,, u ) . This will imply a one - sided bound ( 9.7 ) for A2V ...
... Similarly , ( 8.14b ) follows from ( 8.10 ) . IV.9 Estimates for second - order difference quotients = We will now obtain a bound for second difference quotients A2J △ J ( t ,, u ) . This will imply a one - sided bound ( 9.7 ) for A2V ...
Page 198
... Similarly , the generalized partial derivatives Vt , Vx¡¡ ( if they exist ) are defined as those functions in Loc ( Qo ) such that , for all • € Co ° ( Qo ) , ( 10.2 ) V1Þdxdt = - V + dxdt , ( 10.3 ) Vz1z , Þdxdt = Vz , z , dxdt . Qo ...
... Similarly , the generalized partial derivatives Vt , Vx¡¡ ( if they exist ) are defined as those functions in Loc ( Qo ) such that , for all • € Co ° ( Qo ) , ( 10.2 ) V1Þdxdt = - V + dxdt , ( 10.3 ) Vz1z , Þdxdt = Vz , z , dxdt . Qo ...
Page 377
... Similarly , V. is a viscosity supersolu- tion . We assume that ( 3.1 ) and IV ( 2.2 ) hold . Let V ( t , x ) be the value function for the controlled diffusion process , as in IV ( 2.10 ) . We also make the following assumption , which ...
... Similarly , V. is a viscosity supersolu- tion . We assume that ( 3.1 ) and IV ( 2.2 ) hold . Let V ( t , x ) be the value function for the controlled diffusion process , as in IV ( 2.10 ) . We also make the following assumption , which ...
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 brownian motion C₁ C¹(Q calculus of variations Chapter classical solution consider continuous on Q controlled Markov diffusion convergence convex Corollary D₂V defined definition denote deterministic dynamic programming equation dynamic programming principle Dynkin formula Example exists exit finite formula Hence HJB equation holds implies inequality initial data lateral boundary Lebesgue Lemma lim sup linear Lipschitz continuous Markov chain Markov control policy Markov processes maximum principle minimizing Moreover nonlinear obtain optimal control optimal Markov control partial derivatives partial differential equation proof of Theorem prove reference probability system Remark result satisfies second-order Section semiconvex stochastic control stochastic differential equation Suppose t₁ Theorem 5.1 uniformly continuous unique value function Verification Theorem viscosity solution viscosity subsolution viscosity supersolution yields