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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Results 1-3 of 78
Page 69
... Suppose that We D. Then W is a viscosity solution of ( 3.12 ) in Q if and only if it is a classical solution of ( 3.12 ) . Proof . First suppose that W is a viscosity solution . Since WЄD , w = W is a test function . Moreover every ( t ...
... Suppose that We D. Then W is a viscosity solution of ( 3.12 ) in Q if and only if it is a classical solution of ( 3.12 ) . Proof . First suppose that W is a viscosity solution . Since WЄD , w = W is a test function . Moreover every ( t ...
Page 91
... Suppose ( t , x ) E O * Q . Then proceed exactly as in Step 2 to obtain 1 ( 9.20 ) Þ , ( † , ñ ; 3 , ÿ ) ≤ 1⁄2mv ( 2 ( K1 + 1 ) ( ɛ + 6 ) ) + sup [ W − V ] . მ * Q 3 ' . Suppose that ( 3,7 ) E O * Q and proceed as in Step 3. Then ...
... Suppose ( t , x ) E O * Q . Then proceed exactly as in Step 2 to obtain 1 ( 9.20 ) Þ , ( † , ñ ; 3 , ÿ ) ≤ 1⁄2mv ( 2 ( K1 + 1 ) ( ɛ + 6 ) ) + sup [ W − V ] . მ * Q 3 ' . Suppose that ( 3,7 ) E O * Q and proceed as in Step 3. Then ...
Page 351
... Suppose r2 > 1. Set P1 = { ( x , y ) € 0 : _ = x + y € ( r1 , 1 ) } , x P2 = { ( x , y ) € 0 : _ , € ( 1 , ra ) } . { ( x , y ) x + y Let W1 and W2 be the classical solutions of 1 ——œ2y2Wï‚‚v ( x , y ) + h ( x , y ) = 0 , in Pi , i , yy ...
... Suppose r2 > 1. Set P1 = { ( x , y ) € 0 : _ = x + y € ( r1 , 1 ) } , x P2 = { ( x , y ) € 0 : _ , € ( 1 , ra ) } . { ( x , y ) x + y Let W1 and W2 be the classical solutions of 1 ——œ2y2Wï‚‚v ( x , y ) + h ( x , y ) = 0 , in Pi , i , yy ...
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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 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