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Controlled Markov Diffusions in Rn
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apply assume assumptions boundary condition boundary data bounded brownian motion calculus of variations Chapter classical solution consider continuous function continuous on Q controlled Markov diffusion convergence convex Corollary cylindrical region define definition denote dynamic programming equation dynamic programming principle Dynkin formula Example exists exit finite first-order formula given Hamilton-Jacobi equations Hence HJB equation holds implies inequality ip(x Ishii lateral boundary Lebesgue measure Lemma linear Lipschitz continuous Markov chain Markov control policy Markov processes maximum principle minimizing Moreover nonlinear obtain optimal control optimal control problem optimal Markov control partial derivatives partial differential equation progressively measurable proof of Theorem prove reference probability system Remark result satisfies second-order Section semiconvex semigroup stochastic control stochastic control problem stochastic differential equation Suppose t,ti test function Theorem 5.1 tion to,ti uniformly continuous value function Verification Theorem viscosity solution viscosity subsolution viscosity supersolution yields zero