Generalized Solution in Singular Stochastic Control: The Nondegenerate Problem
Applied Mathematics and Optimization, 1992
Posted: 29 Aug 2009
Date Written: May 6, 1992
This paper is concerned with singular stochastic control for non-degenerate problems. It generalizes the previous work in that the model equation is nonlinear and the cost function need not be convex. The associated dynamic programming equation takes the form of variational inequalities. By combining the principle of dynamic programming and the method of penalization, we show that the value function is characterized as a unique generalized (Sobolev) solution which satisfies the dynamic programming variational inequality in almost everywhere. The approximation for our singular control problem is given in terms of a family of penalized control problems. As a result of such a penalization, we obtain that the value function is also the minimum cost available when only the admissible pairs with uniformly Lipschitz controls are admitted in our cost criterion.
Keywords: Singular stochastic control,Dynamic programming,Variational inequality,Penalization method,Generalized solution,Nearly optimal control
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