Generalized Solution in Singular Stochastic Control: The Nondegenerate Problem

Applied Mathematics and Optimization, 1992

Posted: 29 Aug 2009

See all articles by Steven H. Zhu

Steven H. Zhu

Banc of America Merrill Lynch; Morgan Stanley

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

Suggested Citation

Zhu, Steven H., Generalized Solution in Singular Stochastic Control: The Nondegenerate Problem (May 6, 1992). Applied Mathematics and Optimization, 1992. Available at SSRN:

Steven H. Zhu (Contact Author)

Banc of America Merrill Lynch ( email )

Bank of America Plaza
335 Madison Ave, 5th Floor
New York, NY 10017
United States
646-855-1853 (Phone)


Morgan Stanley ( email )

1585 Broadway
New York, NY 10036
United States

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