## Abstract

The present paper studies the stochastic maximum principle in singular optimal control, where the state is governed by a stochastic differential equation with nonsmooth coefficients, allowing both classical control and singular control. The proof of the main result is based on the approximation of the initial problem, by a sequence of control problems with smooth coefficients. We, then apply Ekeland's variational principle for this approximating sequence of control problems, in order to establish necessary conditions satisfied by a sequence of near optimal controls. Finally, we prove the convergence of the scheme, using Krylov's inequality in the nondegenerate case and the Bouleau-Hirsch flow property in the degenerate one. The adjoint process obtained is given by means of distributional derivatives of the coefficients.

Original language | English (US) |
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Pages (from-to) | 479-494 |

Number of pages | 16 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 355 |

Issue number | 2 |

DOIs | |

State | Published - Jul 15 2009 |

## Keywords

- Adjoint process
- Distributional derivative
- Maximum principle
- Singular control
- Stochastic control
- Stochastic differential equation
- Variational principle

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics