## Two Problems in Control Theory and Applications to Economics

Tacconi, Elisa (2011) *Two Problems in Control Theory and Applications to Economics.* Tesi di Dottorato, LUISS Guido Carli, Department of Economics and Finance > PhD Program in Mathematical Methods for Economics, Business, Finance and Insurance, tutor: Salvatore Federico and Fausto Gozzi and Marco Papi, p. 122. [Doctoral Thesis]

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## Abstract/Index

The thesis is composed by two dierent parts, which are not related each other. The rst part is devoted to study a class of optimal control problems, where the state equation is an ordinary dierential equation with delay in the control variable. This class of problems arises in economic applications, in particular in optimal advertising problems. The control problem is embedded in a suitable Hilbert space and the associated Hamilton-Jacobi-Bellman (HJB) equation considered in this space. It is proved that the value function is continuous with respect to a weak norm and that it solves in the viscosity sense the associated HJB equation. The main result is the proof of a directional C1-regularity result for the value function. This result represents the starting point to dene a feedback map in classical sense going towards a verication theorem yielding optimal feedback controls for the problem. In the second part of the thesis, the techniques of the Malliavin Calculus are applied to a stochastic dierential equation whose coecients depend on a control process, in particular in the special case of Markovian controls. It is calculated the stochastic derivative of the stochastic dierential equation and it is proved that the Malliavin matrix is strictly positive, assuring the results of existence and regularity of densities for the controlled process.

## References

Bibliografia: pp. 117-122.

Item Type: | Doctoral Thesis (PhD) |
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Research documents and activity classification: | LUISS PhD Thesis |

Divisions: | Department of Economics and Finance > PhD Program in Mathematical Methods for Economics, Business, Finance and Insurance |

Thesis Advisor: | Federico, Salvatore and Gozzi, Fausto and Papi, Marco |

Additional Information: | Dottorato di Ricerca in Metodi matematici per l'economia, l'azienda, la finanza e le assicurazioni (XXII ciclo), LUISS Guido Carli, Roma, 2011. Relatore: Prof. Salvatore Federico, Prof. Fausto Gozzi, Prof. Marco Papi. |

Uncontrolled Keywords: | Hamilton Jacobi Bellman. Control theory. Delay equations. Viscosity solution. Regularity. |

MIUR Scientific Area: | Area 13 - Economics and Statistics > SECS-S/06 Mathematics for Economics, Actuarial Studies and Finance |

Deposited By: | Chiara Annulli (admin) |

Deposited On: | 03 Aug 2011 12:53 |

Last Modified: | 28 May 2013 23:05 |

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