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# Singular stochastic control and its relations to Dynkin game and entry-exit problems

Type of Publication: | Dissertation |

URI (citable link): | http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-9329 |

Author: | Boetius, Frederik |

Year of publication: | 2001 |

Title in another language: | Singuläre stochastische Kontrolle und ihre Beziehungen zu Dynkin-Spiel- und -Eintritt-Austritt-Problemen |

Summary: |
We consider a bounded variation singular stochastic control problem
with value V, the associated Dynkin game with value u and an associated entry-exit or optimal switching problem. We establish the relation dV/dx=u known from control of Bronwian motion for a general situation with control of a diffusion and a nonlinear cost functional defined as solution to a BSDE. A saddle point for the Dynkin game is given by the pair of first action times of an optimal control. Through an impulse control approximation scheme we construct a solution to the control problem from solutions to the entry-exit problem, and obtain an integral representation for the value V. As a special case we deduce equivalence of monotone control and optimal stopping. In a Markovian setting we characterize the value of the control problem in n dimensions as the largest viscosity solution to a quasilinear Hamilton-Jacobi-Bellman PDE with gradient constraints. Due to the gradient constraints, the latter has no unique solution in general. The methods are from stochastic analysis and include a priori estimates, pathwise construction, comparison theorems for FSDE and BSDE, Ito formula for convex functions and nonlinear Feynman-Kac formulae. Using this approach we can drop the condition of a ``proper'' operator in the HJB PDE and alter the standard path for comparison towards a global argument. |

Summary in another language: |
We consider a bounded variation singular stochastic control problem
with value V, the associated Dynkin game with value u and an associated entry-exit or optimal switching problem. We establish the relation dV/dx=u known from control of Bronwian motion for a general situation with control of a diffusion and a nonlinear cost functional defined as solution to a BSDE. A saddle point for the Dynkin game is given by the pair of first action times of an optimal control. Through an impulse control approximation scheme we construct a solution to the control problem from solutions to the entry-exit problem, and obtain an integral representation for the value V. As a special case we deduce equivalence of monotone control and optimal stopping. In a Markovian setting we characterize the value of the control problem in n dimensions as the largest viscosity solution to a quasilinear Hamilton-Jacobi-Bellman PDE with gradient constraints. Due to the gradient constraints, the latter has no unique solution in general. The methods are from stochastic analysis and include a priori estimates, pathwise construction, comparison theorems for FSDE and BSDE, Ito formula for convex functions and nonlinear Feynman-Kac formulae. Using this approach we can drop the condition of a ``proper'' operator in the HJB PDE and alter the standard path for comparison towards a global argument. |

Examination date (for dissertations): | Nov 21, 2002 |

Dissertation note: | Doctoral dissertation, University of Konstanz |

MSC Classification: | 93E20; 91A15; 60H10; 60G40; 49L25 |

Subject (DDC): | 510 Mathematics |

Controlled Keywords (GND): | Stochastische optimale Kontrolle, Stochastische Differentialgleichung, Optimales Stoppen, Viskositätslösung, Stochastisches Spiel |

Keywords: | Stochastische Rückwärtsdifferentialgleichung, singuläre Kontrolle, sequentielles Stoppen, Beschränkung des Gradienten, pfadweise Konstruktion, Backward stochastic differential equation, singular control, sequential stopping, gradient constraint, pathwise construction |

Link to License: | In Copyright |

Checksum:
MD5:4859770f041c259eddf0208ed918e495

BOETIUS, Frederik, 2001. Singular stochastic control and its relations to Dynkin game and entry-exit problems [Dissertation]. Konstanz: University of Konstanz

@phdthesis{Boetius2001Singu-584, title={Singular stochastic control and its relations to Dynkin game and entry-exit problems}, year={2001}, author={Boetius, Frederik}, address={Konstanz}, school={Universität Konstanz} }

fbd_pub2.pdf | 764 |