Guo, Yang
(2020)
*A class of two-dimensional strong Markov processes and a continuous-time principal-agent problem with costly renegotiation.*
PhD thesis, London School of Economics and Political Science.

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## Abstract

In the first part of this thesis, we study a continuous-time principal-agent model without precommitment. The agent runs an economic project on behalf of the principal. To this end, the agents apply effort that is costly to them and unobservable by the principal. In return, the agent receives compensation from the principal. The agent is strictly risk-averse and their objective is to maximize their expected utility of compensation minus their expected disutility of effort. The principal is riskneutral and their objective is to maximize their expected utility of income generated by the project minus the compensation paid to the agent. The optimal contract should maximize the principal's expected utility subject to the constraint that it should induce a contractual environment in which it is optimal for the agent to always be truthful. To exclude the requirement of precommitment, the contract allows for costly renegotiation. The optimal contract is fully determined by deriving the explicit solution to a suitable control problem that combines regular stochastic control with singular stochastic control. In the second part of this thesis, we present a study of two-dimensional strong Markov processes whose second component is the running maximum of the first one. The study of such processes has been motivated by recent development in financial mathematics, such as the introduction and the analysis of the π and the watermark options. We first introduce a suitable concept of regularity that generalises the standard regularity assumption of the theory of one-dimensional diffusions/strong Markov process to the two-dimensional setting that we study. Next, we characterise the class of scale functions, namely, the functions that yield local martingales when composed with a Markov process in the family we study. We then show that such a process in natural scale can be represented as a time-changed Brownian motion and its running maximum. Finally, we present a study of associated r-invariant functions. Our analysis makes heavy use of the standard theory of one-dimensional diffusions. The main difficulties arise from the behaviour of the processes on the diagonal where their two components coincide.

Item Type: | Thesis (PhD) |
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Additional Information: | © 2020 Yang Guo |

Library of Congress subject classification: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics |

Supervisor: | Zervos, Mihail |

URI: | http://etheses.lse.ac.uk/id/eprint/4263 |

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