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Free boundary problems in Mathematical Finance : 금융 수학에서의 자유 경계 문제

DC Field Value Language
dc.contributor.advisor강명주-
dc.contributor.author박경현-
dc.date.accessioned2021-11-30T04:50:18Z-
dc.date.available2021-11-30T04:50:18Z-
dc.date.issued2021-02-
dc.identifier.other000000163908-
dc.identifier.urihttps://hdl.handle.net/10371/176028-
dc.identifier.urihttps://dcollection.snu.ac.kr/common/orgView/000000163908ko_KR
dc.description학위논문 (박사) -- 서울대학교 대학원 : 자연과학대학 수리과학부, 2021. 2. 강명주.-
dc.description.abstractThis thesis focuses on theoretically investigating the free boundary problems which arise from two classical problems in mathematical finance: portfolio selection and optimal contract of principal-agent theory. In these two problems, the mathematical structure is not represented as a solely optimal stopping time problem but is formulated in a mixture form in which optimal stopping time is combined with stochastic control or singular control. In this structure, the characterization of the free boundary for optimal stopping time is subtle and the verification theorem of the optimality for the candidate control and stopping time is difficult to prove. To overcome these difficulties, we utilize the dual/martingale method, and then we analyze the variational inequality arising from the dual problem and formally provide the derivation of the solution to the variational inequality. In the thesis, we make a technical contribution by fully characterizing the free boundary and providing the duality and verification theorems. Based on the analytic results, we seek to gain further insight into two classical problems under several realistic model setups.-
dc.description.abstract본 학위 논문은 금융수학의 대표적인 두 가지 문제인 포트폴리오 이론, 최적 계약 이론에서 일어나는 자유 경계 문제 (free boundary problem) 에 대한 해석적 분석을 다룬다. 두 문제에서의 수학적 구조는 단순히 최적 정지 시간 (optimal stopping time) 문제로 나타나는 것이 아니라 확률적 제어 (stochastic control) 혹은 단일 제어 (singular control) 와 함께 결합된 형태로 귀착된다. 이러한 구조는 최적 정지 시간에 대한 자유 경계의 분석이 미묘하여 확률적 제어 및 정지 시간에 대한 최적성 (optimality) 증명이 어렵다. 본 논문은 이중/마팅게일 (dual/martingale) 이론을 활용하여 이중 문제 (dual problem) 에서 발생하는 변분 부등식 (variational inequality) 의 성질을 분석하고 그에 대한 해의 도출과정을 제시한다. 이를 통해 자유 경계를 완전히 특징화하고 이중성 및 최적성 증명을 함께 제시한다. 이러한 해석적 결과에 기반 하여 여러 가지 현실적인 모델에서의 포트폴리오 및 최적 계약 이론 대한 경제학적 의미를 제공한다.-
dc.description.tableofcontentsContents
Abstract i
1 Introduction 1
2 Optimal stopping in portfolio selection 7
2.1 Retirement decision in consumption-leisure and investment
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Model and optimization problem . . . . . . . . . . . . 7
2.1.2 Main results: optimal retirement time and duality theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.3 Variational inequality: ODE analysis . . . . . . . . . . 15
2.1.4 Proof of Proposition 2.1.7 . . . . . . . . . . . . . . . . 17
2.1.5 Proof of Theorem 2.1.8 . . . . . . . . . . . . . . . . . 21
2.2 Risk preference change in consumption and investment problem 24
2.2.1 Model and optimization problem . . . . . . . . . . . . 24
2.2.2 Main results: optimal preference change time and duality theorem . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Variational inequality: PDE analysis . . . . . . . . . . 30
2.2.4 Proof of Theorem 2.2.4 . . . . . . . . . . . . . . . . . 32
2.2.5 Proof of Theorem 2.2.5 . . . . . . . . . . . . . . . . . 35
3 Singular control in portfolio selection 38
3.1 Consumption Ratcheting . . . . . . . . . . . . . . . . . . . . . 38
3.1.1 Model and optimization problem . . . . . . . . . . . . 39
3.1.2 Singular control and optimal stopping . . . . . . . . . 41
3.1.3 Main results: optimal consumption with ratcheting and retirement decision . . . . . . . . . . . . . . . . . 48
3.1.4 Proof of lemmas . . . . . . . . . . . . . . . . . . . . . 50
3.1.5 Proof of Theorem 3.1.10 . . . . . . . . . . . . . . . . . 56
3.1.6 Proof of Theorem 3.1.18 . . . . . . . . . . . . . . . . . 59
3.2 Drawdown Constraint on Consumption . . . . . . . . . . . . . 64
3.2.1 Model and optimization problem . . . . . . . . . . . . 65
3.2.2 Two dimensional singular control problem . . . . . . . 67
3.2.3 Main Results: optimal maximum process and optimal
strategies . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.4 Proof of lemmas . . . . . . . . . . . . . . . . . . . . . 77
3.2.5 Proof of Proposition 3.2.15 . . . . . . . . . . . . . . . 87
3.2.6 Proof of Theorem 3.2.17 . . . . . . . . . . . . . . . . . 90
3.2.7 Proof of Theorem 3.2.18 . . . . . . . . . . . . . . . . . 96
4 Singular control in contract theory 99
4.1 Limited Commitment in Finite time horizon . . . . . . . . . . 99
4.1.1 Model and optimization problem . . . . . . . . . . . . 99
4.1.2 Singular control in nite time-horizon . . . . . . . . . 103
4.1.3 Main results: optimal contract with limited commitment 114
4.1.4 Proof of lemmas . . . . . . . . . . . . . . . . . . . . . 118
4.1.5 Proof of Proposition 4.1.16 . . . . . . . . . . . . . . . 124
4.1.6 Proof of Proposition 4.1.18 . . . . . . . . . . . . . . . 125
4.1.7 Proof of Theorem 4.1.19 . . . . . . . . . . . . . . . . . 126
4.1.8 Proof of Proposition 4.1.21 . . . . . . . . . . . . . . . 131
4.1.9 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . 133
Abstract (in Korean) 141
Acknowledgement (in Korean) 142		-
dc.format.extentiii, 143-
dc.language.isoeng-
dc.publisher서울대학교 대학원-
dc.subjectFree boundary problem-
dc.subjectVariational inequality-
dc.subjectOptimal stopping problem-
dc.subjectSingular control problem-
dc.subjectStochastic control problem-
dc.subjectDuality relationship-
dc.subject자유 경계 문제-
dc.subject변분 부등식-
dc.subject최적 정지 시간-
dc.subject단일 제어-
dc.subject확률적 제어-
dc.subject이중 문제-
dc.subject.ddc510-
dc.titleFree boundary problems in Mathematical Finance-
dc.title.alternative금융 수학에서의 자유 경계 문제-
dc.typeThesis-
dc.typeDissertation-
dc.contributor.AlternativeAuthorKyunghyun Park-
dc.contributor.department자연과학대학 수리과학부-
dc.description.degreeDoctor-
dc.date.awarded2021-02-
dc.identifier.uciI804:11032-000000163908-
dc.identifier.holdings000000000044▲000000000050▲000000163908▲-
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