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free boundary and its applications in finance math : 자유경계와 금융수학에서의 응용

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dc.contributor.advisor이기암-
dc.contributor.author정임용-
dc.date.accessioned2017-07-19T08:58:22Z-
dc.date.available2017-07-19T08:58:22Z-
dc.date.issued2013-02-
dc.identifier.other000000008554-
dc.identifier.urihttps://hdl.handle.net/10371/131461-
dc.description학위논문 (석사)-- 서울대학교 대학원 : 수리과학부, 2013. 2. 이기암.-
dc.description.abstractThis paper is a book which is written based on the contents of references [1] and [2]. First of all, we study the obstacle problem, and define the free boundary of the obstacle problem in chapter 1. In chapter 2, we prove that the existence of solutions of the obstacle problem as three methods-
dc.description.abstractenergy minimizer, singular perturbation method, and viscosity method. And in chapter 3, we study that the $C^{1,1}$ regularity of solutions of the obstacle problem and in the free boundary using the AFC monotonicity formula. Last of all, we show that the regularity of solutions of the American put option in finance math using the previous chapters.-
dc.description.tableofcontents1 Obstacle Problems 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Obstacle Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Free Boundary in the Obstacle Problem . . . . . . . . . . . . . 2
1.4 The Classical Obstacle Problems . . . . . . . . . . . . . . . . . 4
1.5 Model Problems A, B, C . . . . . . . . . . . . . . . . . . . . . . 7
2 Existence of Solution of the Obstacle Problems 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Method 1 (The Existence of Energy Minimizer) . . . . . . . . . 9
2.3 Method 2 (Singular Perturbation Method) . . . . . . . . . . . . 16
2.4 Method 3 (Viscosity Method) . . . . . . . . . . . . . . . . . . . 20
3 Optimal Regularity 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Optimal Regularity in the Classical Obstacle Problem . . . . . . 23
3.3 ACF Monotonicity Formula and Generalizations . . . . . . . . . 26
3.3.1 Harmonic Functions . . . . . . . . . . . . . . . . . . . . 26
3.3.2 ACF Monotonicity Formula . . . . . . . . . . . . . . . . 28
3.3.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Optimal Regularity in Obstacle-Type Problems. . . . . . . . . . 34
3.5 Optimal Regularity up to the Boundary . . . . . . . . . . . . . 37
4 Applications of Free Boundary in Finance Math 42
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 It^o Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4 Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5.2 American Option . . . . . . . . . . . . . . . . . . . . . . 57
4.5.3 Parabolic Notation . . . . . . . . . . . . . . . . . . . . . 58
4.5.4 Mathematical Formulation . . . . . . . . . . . . . . . . . 59
4.5.5 Regularity of the American Put Options . . . . . . . . . 60		-
dc.formatapplication/pdf-
dc.format.extent1164397 bytes-
dc.format.mediumapplication/pdf-
dc.language.isoen-
dc.publisher서울대학교 대학원-
dc.subjectfree boundary-
dc.subject.ddc510-
dc.titlefree boundary and its applications in finance math-
dc.title.alternative자유경계와 금융수학에서의 응용-
dc.typeThesis-
dc.contributor.AlternativeAuthorIm-Yong Jung-
dc.description.degreeMaster-
dc.citation.pages63-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2013-02-
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