Local regularity criteria of the Navier-Stokes equations with slip boundary conditions

배형옥,강경근,김명현 대한수학회 2016 대한수학회지 Vol.53 No.3

We present regularity conditions for suitable weak solutions of the Navier-Stokes equations with slip boundary data near the curved boundary. To be more precise, we prove that suitable weak solutions become regular in a neighborhood boundary points, provided the scaled mixed norm $L^{p,q}_{x,t}$ with $3/p+2/q=2$, $1\leq q<\infty$ is sufficiently small in the neighborhood.

Helmholtz decomposition and semigroup theory to the fluid around a moving body

배형옥 대한수학회 2020 대한수학회보 Vol.57 No.3

To understand the interaction of a fluid and a rigid body, we use the concept of $B$-evolution. Then in a similar way to the usual Navier-Stokes system, we obtain a Helmholtz type decomposition. Using $B$-evolution theory and the decomposition, we work on the semigroup to analyze the linear part of the system.

Analyticity for the Stokes Operator in Besov spaces

배형옥 대한수학회 2003 대한수학회지 Vol.40 No.6

We first show the analyticity of Stokes operator in Besov spaces $B^a_{p,q}(\bR^n_+)$. Then, we estimate the asymptotic behavior of the Stokes solutions. We also show the Hodge decomposition.

Existence of strong mild solution of the Navier-Stokes equations in the half space with nondecaying initial data

배형옥,진범자 대한수학회 2012 대한수학회지 Vol.49 No.1

We construct a mild solutions of the Navier-Stokes equations in half spaces for nondecaying initial velocities. We also obtain the uniform bound of the velocity field and its derivatives.

Time stepwise local volatility

배형옥,임현철 대한수학회 2022 대한수학회보 Vol.59 No.2

We propose a path integral method to construct a time stepwise local volatility for the stock index market under Dupire's model. Our method is focused on the pricing with the Monte Carlo Method (MCM). We solve the problem of randomness of MCM by applying numerical integration. We reconstruct this task as a matrix equation. Our method provides the analytic Jacobian and Hessian required by the nonlinear optimization solver, resulting in stable and fast calculations.

Temporal and spatial decay rates of Navier-Stokes solutions in exterior domains

배형옥,진범자 대한수학회 2007 대한수학회보 Vol.44 No.3

We obtain spatial-temporal decay rates of weak solutions ofincompressible ows in exterior domains. When a domain has a boundary,the pressure term yields diculties since we do not have enough infor-mation on the pressure term near the boundary. For our calculations weprovide an idea which does not require any pressure information. Wealso estimated the spatial and temporal asymptotic behavior for strongsolutions.

박판 스플라인 함수를 이용한 내재변동성 곡면 생성 방법 연구

임현철,배형옥 한국금융공학회 2019 금융공학연구 Vol.18 No.4

In this article, we work on the construction of an implied volatility surface by using radial basis functions (RBFs) based on implied volatility data in equity index markets including KOSPI200, HSCEI, Eurostoxx50. Construction of the implied volatility surface is the ‘missing data problem’ in Statistics. The choice of a 2 dimensional interpolation function for surface construction and the data filtering are important. To solve such problems in Statistics, RBF is used popularly. We first explain multi regression method known for practitioner’s implied volatility surface, and suggest an estimation method using RBF. In particular, we review merits and demerits of the Gaussian RBF, multiquadric RBF, and thin plate spline(TPS) RBF, and study concrete methods for the construction of implied volatility surfaces using these RBFs. After finding volatility data which are seriously different from their neighbors, we remove them. To fix an incomplete shape of the surface caused by missing data, we provide an adapted fitting procedure using the leave one out cross validation (LOOCV) method. Finally, using those we provide an optimal approximation algorithm to the implied volatility surface. In the construction of the implied volatility surface, the most important issue is the inference of missing data. In this work, we show that it is most efficient to estimate the implied volatility from the variance surface obtain by using TPS approximation function. Another important issue in the implied volatility construction is the satisfaction of the no-arbitrage condition (NAC) by volatility data. By applying the quadratic programming with NAC to the TPS-smoothed implied volatility, we solve this problem. 본 논문은 KOSPI200 지수 시장의 내재변동성 데이터를 중심으로 방사기저함수(Radial Basis Function, 이하 RBF)를 이용한 내재변동성 곡면생성을 연구한다. 내재변동성 곡면의 생성은 통계학에서 ‘누락된 데이터 문제’(Missing Data Problem)에 해당한다. 곡면을 근사시키는 2차원 보간 함수의 선택과 데이터의 필터링 방법이 중요하다. 통계학에서 이 문제의 해결 방법으로 RBF가 이용되어왔다. 우리는 먼저 시장에서 실무자의 내재변동성 곡면으로 잘 알려진 다중 회귀분석을 이용한 방법을 소개하고, 이어서 RBF를 이용한 추정 방법을 제시한다. 특히, 가우시안 RBF, 멀티쿼드릭(Multiquadric) RBF 그리고 박판스플라인(Thin Plate Spline, 이하 TPS) RBF의 장단점을 살펴보고, 이들을 사용한 내재변동성 곡면의 구체적인 생성 방법을 연구한다. 비정상적으로 주변의 값과 차이가 나는 변동성 데이터를 찾아 제거하고, 데이터의 누락 때문에 발생하는 불완전한 곡면의 모양을 바로잡기 위하여 데이터 소거에 의한 교차검증법을 응용한 적응적 적합과정을 제시한다. 최종적으로 이들을 사용하여 내재변동성 곡면을 근사하는 최적 알고리즘을 제시한다. 내재변동성 곡면 생성에서 가장 중요한 이슈는 누락 데이터의 추정일 것이다. 본 연구에서 TPS 근사함수를 이용하여 생성된 분산곡면으로 내재변동성을 추정하는 방식이 가장 우수한 결과임을 보인다. 내재변동성 곡면생성의 또 다른 주요 문제는 변동성 데이터의 무위험 차익거래 불가능 조건의 충족이다. 이를 위해, 먼저 TPS 근사에 의한 분산곡면을 구하고, 이로부터 변동성 곡면을 생성한다. 그리고 이를 이용하여 얻은 콜 옵션 가격 곡선들과 관측값과의 오차를 최소화하는 최적화 문제로 바꾸고, 차익거래 불가능 제약조건을 추가하여 이 문제를 해결한다.

A survey on American options: old approaches and new trends

안세륭,배형옥,구형건,이기정 대한수학회 2011 대한수학회보 Vol.48 No.4

This is a survey on American options. An American option allows its owner the privilege of early exercise, whereas a European option can be exercised only at expiration. Because of this early exercise privilege American option pricing involves an optimal stopping problem; the price of an American option is given as a free boundary value problem associated with a Black-Scholes type partial differential equation. Up until now there is no simple closed-form solution to the problem, but there have been a variety of approaches which contribute to the understanding of the properties of the price and the early exercise boundary. These approaches typically provide numerical or approximate analytic methods to find the price and the boundary. Topics included in this survey are early approaches(trees, finite difference schemes,and quasi-analytic methods), an analytic method of lines and randomization, a homotopy method, analytic approximation of early exercise boundaries, Monte Carlo methods, and relatively recent topics such as model uncertainty, backward stochastic differential equations, and real options. We also provide open problems whose answers are expected to contribute to American option pricing.

An Unconditionally Gradient Stable Adaptive Mesh Refinement for the Cahn-Hilliard Equation

김준석,배형옥 한국물리학회 2008 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.53 No.2

We consider a numerical method, the so-called an unconditionally gradient stable adaptive mesh refinement scheme, for solving the Cahn-Hilliard equation representing a model of phase separation in a binary mixture. The continuous problem has a decreasing total energy. We show the same property for the corresponding discrete problem by using eigenvalues of the Hessian matrix of the energy functional. An unconditionally gradient stable time discretization is used to remove the high-order time-step constraints. An adaptive mesh refinement is used to highly resolve narrow interfacial layers.

ELS의 가격 결정을 하기 위한 탈출확률이 연계된 FDM

김용식 ( Yong Sik Kim ),배형옥 ( Hyeong Ohk Bae ),노현석 ( Hyun Seok Roh ) 한국파생상품학회(구 한국선물학회) 2011 선물연구 Vol.19 No.4

주가연계증권의 가격을 결정하는 새로운 유한차분법을 제안한다. 우리의 알고 리듬은 유한차분법에 탈출확률을 사용한다. 수치적 예를 통해서 우리의 알고리듬 이 기존의 유한차분법보다 더 정확함을 보인다. We propose a new numerical algorithm for pricing the Equity linked Security which is a financial derivative. Our algorithm utilizes FDM with the Exit-Probability. Through numerical examples, we validate that our algorithm is more accurate than the conventional method.