The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
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Likelihood Approximation of Diffusion Models through Approximating Brownian Bridge

브라운다리 근사를 통한 확산모형의 우도 근사법

  • Lee, Eun-kyung (Department of Statistics, Ewha Womans University) ;
  • Sim, Songyong (Department of Finance & Information Statistics Hallym University) ;
  • Lee, Yoon Dong (Sogang Business School, Sogang University)
  • 이은경 (이화여자대학교 통계학과) ;
  • 심송용 (한림대학교 금융정보통계학과) ;
  • 이윤동 (서강대학교 경영학부)
  • Received : 2015.06.02
  • Accepted : 2015.10.01
  • Published : 2015.10.31
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Abstract

Diffusion is a mathematical tool to explain the fluctuation of financial assets and the movement of particles in a micro time scale. There are ongoing statistical trials to develop an estimation method for diffusion models based on likelihood. When we estimate diffusion models by applying the maximum likelihood estimation method on data observed at discrete time points, we need to know the transition density of the diffusion. In order to approximate the transition densities of diffusion models, we suggests the method to approximate the path integral of the random process with normal random variables, and compare the numerical properties of the method with other approximation methods.

확산모형은 입자의 운동현상과 금융자산의 미시적 가격변동을 설명하기 위하여 사용되는 수리적 모형이다. 확산모형의 추정방법에 관한 논의는 다양한 분야에서 이루어져 왔다. 통계학적 관점에서 우도적 방법에 기반한 확산모형의 추정방법을 개발하려는 시도가 계속되어 왔다. 이산시간 간격으로 관측된 자료를 이용하여 확산모형을 추정할 때 최대우도 추정법을 적용하기 위해서는 확산모형에 대한 전이확률 밀도함수를 구해야 한다. 본 연구에서는 확산모형의 전이확률밀도를 근사하기 위하여, 정규분포를 따르는 확률변수를 이용하여 브라운다리 확률과정에 대한 경로적분을 대체하는 방법을 제안하고, 그 수치적 성질을 다른 방법들과 비교한다.

Keywords

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