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확산과정의 근사적 (로그)전이확률밀도 함수를 구체적으로 구하는 방법에 대한 최근 연구동향

Recent Development of Closed-form Approximate (Log-)Transition Probability Density Functions of Diffusion Processes

최승문
한국계량경제학회|2020.03|계량경제학보 (Journal of Economic Theory and Econometrics)|제31권 제1호|pp. 97-152|DOI 10.22812/jetem.2020.31.1.004

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참고문헌

키워드

시간-균질 확산과정시간-비균질 확산과정(로그-)전이확률밀도Time-Homogeneous DiffusionTime-Inhomogeneous Diffusion(Log-)Transition Probability Density
주제분류경제학

초록

확산과정 (diffusion)의 전이확률밀도 (transition probability density) 함수 또는 로그-전이확률밀도 함수는 확산과정 모형을 최우추정법 (maximum likelihood estimation)으로 추정할 때 이용될 수 있을뿐 아니라 확산과정 모형의 데이터를 생성할 때 그리고 바탕이 되는 자산의 가격이 확산과정을 따를 때 자산의 가격을 계산할 때 등 여러 모로 매우 유용하게 활용될 수 있다. 그런데 불행히도 손에 꼽을 정도의 몇 가지 확산과정 모형들을 빼고는 확산과정의 전이확률밀도 함수는 보통 알려져 있지 않다. Aït-Sahalia (2002)가 선구적으로 단일변수 시간-균질 확산과정의 전이확률밀도 함수를 근사적으로 그렇지만 구체적인 식으로 구하는 방법을 개발한 것부터 시작되어 지금은 다변수 시간-비균질 점프 확산과정의 전이확률밀도 함수를 구체적인 식으로 근사시키는 방법까지 Choi (2019a), 여러 연구자들이 순차적으로 좀 더 일반적인 확산과정 모형들의 근사적 전이확률밀도 함수나 로그-전이확률밀도 함수를 구하는 방법과 이론들이 확립했다. 이 논문은 확산과정 모형의 근사적 전이확률밀도 함수나 로그-전이확률밀도 함수를 구하기 위해 Aït-Sahalia (2002)의 논문에서 Choi (2013,2015)의 다변수 시간-비균질 확산과정까지 좀 더 일반적인 모형들로 확장하는 과정에서 극복해야 할 문제들이 어떻게 해결됐는 설명하고 있다. 지면의 제약으로 자세한 이론이나 이의 증명을 위한 가정들에 대한 설명은 최대한 줄이고 중요한 결과들을 보여주며 원래 논문들에서 설명하지 않은 암묵적 사실들도 함께 설명하고 있다. 또한 이러한 중심이 되는 연구들에서 파생되거나 이들 연구를 이용한 관련된 논문들도 같이 소개하고 있다.

Abstract

Transition probability density function (TPDF) or log-TPDF of a diffusion is quite useful in many ways. For example, it can be employed not only to estimate a diffusion by the maximum likelihood estimation but also to simulate data from a diffusion or to price an asset when the underlying process follows a diffusion. However, unfortunately, the true TPDF of a diffusion is unknown with a few exceptions in general. Starting from Aït-Sahalia (2002)'s pioneering work on approximate but explicit TPDF of a univariate time-homogeneous diffusion to Choi (2019a)'s recent work on closed-form approximate TPDF of a multivariate time-inhomogeneous jump diffusion, several researchers have subsequently established the way to approximate the TPDFs or log-TPDFs of more general diffusion models. This article explains how people have resolved problems to generalize the method from Aït-Sahalia(2002)'s paper to Choi(2013, 2015)'s multivariate time-inhomogeneous diffusions. Due to space constraints, explanations of detailed theories or assumptions for their proof are reduced to the minimum and we show important results, with tacit facts not described in the original papers. In addition, we also introduce papers derived from and related to those key studies.

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