브라운다리 근사를 통한 확산모형의 우도 근사법 = Likelihood Approximation of Diffusion Models through Approximating Brownian Bridge

국문 초록 (Abstract)

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.

다국어 초록 (Multilingual 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.

참고문헌 (Reference)

1 최영수, "확산모형에 대한 일반화적률추정법의 개선" 한국통계학회 26 (26): 767-783, 2013

2 이윤동, "확산모형에 대한 누율생성함수의 근사와 가우도 추정법" 한국경영과학회 38 (38): 201-216, 2013

3 이은경, "확산모형 전이확률밀도의 급수근사법과 그 계수" 한국통계학회 23 (23): 383-392, 2010

4 A¨ıt-Sahalia, Y., "Transition densities for interest rate and other nonlinear diffusions" 54 : 1361-1395, 1999

5 Lee, Y. D., "The delta expansion for the transition density of diffusion models," 178 : 694-705, 2014

6 Rogers, L., "Smooth transitional densities for one-dimensional diffusions" 17 : 157-161, 1985

7 Hurn, A., "Seeing the wood for the trees : A critical evaluation of methods to estimate the parameters of stochastic differential equations" 5 : 390-455, 2007

8 Chang, J, "On the approximate maximum likelihood estimation for diffusion processes" 39 : 2820-2851, 2011

9 Durham, G., "Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes" 2001

10 A¨ıt-Sahalia, Y., "Maximum-likelihood estimation of discretely-sampled diffusions : A closed-form approximation approach" 70 : 223-262, 2002

11 Li, C., "Maximum-likelihood estimation for diffusion processes via closed-form density expansions" 41 : 1350-1380, 2013

12 Egorov, A. V., "Maximum likelihood estimation of time inhomogeneous diffusions" 114 : 107-139, 2003

13 Eraker, B., "MCMC analysis of diffusion models with application to finance" 19 : 177-191, 2001

14 Elerian, O., "Likelihood inference for discretely observed nonlinear diffusions" 69 : 959-993, 2001

15 Beskos, A., "Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes" 68 : 333-383, 2006

16 Dacunha-Castelle, D., "Estimation of the coefficients of a diffusion from discrete observations" 19 : 263-284, 1986

17 Kessler, M., "Estimation of an ergodic diffusion from discrete observations" 24 : 211-229, 1997

18 Shoji, I., "Estimation for nonlinear stochastic differential equations by a local linearization method" 16 : 733-752, 1998

19 Filipovi´c, D., "Density approximations for multivariate affine jumpdiffusion processes" 176 : 93-111, 2013

20 A¨ıt-Sahalia, Y., "Closed-form likelihood expansions for multivariate diffusions" 36 : 906-937, 2008

21 Chan, K. C., "An empirical comparison of alternative models of the short-term interest rate" 47 : 1209-1227, 1992

22 Pederson, A. R., "A New Approach to maximum likelihood estimation for stochastic differential equations based on discrete observations" 22 : 55-71, 1995