TY - JOUR

T1 - Recent development of closed-form approximate (Log-)transition probability density functions of diffusion processes

AU - Choi, Seungmoon

N1 - Publisher Copyright:
© 2020, Korean Econometric Society. All rights reserved.

PY - 2020/3

Y1 - 2020/3

N2 - 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 es-timate 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, expla-nations 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.

AB - 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 es-timate 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, expla-nations 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.

KW - (Log-)Transition Probability Density

KW - Time-homogeneous Diffusion

KW - Time-inhomogeneous Diffusion

UR - http://www.scopus.com/inward/record.url?scp=85083177724&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85083177724

SN - 1229-2893

VL - 31

SP - 97

EP - 152

JO - Journal of Economic Theory and Econometrics

JF - Journal of Economic Theory and Econometrics

IS - 1

ER -