Since Aït-Sahalia (2002)’s seminal work on obtaining a closed-form approximate transition probability density function (ATPDF) of a univariate time-homogeneous di¤usion process, many researchers employed his idea to extend it to more general cases. Those include ATPDFs of univariate time-inhomogeneous di¤usions (Egorov, Li, and Xu (2003)), likelihood expansions of multivariate time-homogeneous di¤usions (Aït-Sahalia (2008)), ATPDFs of multivariate time-homogeneous jump diffusions (Yu (2007)), likelihood expansions of multivariate time-inhomogeneous di¤usions (Choi (2013)), and ATPDFs of multivariate di¤usions (Choi (2015)). This article considers getting an explicit form of an ATPDF for multivariate time-inhomogeneous jump di¤usion processes which encompass all of the aforementioned models. Using the Kolmogorov partial di¤erential equation (PDE), we …rst …nd PDEs of the coefficients of the ATPDF. These PDEs can be solved and we can get the formulas to retrieve all coefficients of the ATPDF successively when the multivariate time-inhomogeneous jump di¤usion is reducible. However, if it is not reducible we can no longer solve the PDEs. In this case, Taylor-expanding the coefficients and matching the same orders in the PDEs yield an ATPDF for the time-homogeneousjump diffusion. But in the case of time-inhomogeneous jump di¤usion, the similar indeterminacy problem to Choi (2013) occurs. We prove that all of the generally nonzero indeterminate terms are cancelled out in the TPDF expansion. The ATPDF can be utilized to a variety of areas including maximum likelihood estimation, asset pricing, and Bayesian analysis.​

KEY WORDS: Transition Probability Density Function; Multivariate Time-inhomogeneous jump diffusion
JEL codes: C13, C16, G13