Analytical representations for the basic affine jump diffusion

The Basic Affine Jump Diffusion (BAJD) process is widely used in financial modeling. In this paper, we develop an exact analytical representation for its transition density in terms of a series expansion that is uniformly-absolutely convergent on compacts. Computationally, our formula can be evaluated to high level of accuracy by easily adding new terms which are given explicitly. Furthermore, it can be easily generalized to give an analytical expression for the transition density of the subordinate BAJD process which is more realistic than the BAJD process, while existing approaches cannot.