Positive Harris recurrence and exponential ergodicity of the basic affine jump-diffusion

TL;DR

This paper analyzes the basic affine jump-diffusion model, deriving its transition densities, proving its positive Harris recurrence and exponential ergodicity, and characterizing its invariant measure with a closed-form density.

Contribution

It establishes the recurrence and ergodic properties of the BAJD and provides an explicit formula for its invariant measure's density.

Findings

Transition densities of BAJD derived

Proved positive Harris recurrence and exponential ergodicity

Invariant measure density explicitly characterized

Abstract

In this paper we find the transition densities of the basic affine jump-diffusion (BAJD), which is introduced by Duffie and Garleanu [D. Duffie and N. Garleanu, Risk and valuation of collateralized debt obligations, Financial Analysts Journal 57(1) (2001), pp. 41--59] as an extension of the CIR model with jumps. We prove the positive Harris recurrence and exponential ergodicity of the BAJD. Furthermore we prove that the unique invariant probability measure $\pi$ of the BAJD is absolutely continuous with respect to the Lebesgue measure and we also derive a closed form formula for the density function of $\pi$.