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

Abstract

Abstract. In this paper we find the transition densities of the basic affine jump-diffusion (BAJD), which is introduced by Duffie and Gârleanu [D. Duffie and N. Gârleanu, 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 π 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 π.

Exponential Ergodicity of the Jump-Diffusion CIR Process

“…Motivated by some applications in finance, the long-time behavior of the BAJD has been well studied. According to [12,Theorem 3.16] and [10,Proposition 3.1], the BAJD possesses a unique invariant probability measure, whose distributional properties were later investigated in [9,11]. We remark that the results in [10,11] are very general and hold for a large class of affine process with state space R + , where R + denotes the set of all non-negative real numbers.…”

“…The existence and some approximations of the transition densities of the BAJD can be found in [6]. A closed formula of the transition densities of the BAJD was recently derived in [9].…”

“…Under some sharper assumptions we show in this paper that the convergence of the law of the JCIR process to its invariant probability measure under the total variation norm is exponentially fast, which is called the exponential ergodicity. Our method is the same as in [9], namely we show the existence of a Forster-Lyapunov function and then apply the general framework of [16][17][18] to get the exponential ergodicity.…”

Exponential ergodicity of an affine two-factor model based on the α-root process

“…Proof. We basically follow the proof of [16,Theorem 6.3]. The essential idea is to use the so-called Foster-Lyapunov criteria developed in [26] for the geometric ergodicity of Markov chains.…”

Parameter estimation for the subcritical Heston model based on discrete time observations

“…In Section 2 we recall some important results about the existence of a unique strong solution to (1.1), and study its asymptotic properties. In the subcritical case, i.e., when b > 0, we invoke a result due to Cox et al [4] on the unique existence of a stationary distribution, and we slightly improve a result due to Li and Ma [14] and Jin et al [10,Corollary 2.7] and [11,Corollaries 5.9 and 6.4] on the ergodicity of the CIR process (Y t ) t 0 , see Theorem 2.4. We also recall some convergence results for square-integrable martingales.…”