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

Abstract

We study an affine two-factor model introduced by Barczy et al. (2014). One component of this two-dimensional model is the so-called α-root process, which generalizes the well-known Cox–Ingersoll–Ross process. In the α = 2 case, this two-factor model was used by Chen and Joslin (2012) to price defaultable bonds with stochastic recovery rates. In this paper we prove exponential ergodicity of this two-factor model when α ∈ (1, 2). As a possible application, our result can be used to study the parameter estimation problem of the model.

Ergodic and strong Feller properties of affine processes

“…We prove some results on the ergodicity and exponential ergodicity of the processes in total variation distances under natural conditions. Instead of Zhang and Glynn (2018), our approach is based on coupling methods developed in Schilling and Wang (2012) and Wang (2012); see also Li and Ma (2015), which answers the question appeared on Jin et al (2017Jin et al ( ), pp.1145; see also , pp.646.…”

Ergodicity of affine processes on the cone of symmetric positive semidefinite matrices

“…This in turn is true if and only if α = 0, which is a consequence of the particular structure of the boundary ∂S + d . To overcome this difficulty, we use the cone structure to show that ψ(t, u) exp{t B }(u), which requires us to study the first moment of the affine process with admissible parameters (α, b, B, m = 0, μ); see (29). Note that the first moment of affine processes is already studied in Theorem 2.2, where the first moment condition for μ is also explicitly used.…”

“…Finally, let us mention that the long-time behavior of affine processes has previously been studied in many different settings, either based on a detailed study of the characteristic function (see e.g. [46], [39], [20], [47], [34], [32], [36], [30]), by stochastic stability criteria due to Meyn and Tweedie ( [4], [29], [44]), or by coupling techniques ( [19], [18], [40]). One application of such study is towards the estimation of parameters for affine models.…”

On the anisotropic stable JCIR process

“…For subcritical OU-type processes and 1-dimensional continuous-state branching processes with immigration, the exponential ergodicity in total variation has been derived under rather general conditions, see Wang (2012); Li and Ma (2015) and Friesen et al (2019a), all of which used coupling techniques. Other than these two cases, only very few results on ergodicity in total variation are available for multi-dimensional affine processes, except for the models treated in Barczy et al (2014); Jin et al (2017); Mayerhofer et al (2020); Zhang and Glynn (2018). The reason is as follows: in the general case it is not clear if the powerful coupling technique (see Wang, 2012;Li and Ma, 2015) still works; also, it remains a difficult problem to verify the irreducibility of the process when applying the Meyn-Tweedie method (see Meyn and Tweedie, 2009).…”