Stationarity and Ergodicity for an Affine Two-Factor Model

Abstract

We study the existence of a unique stationary distribution and ergodicity for a 2-dimensional affine process. The first coordinate is supposed to be a so-called α-root process with α ∈ (1, 2]. The existence of a unique stationary distribution for the affine process is proved in case of α ∈ (1, 2]; further, in case of α = 2, the ergodicity is also shown.

Conditions for Stationarity and Ergodicity of Two-factor Affine Diffusions

“…Affine processes are joint generalizations of continuous state branching processes and Orstein-Uhlenbeck type processes, and they have applications in financial mathematics, see, e.g., in Duffie et al [7]. The aim of the present paper is to extend the results of Barczy et al [1] for the processes given in (1.1), where the case of β = 0, ̺ = 0, σ 1 = 1, σ 2 = 1, σ 3 = 0 is covered. We give sufficient conditions for the existence of a unique stationary distribution and exponential ergodicity, see Theorems 3.1 and 4.1, respectively.…”

“…which can be checked by standard arguments, see, e.g., in the arXiv version of the proof of Theorem 4.2 in Barczy et al [1]. For all n, p ∈ Z + , using the independence of W , B and L, by Itô's formula, we have d(Y n t X p t ) = nY n−1 t X p t (a − bY t ) dt + σ 1 Y t dW t + pY n t X p−1 t (α − βY t − γX t ) dt + σ 2 Y t (̺ dW t + 1 − ̺ 2 dB t ) + σ 3 dL t…”

“…y ∈ R ++ . For a proof, see, e.g., the proof of Theorem 4.1 in the extended arXiv version of Barczy et al [1]. Now we turn back to the proof that the random vector (4.4) is absolutely continuous with respect to the Lebesgue measure on R 2 with a density function being strictly positive on the set y ∈ R : y > −y 0 − a To prove (c), first we note that, since the sample paths of (Y t , Z t ) t∈R+ are almost surely continuous, for each n ∈ N, the extended generator has the form…”

“…Theorem 5.1. Let us consider the 2-dimensional affine diffusion model 1], and the random vector (Y ∞ , X ∞ ) given by Theorem 3.1. Then all the (mixed) moments of (Y ∞ , X ∞ ) of any order are finite, i.e., we have E(Y n ∞ |X ∞ | p ) < ∞ for all n, p ∈ Z + , and the recursion…”

“…An important observation is that it is enough to prove the results for the special case of ̺ = 0, since there is a non-singular linear transform of a 2-dimensional affine diffusion process which is a special 2-dimensional affine diffusion process with ̺ = 0, see Proposition 2.5. Otherwise, the method of the proofs are the same as in Barczy et al [1].…”