Geometric ergodicity of affine processes on cones

Abstract

For affine processes on finite-dimensional cones, we give criteria for geometric ergodicitythat is exponentially fast convergence to a unique stationary distribution. Ergodic results include both the existence of exponential moments of the limiting distribution, where we exploit the crucial affine property, and finite moments, where we invoke the polynomial property of affine semigroups. Furthermore, we elaborate sufficient conditions for aperiodicity and irreducibility. Our results are applicable to Wishart processes with jumps on the positive semidefinite matrices, continuous-time branching processes with immigration in high dimensions, and classical termstructure models for credit and interest rate risk.

Existence of densities for multi-type continuous-state branching processes with immigration

“…At this point we would like to mention only some recent results on the long-time behavior of CBI processes. Namely, convergence of supercritical CBI processes was recently studied in [BPP18a] and [BPP18b] while convergence in the total variation distance for affine processes on convex cones (including subcritical CBI processes) was recently studied in [MSV18]. Results applicable to the class of affine processes on the canonical state space R d + × R n were obtained in [FJR18c], [GZ18] and [JKR18].…”

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

“…In their work, the S + d -valued affine process is constructed and completely characterized through a set of admissible parameters, and the related generalized Riccati equations are investigated. Subsequent developments complementing the results of [12] can be found in [33], [42], [43], and [44]. Note that the notion of affine processes is not restricted to the state space S + d .…”

“…Based on stochastic stability criteria of Meyn and Tweedie, convergence to the unique invariant distribution in total variation was then studied for a general class of Ornstein-Uhlenbeck processes in [41], while generalized Ornstein-Uhlenbeck processes in dimension one have been investigated in [8] and [37]. First results on exponential ergodicity applicable to a class of affine processes on finite-dimensional cones were recently obtained by Mayerhofer et al [44].…”

“…Additional details are explained in the next section. We would like to mention that our method to show stationarity could also be applied to affine processes on more general convex cones in the spirit of [13] and [44]; however, this would require the introduction of additional notation which would make our key arguments less transparent, so we postpone the detailed discussion of this extension to Section 9.…”

“…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.…”