Stochastic equation and exponential ergodicity in Wasserstein distances for affine processes

Abstract

This work is devoted to the study of conservative affine processes on the canonical state space $D=\mathbb{R}_{+}^{m}\times \mathbb{R}^{n}$, where $m+n>0$. We show that each affine process can be obtained as the pathwise unique strong solution to a stochastic equation driven by Brownian motions and Poisson random measures. Then we study the long-time behavior of affine processes, that is, we show that under first moment condition on the state-dependent and $\log $-moment conditions on the state-independent jump measures, respectively, each subcritical affine process is exponentially ergodic in a suitably chosen Wasserstein distance. Moments of affine processes are studied as well.