# Stochastic equation and exponential ergodicity in Wasserstein distances for affine processes

## TL;DR

This paper investigates the long-term behavior of affine processes, proving exponential ergodicity in Wasserstein distances under certain moment conditions, and establishes their representation as solutions to stochastic equations.

## Contribution

It demonstrates that affine processes can be uniquely represented as solutions to stochastic equations and proves their exponential ergodicity in Wasserstein distances.

## Key findings

- Affine processes are solutions to stochastic equations driven by Brownian motions and Poisson measures.
- Subcritical affine processes exhibit exponential ergodicity in Wasserstein distances.
- Moment conditions are crucial for establishing ergodicity.

## Abstract

This work is devoted to the study of conservative affine processes on the canonical state space $D = $R_+^m \times \R^n$, 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, i.e., 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.