Abstract
Affine jump-diffusion (AJD) processes constitute an important class of continuous time stochastic models that are widely used in finance and econometrics. For instance, many classic models in derivative pricing are special cases of AJD processes: the Ornstein-Uhlenbeck (OU) process (i.e. the Vasicek model), the square-root diffusion process, (i.e. the Cox-Ingersoll-Ross model), and the Heston stochastic volatility model. This class of models is flexible enough to capture various empirical attributes such as stochastic volatility and leverage effects. Its affine structure leads to significant tractability both for computing various expectations and probabilities. Most methods for parameter estimation (e.g. maximum likelihood estimation or generalized methods of moments) of this type of processes generally assume ergodicity in order establish consistency and asymptotic normality of the estimator. In this talk, we present several results on the stochastic stability of AJDs. We establish ergodicity of AJDs by imposing a ‘‘mean reversion’’ assumption and a mild condition on the distribution of the jumps, i.e. the finiteness of a logarithmic moment. As a stronger result, exponential ergodicity is proved if the jumps have a finite moment of a positive order. In addition, we prove strong laws of large numbers and functional central limit theorems for additive functional of this class of models. These limit theorems lay solid foundation for parameter estimation methods of AJDs.
Xiaowei Zhang
Associate Professor
My research research focuses on methodological advances in stochastic simulation and optimization, decision analytics, and reinforcement learning, with applications in service operations management, financial technology, and digital economy.