ASYMPTOTIC EXPANSION FOR STOCHASTIC PROCESSES: AN OVERVIEW AND EXAMPLES

Abstract

The aim of this article is to give an overview of the developments in the theory of the asymptotic expansion for stochastic processes of continuous time. Today we know two typical methods of asymptotic expansion: the martingale approach and the mixing approach. These methods are complementary to each other. The martingale approach was found first and applied to derive an asymptotic expansion for ergodic diffusion processes. However, if the diffusion process satisfies a sufficiently nice mixing condition, then the mixing approach is more effective. On the other hand, the martingale approach is still useful when the higher-order terms do not obey an asymptotic normal law, which makes it impossible to apply the mixing approach. Such examples are seen in a stochastic regression model with a long memory explanatory variable, and in estimation of a volatility parameter over a finite time interval. In the latter example, the data is strongly time dependent, so that it requires a global estimate of the smoothness of random variables. In this sense, the martingale approach is also called the global approach. Contrarily, the mixing approach is called the local approach since the regularity often comes from a local (in time) estimate of the characteristic function. We will focus our attention on the mixing approach in this article. In Section 2, we recall a stochastic process having the “� -Markovian” structure as an underlying stochastic process. The � -Markov model written in continuous time may seem to be complicated, however it has an advantage because nonlinear (Markovian) time series models are included in the present model by natural embedding. Section 3 gives an illustrative application. We demonstrate an ap