Hybrid multi-step estimators for stochastic differential equations based on sampled data

Abstract

We consider an estimation problem of both drift and diffusion coefficient parameters for an ergodic diffusion process based on discrete observations. Hybrid multi-step estimators are proposed and their asymptotic properties, including convergence of moments, are obtained.

Appendix: Markov chain Monte Carlo method

For the target distribution $p(x)dx$ in ${\mathbb {R}}^d$, we run the following Markov chain Monte Carlo method. Fix $h\in (0,1)$ and $\nu >0$, and set $g(x)=(1+|x|^2/\nu )^{-(\nu +d)/2}$.

• For $m=0$. Initialize $x$.

• For $m\ge 1$, iterate

  • Generate $r$ from the inverse gamma distribution with the shape parameter $\nu /2+d/2$ and the rate parameter $\nu /2+|x|^2/2$.

  • Set $y=h^{1/2} x+(1-h)^{1/2}r^{1/2}w$ where $w$ follows the standard normal distribution.

  • Accept $y$ as $x$ with probability $\min \{1,\frac{p(y)g(x)}{p(x)g(y)}\}$. Otherwise, discard $y$.

In this paper, we set $h=0.8$ and $\nu =2$. This is a kind of Metropolis–Hastings algorithm. This MCMC method is efficient for complicated target distribution. The detail of this MCMC method will be described in Kamatani (2014) and is out of the scope of this paper.