Discretely observed diffusions: Approximation of the continuous-time score function (Q2722306)

This paper is about parameter estimation for discretely observed diffusion models with known diffusion function. The idea is to use an approximation of the continuous-time score function as estimating function. The estimating function discussed in this paper is of the form \(\sum_{i=1}^{n}A_{\theta}h(X_{t_{i-1}},\theta),\) where \(A_{\theta}\) is the diffusion generator, \(\theta\) is an unknown \(p\)-dimensional parameter, \(h\) is twice continuously differentiable w.r.t. \(x\), \(X_{t_{i-1}}\) are observations. The main contribution of the paper is to recognize that, with a special choice of \(h\), the estimating function can be interpreted as an approximation to the continuous-time score function. The approximating estimating function is unbiased, it is invariant to data transformations, it provides consistent and asymptotically normal estimators, and it can be explicitly expressed in terms of the drift and diffusion coefficient. The estimating function is also, at least in some cases, available for multidimensional processes. Simulation studies show that the method performs well.