Divergences test statistics for discretely observed diffusion processes

In this paper we propose the use of φ{symbol} - divergences as test statistics to verify simple hypotheses about a one-dimensional parametric diffusion process d Xt = b (Xt, α) d t + σ (Xt, β), α ∈ Rp, β ∈ Rq, p, q > = 1, from discrete observations { Xti, i = 0, ..., n } with ti = i Δn, i = 0, 1, ..., n, under the asymptotic scheme Δn → 0, n Δn → ∞ and n Δn 2 → 0. The class of φ{symbol} - divergences is wide and includes several special members like Kullback-Leibler, Rényi, power and α - divergences. We derive the asymptotic distribution of the test statistics based on the estimated φ{symbol} - divergences. The asymptotic distribution depends on the regularity of the function φ{symbol} and in general it differs from the standard χ2 distribution as in the i.i.d. case. Numerical analysis is used to show the small sample properties of the test statistics in terms of estimated level and power of the test. © 2010 Elsevier B.V. All rights reserved.