Divergences test statistics for discretely observed diffusion processes

In this paper we propose the use of ϕ ‐ divergences as test statistics to verify simple hypotheses about a one-dimensional parametric diffusion process d X t = b ( X t , α ) d t + σ ( X t , β ) , α ∈ R p , β ∈ R q , p , q > = 1 , from discrete observations { X t i , i = 0 , … , n } with t i = i Δ n , i = 0 , 1 , … , n , under the asymptotic scheme Δ n → 0 , n Δ n → ∞ and n Δ n 2 → 0 . The class of ϕ ‐ 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 ‐ divergences . The asymptotic distribution depends on the regularity of the function ϕ and in general it differs from the standard χ 2 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.