Asymptotic analysis of a kernel estimator for stochastic differential equations driven by a mixed sub-fractional Brownian motion

摘要

We consider the problem of estimating the trend function b_t = b(x_t ) for process satisfying stochastic differential equations of the type dX_t = b(X_t)dt + εdS_t^H (a), X₀ = x₀, 0 ≤ t ≤ T, where {S_t^H (a), t ≥ 0} is a mixed sub-fractional Brownian motion with known parameters N, a, and H, such that N ∈ ℕ^*, H ∈ (1/2, 1)^N, and a ∈ ℝ^N{0_N}. We estimate the unknown function b(x_t ) by a kernel estimator b_t and obtain the uniform convergence, rate of convergence and asymptotic normality of the estimator b_t (as ε → 0).