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

Abstract

We consider the problem of estimating the trend function bt = b(xt ) for process satisfying stochastic differential equations of the type\begin{equation*}dX_t = b(X_t)dt + \varepsilon dS_t^H (a),\,X_0 = x_0 ,\,0 \le t \le T,\end{equation*}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\{0N}. We estimate the unknown function b(xt ) by a kernel estimator $\widehat{b_t}$ and obtain the uniform convergence, rate of convergence and asymptotic normality of the estimator $\widehat{b_t}$ (as ε → 0).