Nonparametric estimation of trend function for stochastic differential equations driven by a bifractional Brownian motion

The main objective of this paper is to investigate the problem of estimating the trend function S_t = S(x_t) for process satisfying stochastic differential equations of the type dXt=S(Xt)dt+εdBtH,K, X0=x0, 0≤t≤T,{\rm{d}}{{\rm{X}}_{\rm{t}}} = {\rm{S}}\left( {{{\rm{X}}_{\rm{t}}}} \right){\rm{dt + }}\varepsilon {\rm{dB}}_{\rm{t}}^{{\rm{H,K}}},\,{{\rm{X}}_{\rm{0}}} = {{\rm{x}}_{\rm{0}}},\,0 \le {\rm{t}} \le {\rm{T,}}

where {BtH,K,t≥0{\rm{B}}_{\rm{t}}^{{\rm{H,K}}},{\rm{t}} \ge {\rm{0}}} is a bifractional Brownian motion with known parameters H ∈ (0, 1), K ∈ (0, 1] and HK ∈ (1/2, 1). We estimate the unknown function S(x_t) by a kernel estimator ̂S_t and obtain the asymptotic properties as ε → 0. Finally, a numerical example is provided.