Nonparametric estimation for random effects models driven by fractional Brownian motion using Hermite polynomials

We propose a nonparametric estimation of random effects from the following fractional diffusions $$dX^{j}(t) = \psi _{j}X^{j}(t)d t+X^{j}(t)d W^{H,j}(t), $$ $$~X^j(0)=x^j_0,~t\ge 0, $$ $$ j=1,\ldots ,n,$$ where $$\psi _j$$ are random variables and $$ W^{j,H}$$ are fractional Brownian motions with a common known Hurst index $$H\in (0,1)$$ . We are concerned with the study of Hermite projection and kernel density estimators for the $$\psi _j$$ ’s common density, when the horizon time of observation is fixed or sufficiently large. We corroborate these theoretical results through simulations. An empirical application is made to the real Asian financial data.