Estimation of parameters of SDE driven by fractional Brownian motion with polynomial drift

Abstract

Strongly consistent and asymptotically normal estimators of the Hurst index and volatility parameters of solutions of stochastic differential equations with polynomial drift are proposed. The estimators are based on discrete observations of the underlying processes.

On comparison of the estimators of the Hurst index and the diffusion coefficient of the fractional Gompertz diffusion process

“…The proof of the convergence of σ 2 2,n is analogous to that of c 2 n in [13]. Let us prove that σ 2 1,n a.s.…”

“…Indeed, the asymptotics of the increments of the solution X of the equation (1) are the same as the asymptotics of the increments of the solution of the equation with polynomial drift in [13]. Thus in order to establish the convergence of the estimator H (1) n it suffices to repeat the proof of Theorem 2 in [13]. Further, note that hypotheses (H) and (H1) in [14] are satisfied for the solution of the equation (1), i.e.…”

“…The estimators H (i) n , i = 1, 2, 3, 4, were considered in [13], [12], [3], and [1]. The estimator H (2) n can be used to estimate the Hurst index of the generic form of the SDE with an additional restriction on the diffusion coefficient.…”

On some estimators of the Hurst index of the solution of SDE driven by a fractional Brownian motion

“…Using the same arguments as in [13] it is possible to prove that the estimator H (1) n is strongly consistent and asymptotically normal.…”

“…However, all of these works focused on the strong consistency. The present paper is a generalization of [13] where a special case of (1.1) was considered.…”