Least Squares Estimation for α -Fractional Bridge with Discrete Observations

Abstract EN

We consider a fractional bridge defined as d X t = - α (X t / ( T - t )) d t + d B t H , 0 ≤ t < T , where B H is a fractional Brownian motion of Hurst parameter H > 1 / 2 and parameter α > 0 is unknown.

We are interested in the problem of estimating the unknown parameter α > 0 .

Assume that the process is observed at discrete time t i = i Δ n , i = 0 , … , n , and T n = n Δ n denotes the length of the “observation window.” We construct a least squares estimator α ^ n of α which is consistent; namely, α ^ n converges to α in probability as n → ∞ .