Drift parameter estimation in the double mixed fractional Brownian model via solutions of Fredholm equations with singular kernels

We consider drift parameter estimation in a model driven by the sum of two independent fractional Brownian motions with different Hurst indices. Although the maximum likelihood estimator (MLE) for this model is known theoretically, its computation requires solving an operator equation involving fractional covariance operators. We develop an effective numerical method for approximating the solution of this equation by reformulating it as a Fredholm integral equation of the second kind with a weakly singular kernel. The resulting algorithm enables practical computation of the MLE. Numerical experiments illustrate the performance of the method.

💡 Research Summary

The paper addresses the problem of estimating the drift parameter θ in a continuous‑time stochastic model driven by the sum of two independent fractional Brownian motions (fBms) with distinct Hurst indices H₁ and H₂. The model is
 Xₜ = θ t + B^{H₁}_t + B^{H₂}_t, t ≥ 0,
with H₁∈(½,¾] and H₂∈(H₁,1). Such “double mixed” fBm models capture both short‑term fluctuations (the component with smaller H) and long‑term memory (the component with larger H), making them attractive for financial and econometric applications.

Theoretical background.
Previous work (Mishura, Ralchenko, Yako‑Vliev, 2020) derived the maximum‑likelihood estimator (MLE) for θ under continuous observation on