Two approaches to consistent estimation of parameters of mixed fractional Brownian motion with trend

Alexander Kukush (), Stanislav Lohvinenko (), Yuliya Mishura () and Kostiantyn Ralchenko ()
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Alexander Kukush: Taras Shevchenko National University of Kyiv
Stanislav Lohvinenko: Taras Shevchenko National University of Kyiv
Yuliya Mishura: Taras Shevchenko National University of Kyiv
Kostiantyn Ralchenko: Taras Shevchenko National University of Kyiv

Statistical Inference for Stochastic Processes, 2022, vol. 25, issue 1, No 9, 159-187

Abstract: Abstract We investigate the mixed fractional Brownian motion with trend of the form $$X_t = \theta t + \sigma W_t + \kappa B^H_t$$ X t = θ t + σ W t + κ B t H , driven by a standard Brownian motion W and a fractional Brownian motion $$B^H$$ B H with Hurst parameter H. We develop and compare two approaches to estimation of four unknown parameters $$\theta $$ θ , $$\sigma $$ σ , $$\kappa $$ κ and H by discrete observations. The first algorithm is more traditional: we estimate $$\sigma $$ σ , $$\kappa $$ κ and H using the quadratic variations, while the estimator of $$\theta $$ θ is obtained as a discretization of a continuous-time estimator of maximum likelihood type. This approach has several limitations, in particular, it assumes that $$H

Keywords: Fractional Brownian motion; Wiener process; Mixed power variations; Strong consistency; Mixed model; Ergodic theorem; 60G22; 62F10; 62F12 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11203-021-09252-6

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