We say that a continuous real-valued function $x$ admits the Hurst roughness exponent $H$ if the $p^{\text{th}}$ variation of $x$ converges to zero if $p>1/H$ and to infinity if $p<1/H$. For the sample paths of many stochastic processes, such as fractional Brownian motion, the Hurst roughness exponent exists and equals the standard Hurst parameter. In our main result, we provide a mild condition on the Faber--Schauder coefficients of $x$ under which the Hurst roughness exponent exists and is given as the limit of the classical Gladyshev estimates $\widehat H_n(x)$. This result can be viewed as a strong consistency result for the Gladyshev estimators in an entirely model-free setting, because no assumption whatsoever is made on the possible dynamics of the function $x$. Nonetheless, our proof is probabilistic and relies on a martingale that is hidden in the Faber--Schauder expansion of $x$. Since the Gladyshev estimators are not scale-invariant, we construct several scale-invariant estimators that are derived from the sequence $(\widehat H_n)_{n\in\mathbb{N}}$. We also discuss how a dynamic change in the Hurst roughness parameter of a time series can be detected. Our results are illustrated by means of high-frequency financial time series. This is joint work with Xiyue Han.

Mathematical Finance Seminar

Date

2021-12-09

Time

17:oo

Location:

online

Alexander Schied (U. Waterloo)

## The Hurst roughness exponent and its model-free estimation

Mathematical Finance Seminar

Date

2021-12-16

Time

17:oo

Location:

online

Samuel Drapeau (Shanghai Jiao Tong)

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Mathematical Finance Seminar

Date

2022-01-06

Time

17:oo

Location:

Aurelien Alfonsi (Marne-la-vallée)

## tba

Mathematical Finance Seminar

Date

2022-01-13

Time

17:oo

Location:

online

Max Nendel (Bielefeld)

## tja

Mathematical Finance Seminar

Date

2022-01-20

Time

17:oo

Location:

online

Ralf Wunderlich (BTU Cottbus-Senftenberg)

## tja

Mathematical Finance Seminar

Date

2022-02-03

Time

17:oo

Location:

Jörn Sass (Kaiserslautern)