Page 563 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 563
DELING ROUGHNESS AND LONG-RANGE DEPENDENCE WITH FRACTIONAL
PROCESSES (MS-18)
show the obstacles of the so called turbocharging technique, whose importance is sometimes
overestimated in the literature. To overcome these obstacles we suggest several modifications.
Hypotheses testing of the drift parameter sign for the fractional
Ornstein–Uhlenbeck process
Kostiantyn Ralchenko, k.ralchenko@gmail.com
Taras Shevchenko National University of Kyiv, Ukraine
We study the inference problem for the fractional Ornstein–Uhlenbeck process X = {Xt, t ≥
0}, which is the unique solution of the stochastic differential equation
dXt = θXt dt + dBtH , X0 = x0 ∈ R. (1)
Here θ ∈ R is an unknown drift parameter. The noise is modelled as a fractional Brownian
motion BH = {BtH, t ≥ 0} with a known Hurst index H ∈ (0, 1). First, we investigate
the problem of the drift parameter estimation by continuous and discrete observations of the
trajectory of X. We construct several types of estimators and prove their strong consistency.
It turns out that the methods for constructing the estimators and their asymptotic properties
substantially depend on the sign of the unknown parameter. This motivates the hypothesis
testing of the sign of drift parameter in the model (1). We propose a comparatively simple
test for testing the null hypothesis H0 : θ ≤ 0 against the alternative H1 : θ > 0 and prove its
consistency. Contrary to the previous works, our approach is applicable for all H ∈ (0, 1). The
test is based on the observations of the process X at two points: 0 and T . The distribution
of the test statistic is computed explicitly, and the power of test can be found numerically for
any given simple alternative. Also we consider the hypothesis testing H0 : θ ≥ θ0 against
H1 : θ ≤ 0, where θ0 ∈ (0, 1) is some fixed number. As an auxiliary result of independent
interest we compute the covariance function of the fractional Ornstein–Uhlenbeck process.
References
[1] K. Kubilius, Yu. Mishura, K. Ralchenko, Parameter Estimation in Fractional Diffusion
Models, Bocconi & Springer Series, vol. 8., Springer, 2017
[2] A. Kukush, Yu. Mishura, K. Ralchenko, Hypothesis testing of the drift parameter sign for
fractional Ornstein–Uhlenbeck process, Electron. J. Statist., Vol. 11, No. 1, 2017, pp. 385–
400.
Integration-by-Parts Characterizations of Gaussian Processes
Tommi Sottinen, tommi.sottinen@iki.fi
University of Vaasa, Finland
The Stein’s lemma characterizes the one-dimensional Gaussian distribution via an integration-
by-parts formula. We show that a similar integration-by-parts formula characterizes a wide
class of Gaussian processes, the so-called Gaussian Fredholm processes. Examples include the
Brownian motion and fractional Brownian motions.
561
PROCESSES (MS-18)
show the obstacles of the so called turbocharging technique, whose importance is sometimes
overestimated in the literature. To overcome these obstacles we suggest several modifications.
Hypotheses testing of the drift parameter sign for the fractional
Ornstein–Uhlenbeck process
Kostiantyn Ralchenko, k.ralchenko@gmail.com
Taras Shevchenko National University of Kyiv, Ukraine
We study the inference problem for the fractional Ornstein–Uhlenbeck process X = {Xt, t ≥
0}, which is the unique solution of the stochastic differential equation
dXt = θXt dt + dBtH , X0 = x0 ∈ R. (1)
Here θ ∈ R is an unknown drift parameter. The noise is modelled as a fractional Brownian
motion BH = {BtH, t ≥ 0} with a known Hurst index H ∈ (0, 1). First, we investigate
the problem of the drift parameter estimation by continuous and discrete observations of the
trajectory of X. We construct several types of estimators and prove their strong consistency.
It turns out that the methods for constructing the estimators and their asymptotic properties
substantially depend on the sign of the unknown parameter. This motivates the hypothesis
testing of the sign of drift parameter in the model (1). We propose a comparatively simple
test for testing the null hypothesis H0 : θ ≤ 0 against the alternative H1 : θ > 0 and prove its
consistency. Contrary to the previous works, our approach is applicable for all H ∈ (0, 1). The
test is based on the observations of the process X at two points: 0 and T . The distribution
of the test statistic is computed explicitly, and the power of test can be found numerically for
any given simple alternative. Also we consider the hypothesis testing H0 : θ ≥ θ0 against
H1 : θ ≤ 0, where θ0 ∈ (0, 1) is some fixed number. As an auxiliary result of independent
interest we compute the covariance function of the fractional Ornstein–Uhlenbeck process.
References
[1] K. Kubilius, Yu. Mishura, K. Ralchenko, Parameter Estimation in Fractional Diffusion
Models, Bocconi & Springer Series, vol. 8., Springer, 2017
[2] A. Kukush, Yu. Mishura, K. Ralchenko, Hypothesis testing of the drift parameter sign for
fractional Ornstein–Uhlenbeck process, Electron. J. Statist., Vol. 11, No. 1, 2017, pp. 385–
400.
Integration-by-Parts Characterizations of Gaussian Processes
Tommi Sottinen, tommi.sottinen@iki.fi
University of Vaasa, Finland
The Stein’s lemma characterizes the one-dimensional Gaussian distribution via an integration-
by-parts formula. We show that a similar integration-by-parts formula characterizes a wide
class of Gaussian processes, the so-called Gaussian Fredholm processes. Examples include the
Brownian motion and fractional Brownian motions.
561
