Dzhaparidze and Spreij (Stoch Process Appl, 54:165–174, <CitationRef CitationID="CR5">1994</CitationRef>) showed that the quadratic variation of a semimartingale can be approximated using a randomized periodogram. We show that the same approximation is valid for a special class of continuous stochastic processes. This class contains...</citationref>

Based on observations <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$X_1,\dots ,X_n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo stretchy="false">…</mo> <mo>,</mo> <msub> <mrow> <mi>X</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </mrow> </math> </EquationSource> </InlineEquation> of successive generations of a discrete-parameter Galton–Watson branching process, one wishes to predict whether extinction or explosion will ultimately occur. This problem can be formulated as a simple...</equationsource></equationsource></inlineequation>

The <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">α</mi> </math> </EquationSource> </InlineEquation>-stable L<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$\acute{\mathrm{e}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mover accent="true"> <mi mathvariant="normal">e</mi> <mo>´</mo> </mover> </math> </EquationSource> </InlineEquation>vy motion together with the Poisson process and Brownian motion are the most important examples of L<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\acute{\mathrm{e}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mover accent="true"> <mi mathvariant="normal">e</mi> <mo>´</mo> </mover> </math> </EquationSource> </InlineEquation>vy processes, which form the first class of stochastic processes being studied in the modern...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>

We consider the wavelet-based estimators of mean regression function with long memory moving average errors and investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients. We show that these estimators achieve nearly optimal minimax convergence...

This paper establishes the asymptotic normality of frequency polygons in the context of stationary strongly mixing random fields indexed by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mathbb {Z}^d$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math> </EquationSource> </InlineEquation>. Our method allows us to consider only minimal conditions on the width bins and provides a simple criterion on the mixing...</equationsource></equationsource></inlineequation>

We discuss some inference problems associated with the fractional Ornstein–Uhlenbeck (fO–U) process driven by the fractional Brownian motion (fBm). In particular, we are concerned with the estimation of the drift parameter, assuming that the Hurst parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$H$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>H</mi> </math> </EquationSource> </InlineEquation> is known and is in <InlineEquation ID="IEq2"> <EquationSource...</equationsource></inlineequation></equationsource></equationsource></inlineequation>

Based on Malliavin calculus tools and approximation results, we show how to compute a maximum likelihood type estimator for a rather general differential equation driven by a fractional Brownian motion with Hurst parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$H1/2$$</EquationSource> </InlineEquation>. Rates of convergence for the approximation task are provided,...</equationsource></inlineequation>

Motivated by the availability of continuous event sequences that trace the social behavior in a population e.g. email, we believe that mutually exciting Hawkes processes provide a realistic and informative model for these sequences. For complex mutually exciting processes, the numerical...

This paper presents a goodness-of-fit test for the volatility function of a SDE driven by a Gaussian process with stationary and centered increments. Under rather weak assumptions on the Gaussian process, we provide a procedure for testing whether the unknown volatility function lies in a given...

The paper considers the problem of estimating the parameters in a continuous time regression model with a non-Gaussian noise of pulse type. The vector of unknown parameters is assumed to belong to a compact set. The noise is specified by the Ornstein–Uhlenbeck process driven by the mixture of...