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Approximations for a solution to stochastic heat equation with stable noise

Larysa Pryhara Georgiy Shevchenko

https://doi.org/10.15559/16-VMSTA56

Pub. online: 30 Jun 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 133–144

Abstract

We consider a Cauchy problem for stochastic heat equation driven by a real harmonizable fractional stable process Z with Hurst parameter $H>1/2$ and stability index $\alpha >1$. It is shown that the approximations for its solution, which are defined by truncating the LePage series for Z, converge to the solution.

On fractal faithfulness and fine fractal properties of random variables with independent

Q^{*}

-digits

Muslem Ibragim Grygoriy Torbin

https://doi.org/10.15559/16-VMSTA55

Pub. online: 9 Jun 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 119–131

Abstract

We develop a new technique to prove the faithfulness of the Hausdorff–Besicovitch dimension calculation of the family $\varPhi ({Q}^{\ast })$ of cylinders generated by ${Q}^{\ast }$-expansion of real numbers. All known sufficient conditions for the family $\varPhi ({Q}^{\ast })$ to be faithful for the Hausdorff–Besicovitch dimension calculation use different restrictions on entries $q_{0k}$ and $q_{(s-1)k}$. We show that these restrictions are of purely technical nature and can be removed. Based on these new results, we study fine fractal properties of random variables with independent ${Q}^{\ast }$-digits.

Large deviations for drift parameter estimator of mixed fractional Ornstein–Uhlenbeck process

Dmytro Marushkevych

https://doi.org/10.15559/16-VMSTA54

Pub. online: 7 Jun 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 107–117

Abstract

We investigate large deviation properties of the maximum likelihood drift parameter estimator for Ornstein–Uhlenbeck process driven by mixed fractional Brownian motion.

Erratum to “Asymptotic normality of total least squares estimator in a multivariate errors-in-variables model

A X = B

”

Alexander Kukush Yaroslav Tsaregorodtsev

https://doi.org/10.15559/16-VMSTA53

Pub. online: 16 May 2016 Type: Erratum

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), p. 105

Asymptotics of exponential moments of a weighted local time of a Brownian motion with small variance

Alexei Kulik Daryna Sobolieva

https://doi.org/10.15559/16-VMSTA49

Pub. online: 5 Apr 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 1 (2016), pp. 95–103

Abstract

We prove a large deviation type estimate for the asymptotic behavior of a weighted local time of $\varepsilon W$ as $\varepsilon \to 0$.

Random convolution of inhomogeneous distributions with

O

-exponential tail

Svetlana Danilenko Simona Paškauskaitė Jonas Šiaulys

https://doi.org/10.15559/16-VMSTA52

Pub. online: 4 Apr 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 1 (2016), pp. 79–94

Abstract

Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables (not necessarily identically distributed), and η be a counting random variable independent of this sequence. We obtain sufficient conditions on $\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution function of the random sum $S_{\eta }=\xi _{1}+\xi _{2}+\cdots +\xi _{\eta }$ belongs to the class of $\mathcal{O}$-exponential distributions.

Minimax interpolation of sequences with stationary increments and cointegrated sequences

Maksym Luz Mikhail Moklyachuk

https://doi.org/10.15559/16-VMSTA51

Pub. online: 1 Apr 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 1 (2016), pp. 59–78

Abstract

We consider the problem of optimal estimation of the linear functional $A_{N}\xi ={\sum _{k=0}^{N}}a(k)\xi (k)$ depending on the unknown values of a stochastic sequence $\xi (m)$ with stationary increments from observations of the sequence $\xi (m)+\eta (m)$ at points of the set $\mathbb{Z}\setminus \{0,1,2,\dots ,N\}$, where $\eta (m)$ is a stationary sequence uncorrelated with $\xi (m)$. We propose formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional in the case of spectral certainty, where spectral densities of the sequences are exactly known. We also consider the problem for a class of cointegrated sequences. We propose relations that determine the least favorable spectral densities and the minimax spectral characteristics in the case of spectral uncertainty, where spectral densities are not exactly known while a set of admissible spectral densities is specified.

Asymptotic normality of total least squares estimator in a multivariate errors-in-variables model

A X = B

Alexander Kukush Yaroslav Tsaregorodtsev

https://doi.org/10.15559/16-VMSTA50

Pub. online: 29 Mar 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 1 (2016), pp. 47–57

Abstract

We consider a multivariate functional measurement error model $AX\approx B$. The errors in $[A,B]$ are uncorrelated, row-wise independent, and have equal (unknown) variances. We study the total least squares estimator of X, which, in the case of normal errors, coincides with the maximum likelihood one. We give conditions for asymptotic normality of the estimator when the number of rows in A is increasing. Under mild assumptions, the covariance structure of the limit Gaussian random matrix is nonsingular. For normal errors, the results can be used to construct an asymptotic confidence interval for a linear functional of X.

Equivariant adjusted least squares estimator in two-line fitting model

Sergiy Shklyar

https://doi.org/10.15559/16-VMSTA47

Pub. online: 21 Mar 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 1 (2016), pp. 19–45

Abstract

We consider the two-line fitting problem. True points lie on two straight lines and are observed with Gaussian perturbations. For each observed point, it is not known on which line the corresponding true point lies. The parameters of the lines are estimated.

This model is a restriction of the conic section fitting model because a couple of two lines is a degenerate conic section. The following estimators are constructed: two projections of the adjusted least squares estimator in the conic section fitting model, orthogonal regression estimator, parametric maximum likelihood estimator in the Gaussian model, and regular best asymptotically normal moment estimator.

The conditions for the consistency and asymptotic normality of the projections of the adjusted least squares estimator are provided. All the estimators constructed in the paper are equivariant. The estimators are compared numerically.

Functional limit theorems for additive and multiplicative schemes in the Cox–Ingersoll–Ross model

Yuliia Mishura Yevheniia Munchak

https://doi.org/10.15559/16-VMSTA48

Pub. online: 3 Mar 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 1 (2016), pp. 1–17

Abstract

In this paper, we consider the Cox–Ingersoll–Ross (CIR) process in the regime where the process does not hit zero. We construct additive and multiplicative discrete approximation schemes for the price of asset that is modeled by the CIR process and geometric CIR process. In order to construct these schemes, we take the Euler approximations of the CIR process itself but replace the increments of the Wiener process with iid bounded vanishing symmetric random variables. We introduce a “truncated” CIR process and apply it to prove the weak convergence of asset prices. We establish the fact that this “truncated” process does not hit zero under the same condition considered for the original nontruncated process.

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Online ISSN: 2351-6054
Print ISSN: 2351-6046

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