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Asymptotic behavior of homogeneous additive functionals of the solutions of Itô stochastic differential equations with nonregular dependence on parameter

Grigorij Kulinich Svitlana Kushnirenko Yuliia Mishura

https://doi.org/10.15559/16-VMSTA58

Pub. online: 4 Jul 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 191–208

Abstract

We study the asymptotic behavior of mixed functionals of the form $I_{T}(t)=F_{T}(\xi _{T}(t))+{\int _{0}^{t}}g_{T}(\xi _{T}(s))\hspace{0.1667em}d\xi _{T}(s)$, $t\ge 0$, as $T\to \infty $. Here $\xi _{T}(t)$ is a strong solution of the stochastic differential equation $d\xi _{T}(t)=a_{T}(\xi _{T}(t))\hspace{0.1667em}dt+dW_{T}(t)$, $T>0$ is a parameter, $a_{T}=a_{T}(x)$ are measurable functions such that $\left|a_{T}(x)\right|\le C_{T}$ for all $x\in \mathbb{R}$, $W_{T}(t)$ are standard Wiener processes, $F_{T}=F_{T}(x)$, $x\in \mathbb{R}$, are continuous functions, $g_{T}=g_{T}(x)$, $x\in \mathbb{R}$, are locally bounded functions, and everything is real-valued. The explicit form of the limiting processes for $I_{T}(t)$ is established under very nonregular dependence of $g_{T}$ and $a_{T}$ on the parameter T.

Simulation paradoxes related to a fractional Brownian motion with small Hurst index

Vitalii Makogin

https://doi.org/10.15559/16-VMSTA59

Pub. online: 4 Jul 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 181–190

Abstract

We consider the simulation of sample paths of a fractional Brownian motion with small values of the Hurst index and estimate the behavior of the expected maximum. We prove that, for each fixed N, the error of approximation $\mathbf{E}\max _{t\in [0,1]}{B}^{H}(t)-\mathbf{E}\max _{i=\overline{1,N}}{B}^{H}(i/N)$ grows rapidly to ∞ as the Hurst index tends to 0.

Randomly stopped sums with consistently varying distributions

Edita Kizinevič Jonas Sprindys Jonas Šiaulys

https://doi.org/10.15559/16-VMSTA60

Pub. online: 4 Jul 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 165–179

Abstract

Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables, and η be a counting random variable independent of this sequence. We consider conditions for $\{\xi _{1},\xi _{2},\dots \}$ 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 consistently varying distributions. In our consideration, the random variables $\{\xi _{1},\xi _{2},\dots \}$ are not necessarily identically distributed.

Large deviation principle for one-dimensional SDEs with discontinuous coefficients

Alexei Kulik Daryna Sobolieva

https://doi.org/10.15559/16-VMSTA57

Pub. online: 1 Jul 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 145–164

Abstract

We establish the large deviation principle for solutions of one-dimensional SDEs with discontinuous coefficients. The main statement is formulated in a form similar to the classical Wentzel–Freidlin theorem, but under the considerably weaker assumption that the coefficients have no discontinuities of the second kind.

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.

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