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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.

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-MSTA58

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

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.

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

Vitalii Makogin

https://doi.org/10.15559/16-MSTA59

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

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.

Randomly stopped sums with consistently varying distributions

Edita Kizinevič Jonas Sprindys Jonas Šiaulys

https://doi.org/10.15559/16-MSTA60

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

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.

Large deviation principle for one-dimensional SDEs with discontinuous coefficients

Alexei Kulik Daryna Sobolieva

https://doi.org/10.15559/16-MSTA57

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

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.

Approximations for a solution to stochastic heat equation with stable noise

Larysa Pryhara Georgiy Shevchenko

https://doi.org/10.15559/16-MSTA56

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

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