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Convergence of hitting times for jump-diffusion processes

Georgiy Shevchenko

https://doi.org/10.15559/15-VMSTA32

Pub. online: 23 Sep 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 3 (2015): PRESTO-2015, pp. 203–218

Abstract

We investigate the convergence of hitting times for jump-diffusion processes. Specifically, we study a sequence of stochastic differential equations with jumps. Under reasonable assumptions, we establish the convergence of solutions to the equations and of the moments when the solutions hit certain sets.

Convergence of hitting times for jump-diffusion processes

Georgiy Shevchenko

https://doi.org/10.15559/15-MSTA32

Pub. online: 23 Sep 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 2, Issue 3 (2015): PRESTO-2015, pp. 203–218

Abstract

A group action on increasing sequences of set-indexed Brownian motions

Arthur Yosef

https://doi.org/10.15559/15-VMSTA31

Pub. online: 6 Aug 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 2 (2015), pp. 185–198

Abstract

We prove that a square-integrable set-indexed stochastic process is a set-indexed Brownian motion if and only if its projection on all the strictly increasing continuous sequences are one-parameter G-time-changed Brownian motions. In addition, we study the “sequence-independent variation” property for group stationary-increment stochastic processes in general and for a set-indexed Brownian motion in particular. We present some applications.

A group action on increasing sequences of set-indexed Brownian motions

Arthur Yosef

https://doi.org/10.15559/15-MSTA31

Pub. online: 6 Aug 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 2, Issue 2 (2015), pp. 185–198

Abstract

A Lundberg-type inequality for an inhomogeneous renewal risk model

Ieva Marija Andrulytė Emilija Bernackaitė Dominyka Kievinaitė Jonas Šiaulys

https://doi.org/10.15559/15-VMSTA30

Pub. online: 31 Jul 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 2 (2015), pp. 173–184

Abstract

We obtain a Lundberg-type inequality in the case of an inhomogeneous renewal risk model. We consider the model with independent, but not necessarily identically distributed, claim sizes and the interoccurrence times. In order to prove the main theorem, we first formulate and prove an auxiliary lemma on large values of a sum of random variables asymptotically drifted in the negative direction.

A Lundberg-type inequality for an inhomogeneous renewal risk model

Ieva Marija Andrulytė Emilija Bernackaitė Dominyka Kievinaitė Jonas Šiaulys

https://doi.org/10.15559/15-MSTA30

Pub. online: 31 Jul 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 2, Issue 2 (2015), pp. 173–184

Abstract

Fast

L_{2}

-approximation of integral-type functionals of Markov processes

Iurii Ganychenko

https://doi.org/10.15559/15-VMSTA29

Pub. online: 28 Jul 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 2 (2015), pp. 165–171

Abstract

In this paper, we provide strong $∈_{8}$-rates of approximation of the integral-type functionals of Markov $w ∈ R$ processes by integral sums. We improve the method developed in [2]. Under assumptions on the process formulated only in terms of its transition probability density, we get the accuracy that coincides with that obtained in [3] for a one-dimensional diffusion process.

Fast

L_{2}

-approximation of integral-type functionals of Markov processes

Iurii Ganychenko

https://doi.org/10.15559/15-MSTA29

Pub. online: 28 Jul 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 2, Issue 2 (2015), pp. 165–171

Abstract

In this paper, we provide strong $L_{2}$-rates of approximation of the integral-type functionals of Markov processes by integral sums. We improve the method developed in [2]. Under assumptions on the process formulated only in terms of its transition probability density, we get the accuracy that coincides with that obtained in [3] for a one-dimensional diffusion process.

Construction of maximum likelihood estimator in the mixed fractional–fractional Brownian motion model with double long-range dependence

Yuliya Mishura Ivan Voronov

https://doi.org/10.15559/15-VMSTA28

Pub. online: 20 Jul 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 2 (2015), pp. 147–164

Abstract

We construct an estimator of the unknown drift parameter $\theta \in \mathbb{R}$ in the linear model

\[X_{t}=\theta t+\sigma _{1}{B}^{H_{1}}(t)+\sigma _{2}{B}^{H_{2}}(t),\hspace{0.2778em}t\in [0,T],\]

where ${B}^{H_{1}}$ and ${B}^{H_{2}}$ are two independent fractional Brownian motions with Hurst indices $H_{1}$ and $H_{2}$ satisfying the condition $\frac{1}{2}\le H_{1}<H_{2}<1$. Actually, we reduce the problem to the solution of the integral Fredholm equation of the 2nd kind with a specific weakly singular kernel depending on two power exponents. It is proved that the kernel can be presented as the product of a bounded continuous multiplier and weak singular one, and this representation allows us to prove the compactness of the corresponding integral operator. This, in turn, allows us to establish an existence–uniqueness result for the sequence of the equations on the increasing intervals, to construct accordingly a sequence of statistical estimators, and to establish asymptotic consistency.

Construction of maximum likelihood estimator in the mixed fractional–fractional Brownian motion model with double long-range dependence

Yuliya Mishura Ivan Voronov

https://doi.org/10.15559/15-MSTA28

Pub. online: 20 Jul 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 2, Issue 2 (2015), pp. 147–164

Abstract

We construct an estimator of the unknown drift parameter $\theta \in \mathbb{R}$ in the linear model

\[X_{t}=\theta t+\sigma _{1}{B}^{H_{1}}(t)+\sigma _{2}{B}^{H_{2}}(t),\hspace{0.2778em}t\in [0,T],\]

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