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Integral representation with respect to fractional Brownian motion under a log-Hölder assumption

Taras Shalaiko Georgiy Shevchenko

https://doi.org/10.15559/15-VMSTA35CNF

Pub. online: 25 Sep 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 3 (2015): PRESTO-2015, pp. 219–232

Abstract

We show that if a random variable is the final value of an adapted log-Hölder continuous process, then it can be represented as a stochastic integral with respect to a fractional Brownian motion with adapted integrand. In order to establish this representation result, we extend the definition of the fractional integral.

A multiplicative wavelet-based model for simulation of a random process

Ievgen Turchyn

https://doi.org/10.15559/15-VMSTA33

Pub. online: 24 Sep 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 4 (2015), pp. 309–325

Abstract

We find a multiplicative wavelet-based representation for stochastic processes that can be represented as the exponent of a second-order centered random process. We propose a wavelet-based model for simulation of such a stochastic process and find its rates of convergence to the process in different functional spaces in terms of approximation with given accuracy and reliability. This approach allows us to simulate stochastic processes (including certain classes of processes with heavy tails) with given accuracy and reliability.

Weak approximation rates for integral functionals of Markov processes

Iurii Ganychenko Alexei Kulik

https://doi.org/10.15559/15-VMSTA37CNF

Pub. online: 23 Sep 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 3 (2015): PRESTO-2015, pp. 251–266

Abstract

We obtain weak rates for approximation of an integral functional of a Markov process by integral sums. An assumption on the process is formulated only in terms of its transition probability density, and, therefore, our approach is not strongly dependent on the structure of the process. Applications to the estimates of the rates of approximation of the Feynman–Kac semigroup and of the price of “occupation-time options” are provided.

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.

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 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.

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 $L_{2}$-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.

Identifiability of logistic regression with homoscedastic error: Berkson model

Sergiy Shklyar

https://doi.org/10.15559/15-VMSTA27

Pub. online: 7 Jul 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 2 (2015), pp. 131–146

Abstract

We consider the Berkson model of logistic regression with Gaussian and homoscedastic error in regressor. The measurement error variance can be either known or unknown. We deal with both functional and structural cases. Sufficient conditions for identifiability of regression coefficients are presented.

Conditions for identifiability of the model are studied. In the case where the error variance is known, the regression parameters are identifiable if the distribution of the observed regressor is not concentrated at a single point. In the case where the error variance is not known, the regression parameters are identifiable if the distribution of the observed regressor is not concentrated at three (or less) points.

The key analytic tools are relations between the smoothed logistic distribution function and its derivatives.

On the Feynman–Kac semigroup for some Markov processes

Victoria Knopova

https://doi.org/10.15559/15-VMSTA26

Pub. online: 18 Jun 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 2 (2015), pp. 107–129

Abstract

For a (non-symmetric) strong Markov process X, consider the Feynman–Kac semigroup

\[{T_{t}^{A}}f(x):={\mathbb{E}}^{x}\big[{e}^{A_{t}}f(X_{t})\big],\hspace{1em}x\in {\mathbb{R}}^{n},\hspace{2.5pt}t>0,\]

where A is a continuous additive functional of X associated with some signed measure. Under the assumption that X admits a transition probability density that possesses upper and lower bounds of certain type, we show that the kernel corresponding to ${T_{t}^{A}}$ possesses the density ${p_{t}^{A}}(x,y)$ with respect to the Lebesgue measure and construct upper and lower bounds for ${p_{t}^{A}}(x,y)$. Some examples are provided.

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

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