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Consistency of the total least squares estimator in the linear errors-in-variables regression

Sergiy Shklyar

https://doi.org/10.15559/18-VMSTA104

Pub. online: 30 May 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 3 (2018), pp. 247–295

Abstract

This paper deals with a homoskedastic errors-in-variables linear regression model and properties of the total least squares (TLS) estimator. We partly revise the consistency results for the TLS estimator previously obtained by the author [18]. We present complete and comprehensive proofs of consistency theorems. A theoretical foundation for construction of the TLS estimator and its relation to the generalized eigenvalue problem is explained. Particularly, the uniqueness of the estimate is proved. The Frobenius norm in the definition of the estimator can be substituted by the spectral norm, or by any other unitarily invariant norm; then the consistency results are still valid.

On closeness of two discrete weighted sums

Vydas Čekanavičius Palaniappan Vellaisamy

https://doi.org/10.15559/18-VMSTA103

Pub. online: 21 May 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 2 (2018), pp. 207–224

Abstract

The effect that weighted summands have on each other in approximations of $S={w_{1}}{S_{1}}+{w_{2}}{S_{2}}+\cdots +{w_{N}}{S_{N}}$ is investigated. Here, ${S_{i}}$’s are sums of integer-valued random variables, and ${w_{i}}$ denote weights, $i=1,\dots ,N$. Two cases are considered: the general case of independent random variables when their closeness is ensured by the matching of factorial moments and the case when the ${S_{i}}$ has the Markov Binomial distribution. The Kolmogorov metric is used to estimate the accuracy of approximation.

Large deviations of regression parameter estimator in continuous-time models with sub-Gaussian noise

Alexander V. Ivanov

Igor V. Orlovskyi

https://doi.org/10.15559/18-VMSTA102

Pub. online: 7 May 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 2 (2018), pp. 191–206

Abstract

A continuous-time regression model with a jointly strictly sub-Gaussian random noise is considered in the paper. Upper exponential bounds for probabilities of large deviations of the least squares estimator for the regression parameter are obtained.

Properties of Poisson processes directed by compound Poisson-Gamma subordinators

Khrystyna Buchak Lyudmyla Sakhno

https://doi.org/10.15559/18-VMSTA101

Pub. online: 2 May 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 2 (2018), pp. 167–189

Abstract

In the paper we consider time-changed Poisson processes where the time is expressed by compound Poisson-Gamma subordinators $G(N(t))$ and derive the expressions for their hitting times. We also study the time-changed Poisson processes where the role of time is played by the processes of the form $G(N(t)+at)$ and by the iteration of such processes.

Computation of option greeks under hybrid stochastic volatility models via Malliavin calculus

Bilgi Yilmaz

https://doi.org/10.15559/18-VMSTA100

Pub. online: 24 Apr 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 2 (2018), pp. 145–165

Abstract

This study introduces computation of option sensitivities (Greeks) using the Malliavin calculus under the assumption that the underlying asset and interest rate both evolve from a stochastic volatility model and a stochastic interest rate model, respectively. Therefore, it integrates the recent developments in the Malliavin calculus for the computation of Greeks: Delta, Vega, and Rho and it extends the method slightly. The main results show that Malliavin calculus allows a running Monte Carlo (MC) algorithm to present numerical implementations and to illustrate its effectiveness. The main advantage of this method is that once the algorithms are constructed, they can be used for numerous types of option, even if their payoff functions are not differentiable.

Exponential bounds for the tail probability of the supremum of an inhomogeneous random walk

Dominyka Kievinaitė Jonas Šiaulys

https://doi.org/10.15559/18-VMSTA99

Pub. online: 15 Mar 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 2 (2018), pp. 129–143

Abstract

Let $\{{\xi _{1}},{\xi _{2}},\dots \}$ be a sequence of independent but not necessarily identically distributed random variables. In this paper, the sufficient conditions are found under which the tail probability $\mathbb{P}(\,{\sup _{n\geqslant 0}}\,{\sum _{i=1}^{n}}{\xi _{i}}>x)$ can be bounded above by ${\varrho _{1}}\exp \{-{\varrho _{2}}x\}$ with some positive constants ${\varrho _{1}}$ and ${\varrho _{2}}$. A way to calculate these two constants is presented. The application of the derived bound is discussed and a Lundberg-type inequality is obtained for the ultimate ruin probability in the inhomogeneous renewal risk model satisfying the net profit condition on average.

On backward Kolmogorov equation related to CIR process

Vigirdas Mackevičius Gabrielė Mongirdaitė

https://doi.org/10.15559/18-VMSTA98

Pub. online: 6 Mar 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 1 (2018), pp. 113–127

Abstract

We consider the existence of a classical smooth solution to the backward Kolmogorov equation

\[ \left\{\begin{array}{l@{\hskip10.0pt}l}\partial _{t}u(t,x)=Au(t,x),\hspace{1em}& x\ge 0,\hspace{2.5pt}t\in [0,T],\\{} u(0,x)=f(x),\hspace{1em}& x\ge 0,\end{array}\right.\]

where A is the generator of the CIR process, the solution to the stochastic differential equation

\[ {X_{t}^{x}}=x+{\int _{0}^{t}}\theta \big(\kappa -{X_{s}^{x}}\big)\hspace{0.1667em}ds+\sigma {\int _{0}^{t}}\sqrt{{X_{s}^{x}}}\hspace{0.1667em}dB_{s},\hspace{1em}x\ge 0,\hspace{2.5pt}t\in [0,T],\]

that is, $Af(x)=\theta (\kappa -x){f^{\prime }}(x)+\frac{1}{2}{\sigma }^{2}x{f^{\prime\prime }}(x)$, $x\ge 0$ ($\theta ,\kappa ,\sigma >0$). Alfonsi [1] showed that the equation has a smooth solution with partial derivatives of polynomial growth, provided that the initial function f is smooth with derivatives of polynomial growth. His proof was mainly based on the analytical formula for the transition density of the CIR process in the form of a rather complicated function series. In this paper, for a CIR process satisfying the condition ${\sigma }^{2}\le 4\theta \kappa $, we present a direct proof based on the representation of a CIR process in terms of a squared Bessel process and its additivity property.

Fractional Cox–Ingersoll–Ross process with non-zero «mean»

Yuliya Mishura Anton Yurchenko-Tytarenko

https://doi.org/10.15559/18-VMSTA97

Pub. online: 5 Mar 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 1 (2018), pp. 99–111

Abstract

In this paper we define the fractional Cox–Ingersoll–Ross process as $X_{t}:={Y_{t}^{2}}\mathbf{1}_{\{t<\inf \{s>0:Y_{s}=0\}\}}$, where the process $Y=\{Y_{t},t\ge 0\}$ satisfies the SDE of the form $dY_{t}=\frac{1}{2}(\frac{k}{Y_{t}}-aY_{t})dt+\frac{\sigma }{2}d{B_{t}^{H}}$, $\{{B_{t}^{H}},t\ge 0\}$ is a fractional Brownian motion with an arbitrary Hurst parameter $H\in (0,1)$. We prove that $X_{t}$ satisfies the stochastic differential equation of the form $dX_{t}=(k-aX_{t})dt+\sigma \sqrt{X_{t}}\circ d{B_{t}^{H}}$, where the integral with respect to fractional Brownian motion is considered as the pathwise Stratonovich integral. We also show that for $k>0$, $H>1/2$ the process is strictly positive and never hits zero, so that actually $X_{t}={Y_{t}^{2}}$. Finally, we prove that in the case of $H<1/2$ the probability of not hitting zero on any fixed finite interval by the fractional Cox–Ingersoll–Ross process tends to 1 as $k\to \infty $.

Cliquet option pricing with Meixner processes

Markus Hess

https://doi.org/10.15559/18-VMSTA96

Pub. online: 12 Feb 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 1 (2018), pp. 81–97

Abstract

We investigate the pricing of cliquet options in a geometric Meixner model. The considered option is of monthly sum cap style while the underlying stock price model is driven by a pure-jump Meixner–Lévy process yielding Meixner distributed log-returns. In this setting, we infer semi-analytic expressions for the cliquet option price by using the probability distribution function of the driving Meixner–Lévy process and by an application of Fourier transform techniques. In an introductory section, we compile various facts on the Meixner distribution and the related class of Meixner–Lévy processes. We also propose a customized measure change preserving the Meixner distribution of any Meixner process.

On aggregation of multitype Galton–Watson branching processes with immigration

Mátyás Barczy Fanni K. Nedényi Gyula Pap

https://doi.org/10.15559/18-VMSTA95

Pub. online: 1 Feb 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 1 (2018), pp. 53–79

Abstract

Limit behaviour of temporal and contemporaneous aggregations of independent copies of a stationary multitype Galton–Watson branching process with immigration is studied in the so-called iterated and simultaneous cases, respectively. In both cases, the limit process is a zero mean Brownian motion with the same covariance function under third order moment conditions on the branching and immigration distributions. We specialize our results for generalized integer-valued autoregressive processes and single-type Galton–Watson processes with immigration as well.

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