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

Dominyka Kievinaitė Jonas Šiaulys

https://doi.org/10.15559/18-MSTA99

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.

On backward Kolmogorov equation related to CIR process

Vigirdas Mackevičius Gabrielė Mongirdaitė

https://doi.org/10.15559/18-MSTA98

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

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

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 $.

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

Yuliya Mishura Anton Yurchenko-Tytarenko

https://doi.org/10.15559/18-MSTA97

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

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.

Cliquet option pricing with Meixner processes

Markus Hess

https://doi.org/10.15559/18-MSTA96

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

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.

On aggregation of multitype Galton–Watson branching processes with immigration

Mátyás Barczy Fanni K. Nedényi Gyula Pap

https://doi.org/10.15559/18-MSTA95

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

Confidence regions in Cox proportional hazards model with measurement errors and unbounded parameter set

Oksana Chernova Alexander Kukush

https://doi.org/10.15559/18-VMSTA94

Pub. online: 31 Jan 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 1 (2018), pp. 37–52

Abstract

Cox proportional hazards model with measurement errors is considered. In Kukush and Chernova (2017), we elaborated a simultaneous estimator of the baseline hazard rate $\lambda (\cdot )$ and the regression parameter β, with the unbounded parameter set $\varTheta =\varTheta _{\lambda }\times \varTheta _{\beta }$, where $\varTheta _{\lambda }$ is a closed convex subset of $C[0,\tau ]$ and $\varTheta _{\beta }$ is a compact set in ${\mathbb{R}}^{m}$. The estimator is consistent and asymptotically normal. In the present paper, we construct confidence intervals for integral functionals of $\lambda (\cdot )$ and a confidence region for β under restrictions on the error distribution. In particular, we handle the following cases: (a) the measurement error is bounded, (b) it is a normally distributed random vector, and (c) it has independent components which are shifted Poisson random variables.

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