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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-MSTA102

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.

Properties of Poisson processes directed by compound Poisson-Gamma subordinators

Khrystyna Buchak Lyudmyla Sakhno

https://doi.org/10.15559/18-MSTA101

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

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.

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

Bilgi Yilmaz

https://doi.org/10.15559/18-MSTA100

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

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.

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

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 $.

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