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On infinite divisibility of a class of two-dimensional vectors in the second Wiener chaos

Andreas Basse-O’Connor Jan Pedersen Victor Rohde

https://doi.org/10.15559/20-VMSTA160

Pub. online: 28 Aug 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 3 (2020), pp. 267–289

Abstract

Infinite divisibility of a class of two-dimensional vectors with components in the second Wiener chaos is studied. Necessary and sufficient conditions for infinite divisibility are presented as well as more easily verifiable sufficient conditions. The case where both components consist of a sum of two Gaussian squares is treated in more depth, and it is conjectured that such vectors are infinitely divisible.

Simple approximations for the ruin probability in the risk model with stochastic premiums and a constant dividend strategy

Olena Ragulina

https://doi.org/10.15559/20-VMSTA157

Pub. online: 4 Aug 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 3 (2020), pp. 245–265

Abstract

We deal with a generalization of the risk model with stochastic premiums where dividends are paid according to a constant dividend strategy and consider heuristic approximations for the ruin probability. To be more precise, we construct five- and three-moment analogues to the De Vylder approximation. To this end, we obtain an explicit formula for the ruin probability in the case of exponentially distributed premium and claim sizes. Finally, we analyze the accuracy of the approximations for some typical distributions of premium and claim sizes using statistical estimates obtained by the Monte Carlo methods.

Approximations of the ruin probability in a discrete time risk model

David J. Santana Luis Rincón

https://doi.org/10.15559/20-VMSTA158

Pub. online: 4 Aug 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 3 (2020), pp. 221–243

Abstract

Based on a discrete version of the Pollaczeck–Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber–Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.

Distance from fractional Brownian motion with associated Hurst index

0 < H < 1 / 2

to the subspaces of Gaussian martingales involving power integrands with an arbitrary positive exponent

Oksana Banna

Filipp Buryak Yuliya Mishura

https://doi.org/10.15559/20-VMSTA156

Pub. online: 23 Jun 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 2 (2020), pp. 191–202

Abstract

We find the best approximation of the fractional Brownian motion with the Hurst index $H\in (0,1/2)$ by Gaussian martingales of the form ${\textstyle\int _{0}^{t}}{s^{\gamma }}d{W_{s}}$, where W is a Wiener process, $\gamma >0$.

Irregular barrier reflected BDSDEs with general jumps under stochastic Lipschitz and linear growth conditions

Mohamed Marzougue

Yaya Sagna

https://doi.org/10.15559/20-VMSTA155

Pub. online: 10 Jun 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 2 (2020), pp. 157–190

Abstract

In this paper, a solution is given to reflected backward doubly stochastic differential equations when the barrier is not necessarily right-continuous, and the noise is driven by two independent Brownian motions and an independent Poisson random measure. The existence and uniqueness of the solution is shown, firstly when the coefficients are stochastic Lipschitz, and secondly by weakening the conditions on the stochastic growth coefficient.

Prediction in polynomial errors-in-variables models

Alexander Kukush Ivan Senko

https://doi.org/10.15559/20-VMSTA154

Pub. online: 25 May 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 2 (2020), pp. 203–219

Abstract

A multivariate errors-in-variables (EIV) model with an intercept term, and a polynomial EIV model are considered. Focus is made on a structural homoskedastic case, where vectors of covariates are i.i.d. and measurement errors are i.i.d. as well. The covariates contaminated with errors are normally distributed and the corresponding classical errors are also assumed normal. In both models, it is shown that (inconsistent) ordinary least squares estimators of regression parameters yield an a.s. approximation to the best prediction of response given the values of observable covariates. Thus, not only in the linear EIV, but in the polynomial EIV models as well, consistent estimators of regression parameters are useless in the prediction problem, provided the size and covariance structure of observation errors for the predicted subject do not differ from those in the data used for the model fitting.

Single jump filtrations and local martingales

Alexander A. Gushchin

https://doi.org/10.15559/20-VMSTA153

Pub. online: 25 May 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 2 (2020), pp. 135–156

Abstract

A single jump filtration ${({\mathcal{F}_{t}})_{t\in {\mathbb{R}_{+}}}}$ generated by a random variable γ with values in ${\overline{\mathbb{R}}_{+}}$ on a probability space $(\Omega ,\mathcal{F},\mathsf{P})$ is defined as follows: a set $A\in \mathcal{F}$ belongs to ${\mathcal{F}_{t}}$ if $A\cap \{\gamma >t\}$ is either ∅ or $\{\gamma >t\}$. A process M is proved to be a local martingale with respect to this filtration if and only if it has a representation ${M_{t}}=F(t){\mathbb{1}_{\{t<\gamma \}}}+L{\mathbb{1}_{\{t\geqslant \gamma \}}}$, where F is a deterministic function and L is a random variable such that $\mathsf{E}|{M_{t}}|<\infty $ and $\mathsf{E}({M_{t}})=\mathsf{E}({M_{0}})$ for every $t\in \{t\in {\mathbb{R}_{+}}:\mathsf{P}(\gamma \geqslant t)>0\}$. This result seems to be new even in a special case that has been studied in the literature, namely, where $\mathcal{F}$ is the smallest σ-field with respect to which γ is measurable (and then the filtration is the smallest one with respect to which γ is a stopping time). As a consequence, a full description of all local martingales is given and they are classified according to their global behaviour.

A pure-jump mean-reverting short rate model

Markus Hess

https://doi.org/10.15559/20-VMSTA152

Pub. online: 20 Apr 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 2 (2020), pp. 113–134

Abstract

A new multi-factor short rate model is presented which is bounded from below by a real-valued function of time. The mean-reverting short rate process is modeled by a sum of pure-jump Ornstein–Uhlenbeck processes such that the related bond prices possess affine representations. Also the dynamics of the associated instantaneous forward rate is provided and a condition is derived under which the model can be market-consistently calibrated. The analytical tractability of this model is illustrated by the derivation of an explicit plain vanilla option price formula. With view on practical applications, suitable probability distributions are proposed for the driving jump processes. The paper is concluded by presenting a post-crisis extension of the proposed short and forward rate model.

Alternative probabilistic representations of Barenblatt-type solutions

Alessandro De Gregorio

Roberto Garra

https://doi.org/10.15559/20-VMSTA151

Pub. online: 23 Mar 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 1 (2020), pp. 97–112

Abstract

A general class of probability density functions

\[ u(x,t)=C{t^{-\alpha d}}{\left(1-{\left(\frac{\| x\| }{c{t^{\alpha }}}\right)^{\beta }}\right)_{+}^{\gamma }},\hspace{1em}x\in {\mathbb{R}^{d}},t>0,\]

is considered, containing as particular case the Barenblatt solutions arising, for instance, in the study of nonlinear heat equations. Alternative probabilistic representations of the Barenblatt-type solutions $u(x,t)$ are proposed. In the one-dimensional case, by means of this approach, $u(x,t)$ can be connected with the wave propagation.

Stochastic two-species mutualism model with jumps

Olga Borysenko Oleksandr Borysenko

https://doi.org/10.15559/20-VMSTA150

Pub. online: 3 Mar 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 1 (2020), pp. 1–15

Abstract

The existence and uniqueness are proved for the global positive solution to the system of stochastic differential equations describing a two-species mutualism model disturbed by the white noise, the centered and non-centered Poisson noises. We obtain sufficient conditions for stochastic ultimate boundedness, stochastic permanence, nonpersistence in the mean, strong persistence in the mean and extinction of the solution to the considered system.

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

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