Latest articles of Modern Stochastics: Theory and Applications
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https://www.vmsta.org/https://www.vmsta.org/Latest articles of Modern Stochastics: Theory and Applications
http://www.vmsta.org/journal/VMSTA/feeds/latest
enThu, 13 Aug 2020 10:56:29 +0300<![CDATA[Simple approximations for the ruin probability in the risk model with stochastic premiums and a constant dividend strategy]]>
https://www.vmsta.org/journal/VMSTA/article/183
https://www.vmsta.org/journal/VMSTA/article/183We 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. PDFXML]]>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. PDFXML]]>Olena RagulinaTue, 04 Aug 2020 00:00:00 +0300<![CDATA[Approximations of the ruin probability in a discrete time risk model]]>
https://www.vmsta.org/journal/VMSTA/article/184
https://www.vmsta.org/journal/VMSTA/article/184Based 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. PDFXML]]>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. PDFXML]]>David J. Santana,Luis RincónTue, 04 Aug 2020 00:00:00 +0300<![CDATA[Distance from fractional Brownian motion with associated Hurst index 0]]>
https://www.vmsta.org/journal/VMSTA/article/182
https://www.vmsta.org/journal/VMSTA/article/182We 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, $\gamma >0$. PDFXML]]>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, $\gamma >0$. PDFXML]]>Oksana Banna,Filipp Buryak,Yuliya MishuraTue, 23 Jun 2020 00:00:00 +0300<![CDATA[Irregular barrier reflected BDSDEs with general jumps under stochastic Lipschitz and linear growth conditions]]>
https://www.vmsta.org/journal/VMSTA/article/181
https://www.vmsta.org/journal/VMSTA/article/181In 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. PDFXML]]>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. PDFXML]]>Mohamed Marzougue,Yaya SagnaWed, 10 Jun 2020 00:00:00 +0300<![CDATA[Single jump filtrations and local martingales]]>
https://www.vmsta.org/journal/VMSTA/article/179
https://www.vmsta.org/journal/VMSTA/article/179A 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. PDFXML]]>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. PDFXML]]>Alexander A. GushchinMon, 25 May 2020 00:00:00 +0300<![CDATA[Prediction in polynomial errors-in-variables models]]>
https://www.vmsta.org/journal/VMSTA/article/180
https://www.vmsta.org/journal/VMSTA/article/180A 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. PDFXML]]>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. PDFXML]]>Alexander Kukush,Ivan SenkoMon, 25 May 2020 00:00:00 +0300<![CDATA[A pure-jump mean-reverting short rate model]]>
https://www.vmsta.org/journal/VMSTA/article/178
https://www.vmsta.org/journal/VMSTA/article/178A 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. PDFXML]]>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. PDFXML]]>Markus HessMon, 20 Apr 2020 00:00:00 +0300<![CDATA[Alternative probabilistic representations of Barenblatt-type solutions]]>
https://www.vmsta.org/journal/VMSTA/article/177
https://www.vmsta.org/journal/VMSTA/article/177A general class of probability density functions

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. PDFXML]]>A general class of probability density functions

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. PDFXML]]>Alessandro De Gregorio,Roberto GarraMon, 23 Mar 2020 00:00:00 +0200<![CDATA[Stochastic two-species mutualism model with jumps]]>
https://www.vmsta.org/journal/VMSTA/article/176
https://www.vmsta.org/journal/VMSTA/article/176The 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. PDFXML]]>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. PDFXML]]>Olga Borysenko,Oleksandr BorysenkoTue, 03 Mar 2020 00:00:00 +0200<![CDATA[Pathwise asymptotics for Volterra processes conditioned to a noisy version of the Brownian motion]]>
https://www.vmsta.org/journal/VMSTA/article/175
https://www.vmsta.org/journal/VMSTA/article/175In this paper we investigate a problem of large deviations for continuous Volterra processes under the influence of model disturbances. More precisely, we study the behavior, in the near future after T, of a Volterra process driven by a Brownian motion in a case where the Brownian motion is not directly observable, but only a noisy version is observed or some linear functionals of the noisy version are observed. Some examples are discussed in both cases. PDFXML]]>In this paper we investigate a problem of large deviations for continuous Volterra processes under the influence of model disturbances. More precisely, we study the behavior, in the near future after T, of a Volterra process driven by a Brownian motion in a case where the Brownian motion is not directly observable, but only a noisy version is observed or some linear functionals of the noisy version are observed. Some examples are discussed in both cases. PDFXML]]>Barbara PacchiarottiThu, 27 Feb 2020 00:00:00 +0200