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
enSat, 15 Dec 2018 06:32:22 +0200<![CDATA[Existence and uniqueness of mild solution to fractional stochastic heat equation]]>
https://www.vmsta.org/journal/VMSTA/article/138
https://www.vmsta.org/journal/VMSTA/article/138For a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset {\mathbb{R}^{d}}$ and driven by an ${L^{2}}(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$, a new result on existence and uniqueness of a mild solution is established. Compared to the existing results, the uniqueness in a fully nonlinear case is shown, not assuming the coefficient in front of the noise to be affine. Additionally, the existence of moments for the solution is established. PDFXML]]>For a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset {\mathbb{R}^{d}}$ and driven by an ${L^{2}}(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$, a new result on existence and uniqueness of a mild solution is established. Compared to the existing results, the uniqueness in a fully nonlinear case is shown, not assuming the coefficient in front of the noise to be affine. Additionally, the existence of moments for the solution is established. PDFXML]]>Kostiantyn Ralchenko,Georgiy ShevchenkoWed, 12 Dec 2018 00:00:00 +0200<![CDATA[Studies on generalized Yule models]]>
https://www.vmsta.org/journal/VMSTA/article/137
https://www.vmsta.org/journal/VMSTA/article/137We present a generalization of the Yule model for macroevolution in which, for the appearance of genera, we consider point processes with the order statistics property, while for the growth of species we use nonlinear time-fractional pure birth processes or a critical birth-death process. Further, in specific cases we derive the explicit form of the distribution of the number of species of a genus chosen uniformly at random for each time. Besides, we introduce a time-changed mixed Poisson process with the same marginal distribution as that of the time-fractional Poisson process. PDFXML]]>We present a generalization of the Yule model for macroevolution in which, for the appearance of genera, we consider point processes with the order statistics property, while for the growth of species we use nonlinear time-fractional pure birth processes or a critical birth-death process. Further, in specific cases we derive the explicit form of the distribution of the number of species of a genus chosen uniformly at random for each time. Besides, we introduce a time-changed mixed Poisson process with the same marginal distribution as that of the time-fractional Poisson process. PDFXML]]>Federico PolitoMon, 03 Dec 2018 00:00:00 +0200<![CDATA[Option pricing in time-changed Lévy models with compound Poisson jumps]]>
https://www.vmsta.org/journal/VMSTA/article/136
https://www.vmsta.org/journal/VMSTA/article/136The problem of European-style option pricing in time-changed Lévy models in the presence of compound Poisson jumps is considered. These jumps relate to sudden large drops in stock prices induced by political or economical hits. As the time-changed Lévy models, the variance-gamma and the normal-inverse Gaussian models are discussed. Exact formulas are given for the price of digital asset-or-nothing call option on extra asset in foreign currency. The prices of simpler options can be derived as corollaries of our results and examples are presented. Various types of dependencies between stock prices are mentioned. PDFXML]]>The problem of European-style option pricing in time-changed Lévy models in the presence of compound Poisson jumps is considered. These jumps relate to sudden large drops in stock prices induced by political or economical hits. As the time-changed Lévy models, the variance-gamma and the normal-inverse Gaussian models are discussed. Exact formulas are given for the price of digital asset-or-nothing call option on extra asset in foreign currency. The prices of simpler options can be derived as corollaries of our results and examples are presented. Various types of dependencies between stock prices are mentioned. PDFXML]]>Roman V. Ivanov,Katsunori AnoTue, 27 Nov 2018 00:00:00 +0200<![CDATA[Asymptotics for the sum of three state Markov dependent random variables]]>
https://www.vmsta.org/journal/VMSTA/article/135
https://www.vmsta.org/journal/VMSTA/article/135The insurance model when the amount of claims depends on the state of the insured person (healthy, ill, or dead) and claims are connected in a Markov chain is investigated. The signed compound Poisson approximation is applied to the aggregate claims distribution after $n\in \mathbb{N}$ periods. The accuracy of order $O({n^{-1}})$ and $O({n^{-1/2}})$ is obtained for the local and uniform norms, respectively. In a particular case, the accuracy of estimates in total variation and non-uniform estimates are shown to be at least of order $O({n^{-1}})$. The characteristic function method is used. The results can be applied to estimate the probable loss of an insurer to optimize an insurance premium. PDFXML]]>The insurance model when the amount of claims depends on the state of the insured person (healthy, ill, or dead) and claims are connected in a Markov chain is investigated. The signed compound Poisson approximation is applied to the aggregate claims distribution after $n\in \mathbb{N}$ periods. The accuracy of order $O({n^{-1}})$ and $O({n^{-1/2}})$ is obtained for the local and uniform norms, respectively. In a particular case, the accuracy of estimates in total variation and non-uniform estimates are shown to be at least of order $O({n^{-1}})$. The characteristic function method is used. The results can be applied to estimate the probable loss of an insurer to optimize an insurance premium. PDFXML]]>Gabija Liaudanskaitė,Vydas ČekanavičiusMon, 19 Nov 2018 00:00:00 +0200<![CDATA[Martingale-like sequences in Banach lattices]]>
https://www.vmsta.org/journal/VMSTA/article/134
https://www.vmsta.org/journal/VMSTA/article/134Martingale-like sequences in vector lattice and Banach lattice frameworks are defined in the same way as martingales are defined in [Positivity 9 (2005), 437–456]. In these frameworks, a collection of bounded X-martingales is shown to be a Banach space under the supremum norm, and under some conditions it is also a Banach lattice with coordinate-wise order. Moreover, a necessary and sufficient condition is presented for the collection of $\mathcal{E}$-martingales to be a vector lattice with coordinate-wise order. It is also shown that the collection of bounded $\mathcal{E}$-martingales is a normed lattice but not necessarily a Banach space under the supremum norm. PDFXML]]>Martingale-like sequences in vector lattice and Banach lattice frameworks are defined in the same way as martingales are defined in [Positivity 9 (2005), 437–456]. In these frameworks, a collection of bounded X-martingales is shown to be a Banach space under the supremum norm, and under some conditions it is also a Banach lattice with coordinate-wise order. Moreover, a necessary and sufficient condition is presented for the collection of $\mathcal{E}$-martingales to be a vector lattice with coordinate-wise order. It is also shown that the collection of bounded $\mathcal{E}$-martingales is a normed lattice but not necessarily a Banach space under the supremum norm. PDFXML]]>Haile Gessesse,Alexander MelnikovWed, 07 Nov 2018 00:00:00 +0200<![CDATA[Large deviations for conditionally Gaussian processes: estimates of level crossing probability]]>
https://www.vmsta.org/journal/VMSTA/article/133
https://www.vmsta.org/journal/VMSTA/article/133The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes – the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application. PDFXML]]>The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes – the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application. PDFXML]]>Barbara Pacchiarotti,Alessandro PigliacelliFri, 12 Oct 2018 00:00:00 +0300<![CDATA[Ruin probability for the bi-seasonal discrete time risk model with dependent claims]]>
https://www.vmsta.org/journal/VMSTA/article/132
https://www.vmsta.org/journal/VMSTA/article/132The discrete time risk model with two seasons and dependent claims is considered. An algorithm is created for computing the values of the ultimate ruin probability. Theoretical results are illustrated with numerical examples. PDFXML]]>The discrete time risk model with two seasons and dependent claims is considered. An algorithm is created for computing the values of the ultimate ruin probability. Theoretical results are illustrated with numerical examples. PDFXML]]>Olga Navickienė,Jonas Sprindys,Jonas ŠiaulysMon, 01 Oct 2018 00:00:00 +0300<![CDATA[On generalized stochastic fractional integrals and related inequalities]]>
https://www.vmsta.org/journal/VMSTA/article/131
https://www.vmsta.org/journal/VMSTA/article/131The generalized mean-square fractional integrals ${\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}$ and ${\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}$ of the stochastic process X are introduced. Then, for Jensen-convex and strongly convex stochastic proceses, the generalized fractional Hermite–Hadamard inequality is establish via generalized stochastic fractional integrals. PDFXML]]>The generalized mean-square fractional integrals ${\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}$ and ${\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}$ of the stochastic process X are introduced. Then, for Jensen-convex and strongly convex stochastic proceses, the generalized fractional Hermite–Hadamard inequality is establish via generalized stochastic fractional integrals. PDFXML]]>Hüseyin Budak,Mehmet Zeki SarikayaMon, 24 Sep 2018 00:00:00 +0300<![CDATA[Stochastic models associated to a Nonlocal Porous Medium Equation]]>
https://www.vmsta.org/journal/VMSTA/article/124
https://www.vmsta.org/journal/VMSTA/article/124The nonlocal porous medium equation considered in this paper is a degenerate nonlinear evolution equation involving a space pseudo-differential operator of fractional order. This space-fractional equation admits an explicit, nonnegative, compactly supported weak solution representing a probability density function. In this paper we analyze the link between isotropic transport processes, or random flights, and the nonlocal porous medium equation. In particular, we focus our attention on the interpretation of the weak solution of the nonlinear diffusion equation by means of random flights. PDFXML]]>The nonlocal porous medium equation considered in this paper is a degenerate nonlinear evolution equation involving a space pseudo-differential operator of fractional order. This space-fractional equation admits an explicit, nonnegative, compactly supported weak solution representing a probability density function. In this paper we analyze the link between isotropic transport processes, or random flights, and the nonlocal porous medium equation. In particular, we focus our attention on the interpretation of the weak solution of the nonlinear diffusion equation by means of random flights. PDFXML]]>Alessandro De GregorioWed, 19 Sep 2018 00:00:00 +0300<![CDATA[Drifted Brownian motions governed by fractional tempered derivatives]]>
https://www.vmsta.org/journal/VMSTA/article/125
https://www.vmsta.org/journal/VMSTA/article/125Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann–Liouville type derivatives. For these operators a Marchaud-type form is obtained and a Riesz tempered fractional derivative is examined, together with its Fourier transform. PDFXML]]>Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann–Liouville type derivatives. For these operators a Marchaud-type form is obtained and a Riesz tempered fractional derivative is examined, together with its Fourier transform. PDFXML]]>Mirko D’Ovidio,Francesco Iafrate,Enzo OrsingherWed, 19 Sep 2018 00:00:00 +0300