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Existence and uniqueness of mild solution to fractional stochastic heat equation

Kostiantyn Ralchenko Georgiy Shevchenko

https://doi.org/10.15559/18-VMSTA122

Pub. online: 12 Dec 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 1 (2019), pp. 57–79

Abstract

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.

Studies on generalized Yule models

Federico Polito

https://doi.org/10.15559/18-VMSTA125

Pub. online: 3 Dec 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 1 (2019), pp. 41–55

Abstract

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.

Option pricing in time-changed Lévy models with compound Poisson jumps

Roman V. Ivanov Katsunori Ano

https://doi.org/10.15559/18-VMSTA124

Pub. online: 27 Nov 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 1 (2019), pp. 81–107

Abstract

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.

Asymptotics for the sum of three state Markov dependent random variables

Gabija Liaudanskaitė Vydas Čekanavičius

https://doi.org/10.15559/18-VMSTA123

Pub. online: 19 Nov 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 1 (2019), pp. 109–131

Abstract

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.

Martingale-like sequences in Banach lattices

Haile Gessesse Alexander Melnikov

https://doi.org/10.15559/18-VMSTA120

Pub. online: 7 Nov 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 501–508

Abstract

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.

Large deviations for conditionally Gaussian processes: estimates of level crossing probability

Barbara Pacchiarotti Alessandro Pigliacelli

https://doi.org/10.15559/18-VMSTA119

Pub. online: 12 Oct 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 483–499

Abstract

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.

Ruin probability for the bi-seasonal discrete time risk model with dependent claims

Olga Navickienė Jonas Sprindys Jonas Šiaulys

https://doi.org/10.15559/18-VMSTA118

Pub. online: 1 Oct 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 1 (2019), pp. 133–144

Abstract

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.

On generalized stochastic fractional integrals and related inequalities

Hüseyin Budak Mehmet Zeki Sarikaya

https://doi.org/10.15559/18-VMSTA117

Pub. online: 24 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 471–481

Abstract

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.

Stochastic models associated to a Nonlocal Porous Medium Equation

Alessandro De Gregorio

https://doi.org/10.15559/18-VMSTA112

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 457–470

Abstract

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.

Drifted Brownian motions governed by fractional tempered derivatives

Mirko D’Ovidio Francesco Iafrate Enzo Orsingher

https://doi.org/10.15559/18-VMSTA114

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 445–456

Abstract

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.

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