# Estimation of probability of exceeding a curve by a strictly φ-sub-Gaussian quasi shot noise process

### Abstract

In this paper, we continue to study the properties of a separable strictly φ-sub-Gaussian quasi shot noise process $X(t) = \int_{-\infty}^{+\infty} g(t,u) d\xi(u), t\in\R$, generated by the response function g and the strictly φ-sub-Gaussian process ξ = (ξ(t), t ∈ R) with uncorrelated increments, such that E(ξ(t)−ξ(s))^2 = t−s, t>s ∈ R. We consider the problem of estimating the probability of exceeding some level by such a process on the interval [a;b], a,b ∈ R. The level is given by a continuous function f = {f(t), t ∈ [a;b]}, which satisfies some given conditions. In order to solve this problem, we apply the theorems obtained for random processes from a class V (φ, ψ), which generalizes the class of φ-sub-Gaussian processes. As a result, several estimates for probability of exceeding the curve f by sample pathes of a separable strictly φ-sub-Gaussian quasi shot noise process are obtained. Such estimates can be used in the study of shot noise processes that arise in the problems of financial mathematics, telecommunication networks theory, and other applications.

* Key words*: shot noise processes, φ-sub-Gaussian processes.

* Pages of the article in the issue*: 49 - 56

** Language of the article**: Ukrainian

### References

VASYLYK, O.I. (2019) Properties of strictly φ-sub-Gaussian quasi shot noise processes, Teoriya Imovirnostei ta Matematychna Statystyka, 2(101), p. 49–62.

VASYLYK, O.I. (2019) Estimation of distribution of suprema of a strictly φ-sub-Gaussian quasi shot noise process, Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics & Mathematics Iss.2., p.9 – 18.

VASYLYK, O. I., KOZACHENKO, YU. V., YAMNENKO, R. E. (2008) φ-sub-Gaussian random process, Kyiv: Vydavnycho-Poligrafichnyi Tsentr “Kyivskyi Universytet”, 231 p. (In Ukrainian)

DARIYCHUK, I. V., KOZACHENKO, YU. V., and PERESTYUK, M.M. (2011) Stochastic processes from Orlicz spaces. Chernivtsi: “Zoloti lytavry”, 212 p. (In Ukrainian)

BULDYGIN, V. V., KOZACHENKO, YU. V. (2000) Metric Characterization of Random Variables and Random Processes. American Mathematical Society, Providence, RI, 257 p.

DARIYCHUK, I. V., KOZACHENKO, YU. V. (2009) Some properties of pre-Gaussian shot noise processes. Stochastic Analysis and Random Dynamics. International Conference. Abstracts. Lviv, Ukraine, p.57–59.

DARIYCHUK, I. V., KOZACHENKO, YU. V. (2010) The distribution of the supremum of Θ-pre-Gaussian shot noise processes. Theory of Probability and Mathematical Statistics. No.80, p.85–100.

GIULIANO ANTONINI, R., KOZACHENKO, YU. V., NIKITINA, T. (2003) Space of φ-sub-Gaussian random variables. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 27, p.92–124.

KOOPS, D. T., BOXMA, O. J. AND MANDJES, M. R. H. (2016) Networks of ·/G/∞ Queues with Shot-Noise-Driven Arrival Intensities. Queueing Systems. August 2016, DOI: 10.1007/s11134-017-9520-7.

KOZACHENKO, YU., YAMNENKO, R., VASYLYK, O. (2005) Upper estimate of overrunning by Sub_φ(Ω) random process the level specified by continuous function. Random Oper. Stoch. Equ. 13, no. 2, p.111–128.

KRASNOSEL’SKII, M. A., RUTICKII, YA. B. (1961) Convex Functions and Orlicz Spaces. Moscow, 1958 (in Russian). English translation: P.Noordhoff Ltd, Groningen, 249p., 1961.

SCHMIDT, T. (2014) Catastrophe insurance modeled by shot-noise processes, Risks. ISSN 2227-9091, MDPI, Basel, 2, Iss. 1, p.3–24. http://dx.doi.org/10.3390/risks2010003.

SCHMIDT, T. (2016) Shot-noise processes in finance. arXiv:1612.06616v1.

VASYLYK, O. I. (2017) Strictly φ-sub-Gaussian quasi shot noise processes. Statistics, Optimization and Information Computing. 5, p.109–120.

YAMNENKO, R., VASYLYK O. (2007) Random process from the class V (φ, ψ): Exceeding a curve. Theory of Stochastic Processes. Vol.13 (29), no.4, p. 219–232.

*Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics*, (3), 49-56. https://doi.org/10.17721/1812-5409.2020/3.5

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