The distribution of the supremum of a $\gamma $-reflected stochastic process with an input process belonging to some exponential type Orlicz space

Yamnenko, Rostyslav

doi:10.1090/tpms/1017

Cited by 1 publication

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“…Applying entropy methods for stochastic processes from these classes allows one to investigate the behavior of their extrema, to derive estimates for various functionals of such processes and random fields, to treat their sample paths properties, see, for example, Dozzi, Kozachenko, Mishura, and Ralchenko (2018); Yamnenko (2017); Hopkalo and Sakhno (2021); Sakhno (2022).…”

Section: Introductionmentioning

confidence: 99%

Supremum Distribution of Weighted Sum of Random Processes from Orlicz Spaces of Exponential Type with Continuous Deviation

Tykhonenko,

Yamnenko

2023

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The paper studies distribution of sum of random processes from Orlicz spaces of exponential type weighted by continuous functions, in particular, processes from spaces Subϕ (Ω), SSubϕ (Ω) and class V (ϕ, ψ) are considered. Such spaces and classes of random variables and corresponding stochastic processes provide generalizations of Gaussian and sub-Gaussian random variables and processes and are important for various applications, for example, in queuing theory and financial mathematics. We derive the estimates for the distribution of supremum of weighted sum of such processes deviated by a continuous monotone function using the entropy method. As examples, weighted sum of Wiener and weighted sum of fractional Brownian motion processes with different Hurst indices from classes V (ϕ, ψ) are considered. Corresponding estimates of the probability of exceeding by trajectories of such weighted sums a positive level determined by a linear function are obtained. In the insurance risk theory, such aproblem arises during estimating a ruin probability of the corresponding risk process with a constant premium income, and in the communications theory, it appears for the buffer overflow probability for a single server with a constant service rate.

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