@article{Grigutis2023, 
author = {Andrius Grigutis},
title = {Exact expression of ultimate time survival probability in homogeneous discrete-time risk model},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {3},
pages = {5181-5199},
keywords = {random walk, survival probability, generating function, Vandermonde matrix, homogeneous discrete-time risk model, initial values, ruin theory},
url = {https://www.sciopen.com/article/10.3934/math.2023260},
doi = {10.3934/math.2023260},
abstract = {In this work, we set up the generating function of the ultimate time survival probability    φ  (  u  +  1  ), where     φ  (  u  )  =      P        (                  sup                  n          ⩾          1                            ∑                  i          =          1                          n                            (                              X            i                    −          κ                )            <      u        )    ,   u  ∈            N        0    ,    κ  ∈      N   and the random walk        {                  ∑                  i          =          1                          n                            X        i            ,            n      ∈              N              }   consists of independent and identically distributed random variables        X    i  , which are non-negative and integer-valued. We also give expressions of    φ  (  u  ) via the roots of certain polynomials. The probability    φ  (  u  ) means that the stochastic process     u  +  κ  n  −      ∑          i      =      1              n            X    i  is positive for all    n  ∈      N  , where a certain growth is illustrated by the deterministic part    u  +  κ  n and decrease is given by the subtracted random part        ∑          i      =      1              n            X    i  . Based on the proven theoretical statements, we give several examples of    φ  (  u  ) and its generating function expressions, when random variables        X    i   admit Bernoulli, geometric and some other distributions.}
}