Abstract
In this article we investigate a homogeneous discrete time risk model with a generalized premium income rate which can be any natural number. We derive theorems and give numerical examples for finite and ultimate time survival probability calculation for the mentioned model. Our proved statements for ultimate time survival probability calculation, at some level, are similar to the previously known statements for non-homogeneous risk models, where required initial values of survival probability for some recurrent formulas are gathered by certain limit laws. We also give a simplified proof that a ruin is almost unavoidable with a neutral net profit condition and state several conjectures on a certain type of recurrent matrices non-singularity. All the research done can be interpreted as a possibility that symmetric or asymmetric random walk (r.w.) hits (or not) the line u+κt and that possibility is directly related to the expected value of r.w. generating random variable which might be equal, above or bellow κ.
Highlights
A game of gain and loss occurs in various situations
For generalized premium discrete time risk model (GPDTRM) presented by Formula (3), the finite time ruin probability satisfies the following equations: u +κ −1 φ(u, 1) = H (u + κ − 1), φ(u, T ) =
The last change of r.v. can be well utilized in view of that what is known for discrete time risk model survival or ruin probability calculation, see Section 5 in [15]
Summary
A game of gain and loss occurs in various situations. All individuals have savings, earn income, and face expenses. The finite and ultimate time survival probabilities for the model presented by (3) are correspondingly defined as: φ(u, T ) := P. t =1 where T ∈ N. For GPDTRM presented by Formula (3), the finite time ruin probability satisfies the following equations:. By similar arguments as in proof of Theorem 1, the ultimate time survival probability of the model (3) for all u ∈ N0 satisfies the following relation:.
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