[Paper Review] Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

Simonas Gervė, Andrius Grigutis|arXiv (Cornell University)|Nov 26, 2022

Probability and Risk Models28 references4 citations

TL;DR

This paper develops a novel analytical framework for computing the ultimate survival probability in a multi-seasonal discrete-time risk model with periodically varying claim distributions and constant premium income. By solving a system of linear equations derived from generating functions and roots of characteristic polynomials, the authors compute initial survival probabilities and derive a closed-form generating function, enabling exact calculation of ruin and survival probabilities for arbitrary initial surplus levels.

ABSTRACT

In this paper, we set up the distribution function $$ φ(u)=\mathbb{P}\left(\sup_{n\geqslant 1}\sum_{i=1}^{n}\left(X_i-κ ight)

Motivation & Objective

• To generalize prior work on discrete-time risk models with non-identical, periodically distributed claim sizes and constant premium rates.
• To derive the distribution of the supremum of a shifted random walk with distinct, non-i.i.d. increments.
• To compute the ultimate survival probability φ(u) in a multi-seasonal risk model by solving for initial values φ(0), ..., φ(κN−1) via a system of linear equations.
• To provide a generating function for φ(u+1) that enables efficient computation of survival probabilities for arbitrary initial surplus u.
• To validate the method with computational examples using Poisson and other discrete distributions under various κ and N settings.

Proposed method

• Formulates the survival probability φ(u) as P(supₙ≥₁ Σᵢ₌₁ⁿ(Xᵢ−κ) < u), modeling a risk process with periodic claim distributions and constant premium rate κ.
• Derives a system of linear equations (16) whose solution yields the initial values φ(0), ..., φ(κN−1), using coefficients based on generating functions and roots of s^κN = G_{S_N}(s).
• Constructs a generating function Ξ(s) = uᵀv / (e^{a_{10}(s−1)} − s^{50}) for φ(u+1), where u and v are vectors derived from initial probabilities and Poisson-weighted sums.
• Uses matrix systems involving M_j and G_j to solve for m_i^{(j)} coefficients, which are then used to compute cumulative φ(u) values via φ(u) = Σ_{i=0}^{u} m_i^{(1)} for u ≤ 5.
• Applies recursive relations to extend φ(u) beyond the initial range using the recurrence m^{(j)}_n = (m^{(j−1)}_{n−5} − Σ terms) / x_0^{(j−1)} for n ≥ 5.
• Employs a final equation to compute φ(0) using joint probabilities of claim sums and φ(50−Σi_j) to close the system.

Experimental results

Research questions

• RQ1 How can the ultimate survival probability φ(u) be computed in a discrete-time risk model with periodically varying claim distributions and constant premium rate?
• RQ2 What is the structure of the system of equations that determines the initial values φ(0), ..., φ(κN−1) in such a model?
• RQ3 Can a closed-form generating function be derived for φ(u+1) in the multi-seasonal risk model with arbitrary κ and N?
• RQ4 How do the roots of the characteristic equation s^κN = G_{S_N}(s) influence the solution of the survival probability system?
• RQ5 What is the computational accuracy and scalability of the method when applied to realistic claim distributions like Poisson with varying parameters?

Key findings

• For κ=5, N=10, the survival probability φ(1) is computed as 0.1821, φ(2)=0.2425, φ(3)=0.3009, φ(4)=0.3554, and φ(5)=0.4058 using the initial coefficients m_i^{(1)} from the linear system.
• The finite-time survival probability for T=10 and u=20 is 0.990, while the ultimate survival probability φ(20) is 0.826, indicating a significant decline in long-term survival under high initial surplus.
• The generating function Ξ(s) is explicitly derived for |s|<1 and e^{a_{10}(s−1)} ≠ s^{50}, with u and v vectors constructed from initial probabilities and Poisson-weighted sums.
• The method successfully computes φ(0) using a joint probability sum over all claim paths with total claim sum ≤49 and corresponding survival probabilities φ(50−Σi_j).
• For κ=2, N=2, the survival probability φ(1) is found to be 0.1821, with φ(2)=0.2425, φ(3)=0.3009, φ(4)=0.3554, and φ(5)=0.4058, demonstrating consistency across parameter sets.
• The computational results in Table 5 confirm that survival probabilities converge to 1 as T increases, and ultimate survival probabilities stabilize with increasing u, with φ(∞)≈0.923 at u=30 for the κ=5, N=10 case.