A continuity theorem in the ruin problem
type: Item
instance of: scholarly article
title: A continuity theorem in the ruin problem
zbMATH Open document ID: 1234.60049
DOI: 10.1134/S0037446611040057
author: Igor S. Borisov
author: A. M. Shoisoronov
published in: Siberian Mathematical Journal
publication date: 2011-10-25T00:00:00Z
review text: Consider the random walk \(S(t) = \sum_{k=1}^t \xi_k\), where \(\{\xi_k: k \geq 1\}\) are i.i.d. random variables. The object of interest is the exit time \[ \tau := \min\{k \geq 1: S(k) \notin (-a,b), \min\{S(i): i \leq k\} > -a\} \] from the strip \((-a,b)\) through its upper boundary, where \(a,b > 0\). The ruin probability is the expression \(\psi(u) = \operatorname{P}[\tau < \infty]\). Next, a family of distributions \(\{F_\varepsilon: \varepsilon \geq 0\}\) is considered, denoted by the superscript \(\varepsilon\). It is first shown that if \(\lim_{\varepsilon \to 0}\text{TV}(\xi_1^\varepsilon,\xi_1) = 0\), then also \(\lim_{\varepsilon \to 0}\text{TV}(\tau^\varepsilon,\tau) = 0\), where \(\text{TV}(\cdot,\cdot)\) denotes the total variation distance. The proof is via the estimation of the total variation distance of \(n\)-tuples \((\xi_1,\xi_2, \ldots, \xi_n)\). This implies in particular that \(\lim_{\varepsilon \to 0} \operatorname{P}[\tau^\varepsilon < \infty] = \operatorname{P}[\tau < \infty]\). In the second part, the ruin probabilities are explicitly calculated for the two-sided geometric distribution \[ \operatorname{P}[\xi = k] = \alpha (1-p) p^{k-1} I_{k > 0} + \beta (1-q) q^{-k-1} I_{k < 0} + r I_{k=0}, \] where \(p,q \in (0,1)\) and \(\alpha,\beta > 0\), \(r \geq 0\) such that \(\alpha + \beta + r = 1\); and the two-sided exponential distribution \[ \operatorname{P}[\xi \leq t] = \beta \min\{e^{q t},1\} + (r + \alpha (1-e^{-p t}) I_{t \geq 0}, \] where \(p,q > 0\), and \(\alpha,\beta > 0\), \(r \geq 0\) such that \(\alpha + \beta + r = 1\). First, a formula for the ruin probability is obtained in the case \(\operatorname{E}[\xi] \neq 0\). In the case \(\operatorname{E}[\xi] = 0\), the model is perturbed through \(\alpha_\varepsilon = \alpha + \varepsilon\) and \(\beta_\varepsilon = \beta - \varepsilon\). Then the limit \[ \lim_{\varepsilon \to 0} \operatorname{P}[\tau^\varepsilon < \infty] \] yields the desired result.
reviewed by: Hanspeter Schmidli
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 91B30
zbMATH DE Number: 5963628
zbMATH Keywords: random walk
zbMATH Keywords: exit time
zbMATH Keywords: ruin probability
zbMATH Keywords: two-sided geometric distribution
zbMATH Keywords: two-sided exponential distribution
MaRDI profile type: Publication
cites work: A Note on Dobrushin's Theorem and Couplings in Poisson Approximation in Abelian Groups
cites work: A Note on the Distribution of the Number of Crossings of a Strip by a Random Walk
cites work: Sequential analysis. Tests and confidence intervals
description: scientific article; zbMATH DE number 5963628