A continuity theorem in the ruin problem
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.
scientific article; zbMATH DE number 5963628