Ruin Probabilities as Recurrence Sequences in a Discrete-Time Risk Process

The theory of linear recurrence sequences is applied to obtain an explicit formula for the ultimate ruin probability in a discrete-time risk process. It is assumed that the claims distribution is arbitrary but has finite support \(\varvec{\{0,1,\ldots ,m+1\}}\) $\mathbf{\{}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{\dots }\mathbf{,}\mathit{m}\mathbf{+}\mathbf{1}\mathbf{\}}$ , for some integer \(\varvec{m\ge 1}\) $\mathit{m}\mathbf{\ge }\mathbf{1}$ . The method requires finding the zeroes of an m degree polynomial and solving a system of m linear equations. An approximation is derived and some numerical results and plots are provided as examples.