A solution to the ruin problem for Pareto distributions

An expression is derived for Ψ(u), the probability of ultimate ruin given an initial reserve of u in the case of a Pareto distribution. Laplace transforms and exponential integrals are used to derive this expression, which involves a single integral of real valued functions along the positive real line. Most importantly, the integrand is not of an oscillating kind. This expression for Ψ(u) is new, and may be used to form the basis of a more refined set of asymptotic approximations to Ψ(u). Finally, it shown that Ψ(u) can be expressed as the expected value of a function of a two parameter gamma random variable.