On hitting times for compound Poisson dams with exponential jumps and linear release rate (Q2774457)

For the compound Poisson dam with exponential jumps and linear release rate \(ry\) in state \(y\), the mean and the Laplace transform \(Ee^{-\alpha \tau}\) of the time \(\tau\) for the content \(X_t\) to reach level \(a\) starting from level \(x\) is considered. For the Laplace transform, the approach is to look for a function \(f(y)\) such that \(f(X_t)e^{-\alpha t}\) is a martingale and to apply optional stopping. This leads to Kummer's differential equation having a confluent hypergeometric function (of first or second kind according as \(x>a\) or \(x<a)\) as solution.