Series solution to the first-passage-time problem of a Brownian motion with an exponential time-dependent drift (Q2890740)

The author derives the survival probability and the first-passage-time (FPT) density of a process (a particle position) \(X\) expressed as an arithmetic Brownian motion driven by an exponential time-dependent drift up to an absorbing boundary (a threshold) \(x_{\text{thr}}\) . More precisely, \(X\) satisfies the following SDE NEWLINE\[NEWLINE dX_t=\sigma dB_t+\mu dt + \frac{\epsilon}{\tau_d} e^{-t/\tau_d} dt NEWLINE\]NEWLINE where \(\mu\) is a positive constant, \(\epsilon\) is an intensity coefficient of the exponential time-dependent drift and \(\tau_d\) characterise the intensity. The method employed is to solve the Backward Fokker-Planck equation.