We address some direct and inverse problems, for the first-exit time (FET) \(\tau\) of a drifted Brownian motion with Poissonian resetting \(\mathcal {X}(t)\) from an interval (0, b) and the first-exit area (FEA) A, namely the area swept out by \(\mathcal {X}(t)\) till the time \(\tau\) ; this type of diffusion process \(\mathcal {X}(t)\) is characterized by the fact that a reset to the position \(x_R\) can occur according to a homogeneous Poisson process with rate \(r>0.\) When the initial position \(\mathcal {X}(0)= \eta \in (0,b)\) is deterministic and fixed, the direct FET problem consists in investigating the statistical properties of the FET \(\tau ,\) whilst the direct FEA problem studies the probability distribution of the FEA A. The inverse FET problem regards the case when \(\eta\) is randomly distributed in (0, b) (while r and \(x_R\) are fixed); if F(t) is a given distribution function on the time t axis, the inverse FET problem consists in finding the density g of \(\eta ,\) if it exists, such that \(P[\tau \le t ] = F(t), \ t>0.\) Several explicit examples of solutions to the inverse FET problem are provided.
