Optimal Prediction of the Last-Passage Time of a Transient Diffusion

We identify the integrable stopping time τ∗ with minimal L1-distance from the lastpassage time γz associated with a given level z > 0, for an arbitrary nonnegative time-homogeneous transient diffusion X. We demonstrate that τ∗ is in fact the first time that X assumes a value outside a half-open interval [0, r∗). The upper boundary r∗ > z of this interval is characterized either as the solution for a one-dimensional optimization problem, or as part of the solution for a free-boundary problem. A number of concrete examples illustrate the result.