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# Optimal prediction of the last-passage time of a transient diffusion

## Abstract

We identify the integrable stopping time $\tau_*$ with minimal $L^1$-distance from the last-passage time $\gamma_z$ associated with a given level $z>0$, for an arbitrary nonnegative time-homogeneous transient diffusion $X$. We demonstrate that $\tau_*$ 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.

## Suggested Citation

• Kristoffer Glover & Hardy Hulley, 2014. "Optimal prediction of the last-passage time of a transient diffusion," Published Paper Series 2014-5, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
• Handle: RePEc:uts:ppaper:2014-5
as

File URL: https://epubs.siam.org/doi/abs/10.1137/130950719

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### Keywords

transient diffusions; last-passage times; optimal prediction; optimal stopping; free-boundary problems;

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