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

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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.

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

    1. Baurdoux, Erik J. & Pedraza, José M., 2024. "Lp optimal prediction of the last zero of a spectrally negative Lévy process," LSE Research Online Documents on Economics 119468, London School of Economics and Political Science, LSE Library.
    2. Baurdoux, Erik J. & Pedraza, José M., 2023. "Predicting the last zero before an exponential time of a spectrally negative Lévy process," LSE Research Online Documents on Economics 119290, London School of Economics and Political Science, LSE Library.

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