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OPTIMAL STOPPING FOR THE LAST EXIT TIME
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 1 / February 2019
- Published online by Cambridge University Press:
- 04 October 2018, pp. 148-160
- Print publication:
- February 2019
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- Article
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- You have access
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Given a one-dimensional downwards transient diffusion process
$X$, we consider a random time 
$\unicode[STIX]{x1D70C}$, the last exit time when 
$X$ exits a certain level 
$\ell$, and detect the optimal stopping time for it. In particular, for this random time 
$\unicode[STIX]{x1D70C}$, we solve the optimisation problem 
$\inf _{\unicode[STIX]{x1D70F}}\mathbb{E}[\unicode[STIX]{x1D706}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70C})_{+}+(1-\unicode[STIX]{x1D706})(\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70F})_{+}]$ over all stopping times 
$\unicode[STIX]{x1D70F}$. We show that the process should stop optimally when it runs below some fixed level 
$\unicode[STIX]{x1D705}_{\ell }$ for the first time, where 
$\unicode[STIX]{x1D705}_{\ell }$ is the unique solution in the interval 
$(0,\unicode[STIX]{x1D706}\ell )$ of an explicitly defined equation.
