On the area swept by a biased diffusion till its first-exit time: Martingale approach and gambling opportunities

Yonathan Sarmiento; Debraj Das; Édgar Roldán

doi:10.1007/s12648-024-03182-8

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arXiv · 2401.00895

On the area swept by a biased diffusion till its first-exit time: Martingale approach and gambling opportunities

Yonathan Sarmiento·Debraj Das·Édgar Roldán

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Abstract

Using martingale theory, we compute, in very few lines, exact analytical expressions for various first-exit-time statistics associated with one-dimensional biased diffusion. Examples include the distribution for the first-exit time from an interval, moments for the first-exit site, and functionals of the position, which involve memory and time integration. As a key example, we compute analytically the mean area swept by a biased diffusion until it escapes an interval that may be asymmetric and have arbitrary length. The mean area allows us to derive the hitherto unexplored cross-correlation function between the first-exit time and the first-exit site, which vanishes only for exit problems from symmetric intervals. As a colophon, we explore connections of our results with gambling, showing that betting on the time-integrated value of a losing game it is possible to design a strategy that leads to a net average win.

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Yonathan Sarmiento, Debraj Das, Édgar Roldán. 2023-12-31. On the area swept by a biased diffusion till its first-exit time: Martingale approach and gambling opportunities. https://doi.org/10.1007/s12648-024-03182-8

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