First-Passage Times with Brownian Motion and Power-Law Waiting

Summary

The note asks how power-law distributed pauses would change the first-passage-time density of Brownian motion with drift. It frames the process as motion that advances according to a drifted Brownian model but intermittently waits, and proposes convolving the usual inverse Gaussian first-passage density with a power-law waiting-time distribution.

No derivation, formula, parameterization, or answer is supplied, so the convolution is only a suggested approach rather than an established result in the document. The exact density would depend on how waiting periods are incorporated into the motion, including whether pauses occur independently and how their durations are modeled. The note is a mathematical research question, with no trading application or empirical evidence presented.

Key ideas

• The problem concerns first-passage times for Brownian motion with drift and intermittent pauses.
• The suggested construction is a convolution involving an inverse Gaussian density and power-law waiting times.
• The note does not derive or state a resulting density function.
• A solution would need to specify how waiting periods interact with the underlying motion.

Tags

# Convolution of inverse gaussian and power law distributions


# Convolution of inverse gaussian and power law distributions












I am trying to understand how the first passage time density of Brownian motion with drift is modified by the presence of waiting times that are distributed as a power law

In other words, what is the density function for first passage time of a Brownian motion with drift when the movement "pauses" for time intervals that follow a power law distribution

I believe the density function would be the convolution of inverse Gaussian and power law distributions. Is there an expression for this? Any help would be much appreciated

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.