Expected Count of Brownian Motion Threshold Hits

Summary

The note defines a sequence of stopping times for standard Brownian motion by recording each first passage at which the process moves more than a fixed distance from its value at the previous stopping time. Starting at time zero, it asks for the expected number of these events that occur before a specified horizon.

This is a first-passage and renewal-style question about repeated threshold crossings, rather than a trading method or market analysis. The document does not provide a solution, assumptions beyond the stated stopping-time construction, or supporting evidence. In particular, it leaves open how to derive the expected count from the distribution of the time between successive hits and how the finite horizon affects that expectation.

Key ideas

• Each stopping time marks a move exceeding a fixed threshold from the process value at the preceding stopping time.
• The count includes stopping times that occur before a given time horizon.
• The question concerns the expected number of repeated Brownian threshold hits.
• No derivation or result for the expected count is provided.

Tags

# Expectation of number of hits by a brownian motion


# Expectation of number of hits by a brownian motion












If we denote $\tau_i$ the sequence of stopping times defined by: $\tau_i = \inf(t>\tau_{i-1} : |B(t)-B(\tau_{i-1})| > a)$, $\tau_0=0$.

If we denote N the number of stopping times below T.

What is the expectation of N (i.e. the average number of stopping times below T)?

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