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TL;DR

First Passage Time of Brownian Motion: A canonical quantitative trading interview question at olympiad difficulty. Commonly asked at Two Sigma, DE Shaw, Citadel, Jane Street.

By Valenke Exam Prep Team·Last updated 2026-06-01
olympiadStochastic Processes & Calculus

First Passage Time of Brownian Motion

Asked at: Two Sigma, DE Shaw, Citadel, Jane Street

Problem
Let WtW_t be standard Brownian motion and  aua=inf⁡{t>0:Wt=a}\ au_a = \inf\{t > 0 : W_t = a\} for a>0a > 0. Show that E[ aua]=∞E[\ au_a] = \infty despite P( aua<∞)=1P(\ au_a < \infty) = 1. Find the distribution of  aua\ au_a.
Related concepts
Brownian MotionFirst Passage Time

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