Random Walk on a Bridge: Steps & Probabilities

This question presents a one-dimensional symmetric random walk on a finite interval, interpreted as a person wandering back and forth on a bridge with absorbing endpoints. The candidate must determine both the probability of hitting one endpoint before the other, and the expected time to reach either endpoint, starting from an interior point. It is a classic hitting-probability and hitting-time problem for a simple random walk with boundaries, typical of stochastic calculus and probability interviews for quant and trading roles at banks and hedge funds.

Solving it leans heavily on martingale ideas and the optional stopping theorem, as well as discrete-time Markov chain intuition. The interviewer expects the candidate to recognize that the walk has no drift and that certain functions of the position form martingales whose expectations can be related at stopping times. It also tests the ability to set up and solve simple difference equations, interpret absorbing boundaries correctly, and reason about how variance growth links to expected stopping time. Clear explanation, careful conditioning, and correct handling of boundary conditions are key signals of a strong answer.

What it tests

Problems involving a symmetric random walk with absorbing barriers are governed by the principle that the process's expected value and, more generally, certain functions of the process (like its square) remain constant in expectation until absorption. This is the martingale property: for a fair walk, the expected position at the stopping time equals the starting position, and the expected squared displacement relates directly to the expected number of steps. The boundaries serve as absorbing states, and the probability of absorption at each boundary is determined by the relative distances from the starting point to each barrier. This structure holds because, in a fair random walk, there is no drift, so the only bias comes from the initial position's proximity to the boundaries, and the process's future is independent of its past except for its current state. The expected time to absorption depends on the product of the distances to the two boundaries, reflecting the quadratic growth of variance in random walks.