The Asymmetric Barrier

Expected stopping time in an asymmetric random walk

This is a medium-difficulty probability question that tests your ability to compute expected hitting times in a bounded random walk. It appears frequently in quant interviews because it requires both structural insight and careful algebra—traits that matter when pricing financial derivatives or analysing trading algorithms.

The setup is a symmetric random walk (equal odds of up or down at each step) but with asymmetric barriers: one is closer than the other. To solve it, you need to set up a system of linear equations for the expected stopping time from each position, then solve consistently. The key is recognising which positions matter, writing out their recurrence relations, and handling the boundary conditions correctly.

• Markov chain stopping times and first-passage problems
• Setting up and solving recurrence relations
• Harmonic functions and the optional stopping theorem