Asymmetric Random Walk Absorption

Solving absorption probabilities in a biased random walk

This is a hard probability question testing mastery of random walks and absorbing barriers—a foundational topic in quantitative finance and applied probability. Quant trading desks use these models to reason about price paths and stopping times; getting the setup right is essential.

The problem asks you to find the probability that a biased random walk, starting at the origin and taking steps right (heads) or left (tails) with unequal probabilities, hits a positive barrier before a negative one. The key is recognizing that this is a gambler's ruin scenario and setting up a recurrence relation for the absorption probability at each position. You will need to solve the resulting difference equation and apply boundary conditions to isolate the answer.

• Difference equations and their characteristic roots
• Boundary conditions and absorbing states
• The role of the odds ratio in asymmetric walks
• Why the fair-game assumption () creates a degenerate case